diff --git a/LICENSE b/LICENSE --- a/LICENSE +++ b/LICENSE @@ -1,4 +1,4 @@ -Copyright (c) 2015-2017, Balazs Komuves +Copyright (c) 2015-2021, Balazs Komuves All rights reserved. Redistribution and use in source and binary forms, with or without diff --git a/README.md b/README.md new file mode 100644 --- /dev/null +++ b/README.md @@ -0,0 +1,54 @@ + +Characteristic classes of coincident root loci +============================================== + +Coincident root loci (or discriminant strata) are subsets +of the space of homogeneous polynomials in two variables defined by +root multiplicities: A nonzero degree _n_ polynomial has _n_ roots in the +complex projective line P^1, but some of these can coincide, which gives us a +partition of _n_. Hence for each partition _lambda_ we get a set of polynomials +(those with root multiplicities given by _lambda_), +which together stratify the space of these polynomials, which (modulo multiplying by scalars) +is P^n. These are quasi-projective varieties, invariant under the action of GL(2); +their closures are highly singular projective varieties, making them a good +example for studying invariants of singular varieties. + +This package contains a number of different algorithms to compute invariants +and characteristic classes of these varieties: + +- degree +- Euler characteristic +- the fundamental class in equivariant cohomology +- Chern-Schwartz-MacPherson (CSM) class, Segre-SM class +- equivariant CSM class +- Hirzebruch Chi-y genus +- Todd class, motivic Hirzebruch class +- motivic Chern class +- equivariant motivic Chern class + +Some of the algorithms are implemented in Mathematica +instead of (or in addition to) Haskell. + +Another (better organized) Mathematica implementation is available at +. + + +Example usage +============= + +For example if you want to know what is the equivariant CSM class of the +(open) loci corresponding to the partition [2,2,1,1], you can use the following +piece of code: + + {-# LANGUAGE TypeApplications #-} + + import Math.Combinat.Partitions + import Math.RootLoci.Algebra.SymmPoly ( AB ) + import Math.Algebra.Polynomial.Pretty ( pretty ) + import Math.RootLoci.CSM.Equivariant.Umbral + + csm ps = umbralOpenCSM @AB (mkPartition ps) + + main = do + putStrLn $ pretty $ csm [2,2,1,1] + diff --git a/coincident-root-loci.cabal b/coincident-root-loci.cabal --- a/coincident-root-loci.cabal +++ b/coincident-root-loci.cabal @@ -1,100 +1,114 @@ -Name: coincident-root-loci -Version: 0.2 -Synopsis: Equivariant CSM classes of coincident root loci +Cabal-Version: 2.4 +Name: coincident-root-loci +Version: 0.3 +Synopsis: Equivariant CSM classes of coincident root loci -Description: This library contians a set of function to compute, among - others, the @GL(2)@-equivariant Chern-Schwartz-MacPherson - classes of coincident root loci, which are subvarieties - of the space of unordered @n@-tuples of points in the complex - projective line. To such an @n@-tuples we can associate - a partition of @n@ given by the multiplicities of the distinct - points; this stratifies the set of all @n@-tuples, and we - call these strata \"coincident root loci\". +Description: This library contians a set of function to compute, among + others, the @GL(2)@-equivariant Chern-Schwartz-MacPherson + classes of coincident root loci, which are subvarieties + of the space of unordered @n@-tuples of points in the complex + projective line. To such an @n@-tuples we can associate + a partition of @n@ given by the multiplicities of the distinct + points; this stratifies the set of all @n@-tuples, and we + call these strata \"coincident root loci\". + + This package is supplementary software for a forthcoming paper. - This package is supplementary software for a forthcoming paper. +License: BSD-3-Clause +License-file: LICENSE +Author: Balazs Komuves +Copyright: (c) 2015-2021 Balazs Komuves +Maintainer: bkomuves (plus) hackage (at) gmail (dot) com +Homepage: https://hub.darcs.net/bkomuves/coincident-root-loci +Stability: Experimental +Category: Math +Tested-With: GHC == 8.6.5 +Build-Type: Simple -License: BSD3 -License-file: LICENSE -Author: Balazs Komuves -Copyright: (c) 2015-2017 Balazs Komuves -Maintainer: bkomuves (plus) hackage (at) gmail (dot) com -Homepage: http://code.haskell.org/~bkomuves/ -Stability: Experimental -Category: Math -Tested-With: GHC == 8.0.2 -Cabal-Version: >= 1.18 -Build-Type: Simple +extra-source-files: README.md + mathematica/equiv_motivic_chern.nb + mathematica/equiv_motivic_chern.src + mathematica/equivariant_CSM_via_motivic.nb + mathematica/equivariant_CSM_via_motivic.src + slides/motivic_slides_2018.pdf + slides/csm_slides_2017.pdf +source-repository head + type: darcs + location: https://hub.darcs.net/bkomuves/coincident-root-loci + -------------------------------------------------------------------------------- Library - Build-Depends: base >= 4 && < 5, - array >= 0.5, containers, random, transformers, - combinat >= 0.2.8.2 + Build-Depends: base >= 4 && < 5, + array >= 0.5, containers >= 0.5, random, transformers, + combinat >= 0.2.10.0, polynomial-algebra >= 0.1 Exposed-Modules: - -- Math.RootLoci - Math.RootLoci.Classic - -- Math.RootLoci.Dual - Math.RootLoci.Dual.Restriction - Math.RootLoci.Dual.Localization - -- Math.RootLoci.CSM - -- Math.RootLoci.CSM.Equivariant - Math.RootLoci.CSM.Equivariant.Direct - Math.RootLoci.CSM.Equivariant.Recursive - Math.RootLoci.CSM.Equivariant.Ordered - Math.RootLoci.CSM.Equivariant.PushForward - Math.RootLoci.CSM.Equivariant.Umbral - Math.RootLoci.CSM.Aluffi - Math.RootLoci.CSM.Projective - Math.RootLoci.Geometry - Math.RootLoci.Geometry.Forget - Math.RootLoci.Geometry.Cohomology - Math.RootLoci.Geometry.Mobius - -- Math.RootLoci.Applications - -- Math.RootLoci.Applications.FlexLines - Math.RootLoci.Algebra - Math.RootLoci.Algebra.FreeMod - Math.RootLoci.Algebra.Polynomial - Math.RootLoci.Algebra.SymmPoly - Math.RootLoci.Misc - Math.RootLoci.Misc.Pretty - Math.RootLoci.Misc.PTable - Math.RootLoci.Misc.Common + -- Math.RootLoci + Math.RootLoci.Classic + -- Math.RootLoci.Dual + Math.RootLoci.Dual.Restriction + Math.RootLoci.Dual.Localization + -- Math.RootLoci.CSM + -- Math.RootLoci.CSM.Equivariant + Math.RootLoci.CSM.Equivariant.Direct + Math.RootLoci.CSM.Equivariant.Recursive + Math.RootLoci.CSM.Equivariant.Ordered + Math.RootLoci.CSM.Equivariant.PushForward + Math.RootLoci.CSM.Equivariant.Umbral + Math.RootLoci.CSM.Aluffi + Math.RootLoci.CSM.Projective + Math.RootLoci.Segre.Equivariant + Math.RootLoci.Geometry + Math.RootLoci.Geometry.Forget + Math.RootLoci.Geometry.Cohomology + Math.RootLoci.Geometry.Mobius + -- Math.RootLoci.Motivic + Math.RootLoci.Motivic.Classes + Math.RootLoci.Motivic.Abstract + Math.RootLoci.Motivic.Homology + -- Math.RootLoci.Motivic.Formula + Math.RootLoci.Algebra + Math.RootLoci.Algebra.SymmPoly + Math.RootLoci.Misc + Math.RootLoci.Misc.PTable + Math.RootLoci.Misc.Common - Default-Extensions: CPP, BangPatterns - Other-Extensions: MultiParamTypeClasses, ScopedTypeVariables, - GeneralizedNewtypeDeriving + Default-Extensions: CPP, BangPatterns + Other-Extensions: MultiParamTypeClasses, ScopedTypeVariables, + GeneralizedNewtypeDeriving - Default-Language: Haskell2010 + Default-Language: Haskell2010 - Hs-Source-Dirs: src + Hs-Source-Dirs: src - ghc-options: -fwarn-tabs -fno-warn-unused-matches -fno-warn-name-shadowing -fno-warn-unused-imports + ghc-options: -fwarn-tabs -fno-warn-unused-matches -fno-warn-name-shadowing -fno-warn-unused-imports -------------------------------------------------------------------------------- test-suite test - default-language: Haskell2010 - type: exitcode-stdio-1.0 - hs-source-dirs: test - main-is: testSuite.hs + default-language: Haskell2010 + type: exitcode-stdio-1.0 + hs-source-dirs: test + main-is: testSuite.hs - build-depends: base >= 4 && < 5, containers >= 0.4, array >= 0.5, - tasty >= 0.11, tasty-hunit >= 0.9, - combinat >= 0.2.8.2, - 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b/mathematica/equiv_motivic_chern.src @@ -0,0 +1,504 @@ + +(* ===== EQUIVARIANT MOTIVIC CHERN CLASSES OF COINCIDENT ROOT LOCI ==== *) + +<< Combinatorica` + +(* === NORMALIZATION in K(P^n) === *) + +(* Weights of Sym^n C^2 *) +Clear[L, X, Y] +WT[n_, i_] := X^(n - i)*Y^i + +(* the relation in K (P^n). Convention: +L is the tautological line bundle, + c_ 1(X) = -alpha, c_ 1(Y) = -beta *) + +KREL[n_] := Product[1 - L*WT[n, i], {i, 0, n}] + +(* normal form of L^(n+1) *) +Clear[Lpow$nplus1] +Lpow$nplus1[n_] := + Lpow$nplus1[n] = + Expand[L^(n + 1) - (-1)^(n + 1) KREL[n]/(X*Y)^Binomial[n + 1, 2]] + +(* (unefficiently) normalize a polynomial in L *) +Clear[KnormalizeSlow] +KnormalizeSlow[n_, Z0_] := Module[{Z = Expand[Z0], m}, + m = Exponent[Z, L]; + If[m <= n, Z, + KnormalizeSlow[n, Z /. {L^m -> L^(m - n - 1)*Lpow$nplus1[n]}]] + ] + +(* Table of normal forms of L^p *) +Clear[LPowXY] +LPowXY[n_, k_] := LPowXY[n, k] = Expand[KnormalizeSlow[n, L^k]] + +(* More efficient normalization in K^(P^n) +usage: KnormalizeVarXY[dim,L,expr] *) +Clear[KnormalizeVarXY] +KnormalizeVarXY[n_, uuu_, Z0_] := Module[ + {m, Z}, + Z = Expand[Z0]; + m = Exponent[Z, uuu]; + Expand[Sum[Coefficient[Z, uuu, k]*(LPowXY[n, k] /. {L -> uuu}), {k, 0, m}]] + ] + +(* === PUSHFORWARD ALONG PSI (THE MULTIPLICATION MAP) === *) + +(* formulas for (psi_{n,1})_! (L1^p*L2^q) *) +Clear[PSI$N1f] +PSI$N1f[n_, 0, 1] := L*(X^(n + 1) - Y^(n + 1))/(X - Y) +PSI$N1f[n_, p_, 0] := + L^p*(X^(p + 1) - Y^(p + 1))/(X - Y) + + L^(p + 1) (X^(n + 1)*Y^(p + 1) - Y^(n + 1) X^(p + 1))/(X - Y) +PSI$N1f[n_, p_, q_] := If[p >= q, + L^q*PSI$N1f[n, p - q, 0], + L^p*PSI$N1f[n, 0, q - p] + ] + +Clear[PSI$N1] +PSI$N1[n_, p_, q_] := PSI$N1[n, p, q] = Expand[Factor[PSI$N1f[n, p, q]]] + +Clear[newPsiBang2] +newPsiBang2[{n_, 1}, {var1_, var2_}, outvar_, ZZ_] := Module[ + {Z}, + Z = Expand[ZZ]; + Expand[Sum[ + Coefficient[Coefficient[Z, var1, i], var2, + j]*(PSI$N1[n, i, j] /. {L -> outvar}), {i, 0, n}, {j, 0, 1}]] + ] + +(* find elements in K(P^m x P^1) such that their pushforward to K(P^(m+1)) is \ +{1,L,L^2,...L^(m+1)} *) +findBasis$M1[m_] := findBasis$M1[m] = + Module[{vars, A, B, list, i, j, k, eqs, sols, sol}, + vars = Table[Subscript[a, i], {i, 0, m + 1}]; + A = Sum[Subscript[a, i]*L1^i, {i, 0, m}] + + Subscript[a, m + 1]*L1^m*L2; + B = newPsiBang2[{m, 1}, {L1, L2}, L, A]; + list = Table[Null, {i, 0, m + 1}]; + For[k = 0, k <= m + 1, k++, + eqs = Table[Coefficient[B, L, i] == If[k == i, 1, 0], {i, 0, m + 1}]; + sols = Solve[eqs, vars]; + sol = sols[[1]]; + list[[k + 1]] = A /. sol; + ]; + list + ] + + +Clear[PSI$NM, S, T] +PSI$NM[n_, 0, p_, 0] := L^p +PSI$NM[0, m_, 0, q_] := L^q +PSI$NM[n_, 1, p_, q_] := PSI$NM[n, 1, p, q] = PSI$N1[n, p, q] +PSI$NM[n_, mplus1_, p_, q_] := PSI$NM[n, mplus1, p, q] = Module[ + {m = mplus1 - 1, basis, A, B, C}, + basis = findBasis$M1[m]; + A = basis[[q + 1]] /. {L1 -> L2, L2 -> T}; + B = newPsiBang2[{n, m}, {L1, L2}, S, L1^p*A]; + C = newPsiBang2[{n + m, 1}, {S, T}, L, B]; + Expand[Factor[C]] + ] + +newPsiBang2[{n_, m_}, {var1_, var2_}, outvar_, ZZ_] := Module[ + {Z, deg1, deg2}, + Z = Expand[ZZ]; + deg1 = Exponent[Z, var1]; + deg2 = Exponent[Z, var2]; + Expand[Sum[ + Coefficient[Coefficient[Z, var1, i], var2, + j]*(PSI$NM[n, m, i, j] /. {L -> outvar}), {i, 0, deg1}, {j, 0, deg2}]] + ] + +newPsiBangMany[{n_}, {uuu_}, www_, Z_] := Z /. {uuu -> www} +newPsiBangMany[ns_, uuus_, www_, Z_] := Module[{ttt, vvvs, ms}, + If[Length[uuus] == 1, Z /. {uuus[[1]] -> www}, + If[Length[uuus] == 2, newPsiBang2[ns, uuus, www, Z], + ms = Join[{ns[[1]] + ns[[2]]}, Drop[ns, 2]]; + vvvs = Join[{ttt}, Drop[uuus, 2]]; + newPsiBangMany[ms, vvvs, www, + newPsiBang2[Take[ns, 2], Take[uuus, 2], ttt, Z]]]]] + +(* === PUSHFORwARD ALONG THE DIAGONAL MAP === *) + +notk[n_, k_] := Select[Range[0, n], # != k &] + +(* class of the diagonal in K(P^n x P^n) *) +Clear[DELTA$CLASS$2] +DELTA$CLASS$2[n_] := DELTA$CLASS$2[n] = + Factor[Sum[ + Product[ (1 - + L1*WT[n, i]) (1 - L2*WT[n, i])/(1 - WT[n, i]/WT[n, k]) , {i, + notk[n, k]}] , {k, 0, n}]] + +newDeltaBang2[n_, invar_, {var1_, var2_}, ZZ_] := Module[ + {Z = Expand[ZZ], m, A}, + m = Exponent[Z, invar]; + A = Expand[(DELTA$CLASS$2[n] /. {L1 -> var1, L2 -> var2})* + Sum[var1^p*Coefficient[Z, invar, p], {p, 0, m}]]; + KnormalizeVarXY[n, var1, A] + ] + +(* usage: deltaBangMany[dim,L,{L1,L2,L3},X] *) + +newDeltaBangMany[n_, uuu_, vvvs_, Z_] := Module[{ttt}, + If[Length[vvvs] == 1, Z /. {uuu -> vvvs[[1]]}, + If[Length[vvvs] == 2, newDeltaBang2[n, uuu, vvvs, Z], + newDeltaBangMany[n, ttt, Drop[vvvs, 1], + newDeltaBang2[n, uuu, {vvvs[[1]], ttt}, Z]]]]] + + +(* PUSHFORWARD ALONG THE POWER MAP *) + +Clear[OMEGA$ND] +OMEGA$ND[n_, d_, p_] := OMEGA$ND[n, d, p] = Module[{vars, ns, A, B}, + vars = Table[Subscript[TMP$X, i], {i, 1, d}]; + ns = Table[n, {i, 1, d}]; + A = newDeltaBangMany[n, L, vars, L^p]; + B = newPsiBangMany[ns, vars, L, Expand[A]]; + Expand[B] + ] + +newOmegaBang[n_, 0, var1_, var2_, ZZ_] := Coefficient[ZZ, var1, 0] +newOmegaBang[n_, 1, var1_, var2_, ZZ_] := ZZ /. {var1 -> var2} +newOmegaBang[n_, d_, var1_, var2_, ZZ_] := Module[ + {Z = Expand[ZZ], m, A}, + m = Exponent[Z, var1]; + A = Sum[(OMEGA$ND[n, d, p] /. {L -> var2})*Coefficient[Z, var1, p], {p, 0, + m}]; + Expand[A] + ] + +(* P^n1 x P^n2 x P^n3 -> P^(d1*n1) x P^(d2*n2) x P^(d3*n3) -> P^(d1*n1 + \ +d2*n2 + d3*n3 *) +newOmegaBangLam[ns_, ds_, uuus_, www_, Z0_] := Module[ + {Z = Expand[Z0], + m = Length[ns], + vars, W, nds, ttt, i}, + vars = Table[Subscript[ttt, i], {i, 1, m}]; + nds = Table[ ns[[i]]*ds[[i]], {i, 1, m}]; + W = Z; + For[i = 1, i <= m, i++, + W = newOmegaBang[ns[[i]], ds[[i]], uuus[[i]], vars[[i]], W]]; + newPsiBangMany[nds, vars, www, W] + ] + + +(*only works for lists of equal length!!*) +Zip[as_, bs_] := MapThread[{#1, #2} &, {as, bs}] +Zip3[as_, bs_, cs_] := MapThread[{#1, #2, #3} &, {as, bs, cs}] + +Fst[pair_] := pair[[1]] +Snd[pair_] := pair[[2]] + +extendListWithZeros[L_, n_] := Join[L, Table[0, {i, 1, n - Length[L]}]] + +EmptyPartQ[part_] := Length[part] == 0 + +DualPart[{}] := {} +DualPart[lam_] := + With[{m = lam[[1]]}, Table[Length[Select[lam, # >= i &]], {i, 1, m}]] + +toExpoForm[{}] := {} +toExpoForm[part_] := + Module[{k = Max[part]}, Table[Length[Select[part, # == j &]], {j, 1, k}]] + +posVectorQ[as_] := Map[# >= 0 &, as] /. {List -> And}; +kdeTriples[p_, ns_] := Module[ + {m = Length[ns], + posQ, oneK, A}, + oneK[k_] := + Table[{k, ns - es, es}, {es, Combinatorica`Compositions[p - k, m]}]; + A = Table [oneK[k], {k, 0, p - 1}]; + A = Select[Flatten[A, 1], posVectorQ[Snd[#]] &]; + A] + + +(* motivic Chern class of P^n *) +Clear[mcPn]; +mcPn[n_] := mcPn[n] = + Module[{rel, tmp}, rel = Product[1 - L*X^(n - i)*Y^i, {i, 0, n}]; + tmp = Product[1 + t*L*X^(n - i)*Y^i, {i, 0, n}]; + Factor[(tmp - (-t)^(n + 1) rel)/(1 + t)]] + +Clear[L, S] + +LL[i_] := Subscript[L, i]; +SS[i_] := Subscript[S, i]; +LLs[n_] := Table[LL[i], {i, 1, n}] +SSs[n_] := Table[SS[i], {i, 1, n}] + +Clear[mcXLam, mcDisj1, mcDisj, mcDisjSorted] + +mcXLam[{}] := mcXLam[{}] = 1 +mcXLam[{1}] := mcXLam[{1}] = mcPn[1] + +mcDisj1[0] := mcDisj1[0] = 1 +mcDisj1[1] := mcDisj1[1] = mcPn[1] + +mcDisj[{}] := mcDisj[{}] = 1 + +mcDisjSorted[{}] := mcDisjSorted[{}] = 1 + +(* equivariant motivic chern class of D(n)=X(1^n) *) + +mcDisj1[n_] := mcDisj1[n] = Module[ + {parts = Select[Combinatorica`Partitions[n], Length[#] < n &]}, + Expand[mcPn[n] - Sum[mcXLam[p], {p, parts}]] + ] + +(* equivariant motivic chern class of X_lambda *) + +mcXLam[lambda_] := mcXLam[lambda] = Module[ + {es = toExpoForm[lambda], m, m1, ns, pairs, Z}, + m = Length[es]; + ns = Range[m]; + pairs = Zip[ns, es]; (* i^e_ *) + + pairs = Select[pairs, Snd[#] > 0 &]; (* !!! *) + m1 = Length[pairs]; + ns = Map[Fst, pairs]; + es = Map[Snd, pairs]; + Z = mcDisj[es]; + (* Print["xlam1 - ",pairs]; + Print["xlam2 - ",es," | ",ns," | ",uus[m1]," | ",X]; *) + + Expand[newOmegaBangLam[es, ns, LLs[m1], L, Z]] + ] + +(* equivariant motivic chern class of D(d1,d2,...) *) + +mcDisj[{n_}] := mcDisj[{n}] = mcDisj1[n] /. {L -> LL[1]} +mcDisj[ns0_] := mcDisj[ns0] = Module[ + {m = Length[ns0], + nis0, nis1, ns1, idxs, X, ttt}, + nis0 = Zip[ns0, Range[m]]; + nis1 = SortBy[nis0, -Fst[#] &]; + idxs = Map[Snd, nis1]; + ns1 = Select[Map[Fst, nis1], # > 0 &]; + (* Print["nis1 - ",nis0," | ",nis1]; + Print["nis2 - ",ns1]; *) + X = mcDisjSorted[ns1]; + X = X /. Table[LL[i] -> Subscript[ttt, i], {i, 1, m}]; + X /. Table[Subscript[ttt, i] -> LL[idxs[[i]]], {i, 1, m}] + ] + +(* a single term corresponding to a triple (k,ds,es) *) +Clear[singleKDE] +singleKDE[{k_, ds_, es_}] := ssingleKDE[{k, ds, es}] = Module[ + {A, B, + m, vars, dims, + pp, qq, rr, ss, + pps, qqs, rrs, sss + }, + m = Length[ds]; + pp[i_] := Subscript[pp$p, i]; + qq[i_] := Subscript[qq$q, i]; + rr[i_] := Subscript[rr$r, i]; + ss[i_] := Subscript[ss$s, i]; + pps = Table[pp[i], {i, 1, m}]; + qqs = Table[qq[i], {i, 1, m}]; + rrs = Table[rr[i], {i, 1, m}]; + sss = Table[ss[i], {i, 1, m}]; + vars = Join[{zzz}, pps, qqs]; + dims = Join[{k}, ds, es]; + A = mcDisj[dims] /. Table[LL[i] -> vars[[i]], {i, 1, 2 m + 1}]; + B = A; + For[i = 1, i <= m, i++, + B = Expand[newDeltaBang2[es[[i]], qq[i], {rr[i], ss[i]}, B]]]; + For[i = 1, i <= m, i++, + B = Expand[newPsiBang2[{ds[[i]], es[[i]]}, {pp[i], ss[i]}, LL[i], B]]]; + B = newPsiBangMany[Join[{k}, es], Join[{zzz}, rrs], z, B]; + B + ] + +(* equiv mc of D(d1,d2,...), but we require d1>=d2>=d3>=...>=dn>0 *) + +mcDisjSorted[{n_}] := mcDisjSorted[{n}] = mcDisj1[n] /. {L -> LL[1]} +mcDisjSorted[pns_] := mcDisjSorted[pns] = Module[ + {p = pns[[1]], + ns = Drop[pns, 1], + A, B, rest, + KDE + }, + KDE = kdeTriples[p, ns]; + (* Print["sorted1 - ",p," | ",ns]; + Print["sorted2 - ",KDE]; *) + A = (mcDisj1[p] /. {L -> z})*mcDisj[ns]; + rest = Sum[singleKDE[kde], {kde, KDE}]; + B = Expand[A - rest]; + B = B /. Table[LL[i] -> LL[i + 1], {i, 1, Length[ns]}]; + B = B /. {z -> LL[1]} + ] + + +(* export the classes of X(lambda) for |lambda|<=n *) + +ExportMC[n_] := Module[ + {h, i, p, parts, k, m, s, j, A}, + h = OpenWrite["equivariant_mc_classes.txt"]; + For[i = 1, i <= n, i++, + Print["\nn = ", i]; + WriteString[h, "\n(* =================== *)"]; + WriteString[h, "\n(* ---- n = " <> ToString[i] <> " ---- *)\n\n"]; + parts = Partitions[i]; + m = Length[parts]; + For[j = 1, j <= m, j++, + p = parts[[j]]; + Print["part = ", p]; + A = Expand[mcXLam[p]]; + (* Print[A]; *) + + s = ToString[A, FormatType -> InputForm, PageWidth -> Infinity, + TotalWidth -> Infinity]; + s = StringJoin["mc[", ToString[p], "] = ", s, " ;\n\n"]; + (* Print[s]; *) + WriteString[h, s]; + ] + ] + Close[h]; + ] + + +mcXLam[{1, 1, 1}] + +t - t^3 - L t X^3 + L t^3 X^3 - L X^2 Y - 2 L t X^2 Y + L t^3 X^2 Y + + L^2 t X^5 Y - L^2 t^3 X^5 Y - L X Y^2 - 2 L t X Y^2 + L t^3 X Y^2 + + L^2 X^4 Y^2 + L^2 t X^4 Y^2 - L^2 t^2 X^4 Y^2 - L^2 t^3 X^4 Y^2 - L t Y^3 + + L t^3 Y^3 + L^2 X^3 Y^3 + 2 L^2 t X^3 Y^3 - L^2 t^2 X^3 Y^3 - + 2 L^2 t^3 X^3 Y^3 + L^3 t^2 X^6 Y^3 + L^3 t^3 X^6 Y^3 + L^2 X^2 Y^4 + + L^2 t X^2 Y^4 - L^2 t^2 X^2 Y^4 - L^2 t^3 X^2 Y^4 + L^3 t X^5 Y^4 + + 2 L^3 t^2 X^5 Y^4 + L^3 t^3 X^5 Y^4 + L^2 t X Y^5 - L^2 t^3 X Y^5 + + L^3 t X^4 Y^5 + 2 L^3 t^2 X^4 Y^5 + L^3 t^3 X^4 Y^5 + L^3 t^2 X^3 Y^6 + + L^3 t^3 X^3 Y^6 + +mcXLam[{1, 1, 1, 1, 1}] + +t^3 - t^5 - L t^3 X^5 + L t^5 X^5 - L t^3 X^4 Y + L t^5 X^4 Y + + L^2 t^3 X^9 Y - L^2 t^5 X^9 Y + L t X^3 Y^2 + L t^2 X^3 Y^2 - L t^3 X^3 Y^2 + + L t^5 X^3 Y^2 + L^2 t^3 X^8 Y^2 - L^2 t^5 X^8 Y^2 + L t X^2 Y^3 + + L t^2 X^2 Y^3 - L t^3 X^2 Y^3 + L t^5 X^2 Y^3 - L^2 t X^7 Y^3 + + 3 L^2 t^3 X^7 Y^3 - 2 L^2 t^5 X^7 Y^3 - L^3 t^3 X^12 Y^3 + L^3 t^5 X^12 Y^3 - + L t^3 X Y^4 + L t^5 X Y^4 + L^2 t^2 X^6 Y^4 + 3 L^2 t^3 X^6 Y^4 - + 2 L^2 t^5 X^6 Y^4 - L^3 t X^11 Y^4 - 2 L^3 t^2 X^11 Y^4 - + 2 L^3 t^3 X^11 Y^4 + L^3 t^5 X^11 Y^4 - L t^3 Y^5 + L t^5 Y^5 + L^2 X^5 Y^5 + + L^2 t^2 X^5 Y^5 + 5 L^2 t^3 X^5 Y^5 - 3 L^2 t^5 X^5 Y^5 - L^3 X^10 Y^5 - + 2 L^3 t X^10 Y^5 - 4 L^3 t^2 X^10 Y^5 - 5 L^3 t^3 X^10 Y^5 + + 2 L^3 t^5 X^10 Y^5 + L^2 t^2 X^4 Y^6 + 3 L^2 t^3 X^4 Y^6 - + 2 L^2 t^5 X^4 Y^6 - L^3 X^9 Y^6 - 3 L^3 t X^9 Y^6 - 6 L^3 t^2 X^9 Y^6 - + 7 L^3 t^3 X^9 Y^6 + 3 L^3 t^5 X^9 Y^6 + L^4 t^3 X^14 Y^6 - L^4 t^5 X^14 Y^6 - + L^2 t X^3 Y^7 + 3 L^2 t^3 X^3 Y^7 - 2 L^2 t^5 X^3 Y^7 - L^3 X^8 Y^7 - + 3 L^3 t X^8 Y^7 - 7 L^3 t^2 X^8 Y^7 - 8 L^3 t^3 X^8 Y^7 + 3 L^3 t^5 X^8 Y^7 + + L^4 t X^13 Y^7 + 2 L^4 t^2 X^13 Y^7 + L^4 t^3 X^13 Y^7 - L^4 t^4 X^13 Y^7 - + L^4 t^5 X^13 Y^7 + L^2 t^3 X^2 Y^8 - L^2 t^5 X^2 Y^8 - L^3 X^7 Y^8 - + 3 L^3 t X^7 Y^8 - 7 L^3 t^2 X^7 Y^8 - 8 L^3 t^3 X^7 Y^8 + 3 L^3 t^5 X^7 Y^8 + + L^4 X^12 Y^8 + 2 L^4 t X^12 Y^8 + 4 L^4 t^2 X^12 Y^8 + 4 L^4 t^3 X^12 Y^8 - + L^4 t^4 X^12 Y^8 - 2 L^4 t^5 X^12 Y^8 + L^2 t^3 X Y^9 - L^2 t^5 X Y^9 - + L^3 X^6 Y^9 - 3 L^3 t X^6 Y^9 - 6 L^3 t^2 X^6 Y^9 - 7 L^3 t^3 X^6 Y^9 + + 3 L^3 t^5 X^6 Y^9 + L^4 X^11 Y^9 + 5 L^4 t X^11 Y^9 + 8 L^4 t^2 X^11 Y^9 + + 5 L^4 t^3 X^11 Y^9 - L^4 t^4 X^11 Y^9 - 2 L^4 t^5 X^11 Y^9 - L^3 X^5 Y^10 - + 2 L^3 t X^5 Y^10 - 4 L^3 t^2 X^5 Y^10 - 5 L^3 t^3 X^5 Y^10 + + 2 L^3 t^5 X^5 Y^10 + 2 L^4 X^10 Y^10 + 5 L^4 t X^10 Y^10 + + 9 L^4 t^2 X^10 Y^10 + 8 L^4 t^3 X^10 Y^10 - L^4 t^4 X^10 Y^10 - + 3 L^4 t^5 X^10 Y^10 + L^5 t^4 X^15 Y^10 + L^5 t^5 X^15 Y^10 - + L^3 t X^4 Y^11 - 2 L^3 t^2 X^4 Y^11 - 2 L^3 t^3 X^4 Y^11 + L^3 t^5 X^4 Y^11 + + L^4 X^9 Y^11 + 5 L^4 t X^9 Y^11 + 8 L^4 t^2 X^9 Y^11 + 5 L^4 t^3 X^9 Y^11 - + L^4 t^4 X^9 Y^11 - 2 L^4 t^5 X^9 Y^11 + L^5 t^2 X^14 Y^11 + + 2 L^5 t^3 X^14 Y^11 + 2 L^5 t^4 X^14 Y^11 + L^5 t^5 X^14 Y^11 - + L^3 t^3 X^3 Y^12 + L^3 t^5 X^3 Y^12 + L^4 X^8 Y^12 + 2 L^4 t X^8 Y^12 + + 4 L^4 t^2 X^8 Y^12 + 4 L^4 t^3 X^8 Y^12 - L^4 t^4 X^8 Y^12 - + 2 L^4 t^5 X^8 Y^12 + L^5 t X^13 Y^12 + 2 L^5 t^2 X^13 Y^12 + + 3 L^5 t^3 X^13 Y^12 + 3 L^5 t^4 X^13 Y^12 + L^5 t^5 X^13 Y^12 + + L^4 t X^7 Y^13 + 2 L^4 t^2 X^7 Y^13 + L^4 t^3 X^7 Y^13 - L^4 t^4 X^7 Y^13 - + L^4 t^5 X^7 Y^13 + L^5 t X^12 Y^13 + 2 L^5 t^2 X^12 Y^13 + + 3 L^5 t^3 X^12 Y^13 + 3 L^5 t^4 X^12 Y^13 + L^5 t^5 X^12 Y^13 + + L^4 t^3 X^6 Y^14 - L^4 t^5 X^6 Y^14 + L^5 t^2 X^11 Y^14 + + 2 L^5 t^3 X^11 Y^14 + 2 L^5 t^4 X^11 Y^14 + L^5 t^5 X^11 Y^14 + + L^5 t^4 X^10 Y^15 + L^5 t^5 X^10 Y^15 + +mcXLam[{2, 2, 1, 1}] + +-t - t^2 + t^3 + t^4 + L t X^6 + L t^2 X^6 - L t^3 X^6 - L t^4 X^6 + + L X^5 Y + 2 L t X^5 Y + L t^2 X^5 Y - L t^3 X^5 Y - L t^4 X^5 Y - + L^2 t X^11 Y - L^2 t^2 X^11 Y + L^2 t^3 X^11 Y + L^2 t^4 X^11 Y + + 2 L t X^4 Y^2 + 2 L t^2 X^4 Y^2 - L t^3 X^4 Y^2 - L t^4 X^4 Y^2 - + L^2 X^10 Y^2 - 3 L^2 t X^10 Y^2 - 2 L^2 t^2 X^10 Y^2 + L^2 t^3 X^10 Y^2 + + L^2 t^4 X^10 Y^2 + L t X^3 Y^3 + L t^2 X^3 Y^3 - L t^3 X^3 Y^3 - + L t^4 X^3 Y^3 - 2 L^2 X^9 Y^3 - 7 L^2 t X^9 Y^3 - 6 L^2 t^2 X^9 Y^3 + + L^2 t^3 X^9 Y^3 + 2 L^2 t^4 X^9 Y^3 + L^3 t X^15 Y^3 + L^3 t^2 X^15 Y^3 - + L^3 t^3 X^15 Y^3 - L^3 t^4 X^15 Y^3 + 2 L t X^2 Y^4 + 2 L t^2 X^2 Y^4 - + L t^3 X^2 Y^4 - L t^4 X^2 Y^4 - 3 L^2 X^8 Y^4 - 11 L^2 t X^8 Y^4 - + 10 L^2 t^2 X^8 Y^4 + 2 L^2 t^4 X^8 Y^4 + L^3 X^14 Y^4 + 4 L^3 t X^14 Y^4 + + 3 L^3 t^2 X^14 Y^4 - L^3 t^3 X^14 Y^4 - L^3 t^4 X^14 Y^4 + L X Y^5 + + 2 L t X Y^5 + L t^2 X Y^5 - L t^3 X Y^5 - L t^4 X Y^5 - 5 L^2 X^7 Y^5 - + 15 L^2 t X^7 Y^5 - 13 L^2 t^2 X^7 Y^5 + 3 L^2 t^4 X^7 Y^5 + 3 L^3 X^13 Y^5 + + 11 L^3 t X^13 Y^5 + 11 L^3 t^2 X^13 Y^5 + L^3 t^3 X^13 Y^5 - + 2 L^3 t^4 X^13 Y^5 + L t Y^6 + L t^2 Y^6 - L t^3 Y^6 - L t^4 Y^6 - + 5 L^2 X^6 Y^6 - 16 L^2 t X^6 Y^6 - 14 L^2 t^2 X^6 Y^6 + 3 L^2 t^4 X^6 Y^6 + + 7 L^3 X^12 Y^6 + 21 L^3 t X^12 Y^6 + 19 L^3 t^2 X^12 Y^6 + + 3 L^3 t^3 X^12 Y^6 - 2 L^3 t^4 X^12 Y^6 - L^4 t X^18 Y^6 - L^4 t^2 X^18 Y^6 + + L^4 t^3 X^18 Y^6 + L^4 t^4 X^18 Y^6 - 5 L^2 X^5 Y^7 - 15 L^2 t X^5 Y^7 - + 13 L^2 t^2 X^5 Y^7 + 3 L^2 t^4 X^5 Y^7 + 9 L^3 X^11 Y^7 + + 30 L^3 t X^11 Y^7 + 30 L^3 t^2 X^11 Y^7 + 6 L^3 t^3 X^11 Y^7 - + 3 L^3 t^4 X^11 Y^7 - L^4 X^17 Y^7 - 4 L^4 t X^17 Y^7 - 3 L^4 t^2 X^17 Y^7 + + L^4 t^3 X^17 Y^7 + L^4 t^4 X^17 Y^7 - 3 L^2 X^4 Y^8 - 11 L^2 t X^4 Y^8 - + 10 L^2 t^2 X^4 Y^8 + 2 L^2 t^4 X^4 Y^8 + 13 L^3 X^10 Y^8 + + 38 L^3 t X^10 Y^8 + 36 L^3 t^2 X^10 Y^8 + 9 L^3 t^3 X^10 Y^8 - + 2 L^3 t^4 X^10 Y^8 - 3 L^4 X^16 Y^8 - 10 L^4 t X^16 Y^8 - 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Subscript[u, i] + +(* only works for lists of equal length!! *) +Zip[as_, bs_] := MapThread[{#1, #2} &, {as, bs}] +Zip3[as_, bs_, cs_] := MapThread[{#1, #2, #3} &, {as, bs, cs}] + +Fst[pair_] := pair[[1]] +Snd[pair_] := pair[[2]] + +extendListWithZeros[L_, n_] := Join[L, Table[0, {i, 1, n - Length[L]}]] + +SumList[L_] := Sum[x, {x, L}] + +(* === EQUIVARIANT COHOMOLOGY RING OF P^n === *) + +(* Weights of Sym^n C^2 *) +wt[n_, i_] := (n - i)*\[Alpha] + i*\[Beta] + +(* the relation in H^*(P^n). Convention: +u = -c1(L) where L is the tautological line bundle *) + +rel[n_] := Product[u + wt[n, i], {i, 0, n}] + +Clear[upow$nplus1] +upow$nplus1[n_] := upow$nplus1[n] = Expand[u^(n + 1) - rel[n]] +(* (unefficiently) normalize a /polynomial/ in u *) + +Clear[normalizeSlow] +normalizeSlow[n_, X0_] := Module[{X = Expand[X0], m}, + m = Exponent[X, u]; + If[m <= n, X, normalizeSlow[n, X /. {u^m -> u^(m - n - 1)*upow$nplus1[n]}]] + ] + +(* table of normalized powers of u *) +Clear[UPowAB] +UPowAB[n_, k_] := UPowAB[n, k] = Expand[normalizeSlow[n, u^k]] + +Clear[normalizeVarAB] +normalizeVarAB[n_, uuu_, X_] := Module[ + {m = Exponent[X, uuu]}, + Expand[Sum[Coefficient[X, uuu, k]*(UPowAB[n, k] /. {u -> uuu}), {k, 0, m}]] + ] + +(* === pushforward along the multiplication (single monom) === *) + +(* Psi_* (u^k * v^l) = ? cohomology indexing *) +Clear[psiStarAB] +psiStarAB[n_, m_, 0, 0] := psiStarAB[n, m, 0, 0] = Binomial[n + m, n] +psiStarAB[n_, m_, k_, l_] := psiStarAB[n, m, k, l] = Module[ + {A, B, AB, F, R, IJ, sel, fun, cft}, + A = Product[u + wt[n, i], {i, 0, k - 1}]; + B = Product[v + wt[m, j], {j, 0, l - 1}]; + AB = Expand[A*B]; + sel[{i_, j_}] := (i < k) || (j < l); + cft[{i_, j_}] := Coefficient[Coefficient[AB, u, i], v, j]; + fun[{i_, j_}] := psiStarAB[n, m, i, j]; + IJ = Flatten[Table[{i, j}, {i, 0, k}, {j, 0, l}], 1]; + IJ = Select[ IJ, sel]; + (* Print[IJ]; + Print[AB]; *) + + F = Binomial[n + m - k - l, n - k]* + Product[w + wt[n + m, i], {i, 0, k + l - 1}]; + R = Sum[ cft[ij]*fun[ij], {ij, IJ}]; + Expand[F - R] + ] + +Clear[psiStarChern] +psiStarChern[n_, m_, k_, l_] := psiStarChern[n, m, k, l] = + SymmetricReduction[psiStarAB[n, m, k, l], {\[Alpha], \[Beta]}, {c, d}][[1]] + + +(* === class of the diagonal === *) + +notk[n_, k_] := Select[Table[i, {i, 0, n}], # != k &] + +(* class of the diagonal in P^n x P^n *) +Clear[deltaClassAB] +deltaClassAB[n_] := deltaClassAB[n] = + Expand[Factor[ + Sum[ Product[(v + wt[n, i]) (w + wt[n, i])/(wt[n, i] - wt[n, k]), {i, + notk[n, k]}], {k, 0, n}]]] + +Clear[deltaClassCh] +deltaClassCh[n_] := deltaClassCh[n] = + Expand[SymmetricReduction[deltaClassAB[n], {\[Alpha], \[Beta]}, {c, d}][[1]]] + +(* small diagonal in P^n x P^n x P^n *) +deltaClassTri[n_] := deltaClassTri[n] = + Expand[Factor[ + Sum[ Product[(uu[1] + wt[n, i]) (uu[2] + + wt[n, i]) (uu[3] + wt[n, i])/(wt[n, i] - wt[n, k]), {i, + notk[n, k]}], {k, 0, n}]]] + +(* === pushforward along the diagonal map (single monom) === *) + +(* pushforward of u^k along Delta : P^n -> P^n x P^n *) +Clear[deltaStarAB] +deltaStarAB[n_, 0] := deltaStarAB[n, 0] = deltaClassAB[n] +deltaStarAB[n_, k_] := deltaStarAB[n, k] = Module[ + {prod, Y, preY, Delta, rest}, + Y = Expand[Product[v + wt[n, i], {i, 0, k - 1}]]; + preY = Y /. {v -> u}; + Delta = deltaClassAB[n]; + prod = normalizeVarAB[n, v, Y*Delta]; + rest = Sum[Coefficient[preY, u, i]*deltaStarAB[n, i], {i, 0, k - 1} ]; + Expand[ prod - rest] + ] + +Clear[deltaStarCh] +deltaStarCh[n_] := deltaStarCh[n] = + Expand[SymmetricReduction[deltaStarAB[n], {\[Alpha], \[Beta]}, {c, d}][[1]]] + + +(* === pushforwards for polynomials (not just single monoms) === *) + +delta2[n_, uuu_, {vvv_, www_}, X0_] := Module[{X = Expand[X0]}, + Sum[Coefficient[X, uuu, i]*(deltaStarAB[n, i] /. {v -> vvv, w -> www}) , {i, + 0, n}] + ] + +deltaMany[n_, uuu_, vvvs_, X_] := Module[{ttt}, + If[Length[vvvs] == 1, X /. {uuu -> vvvs[[1]]}, + If[Length[vvvs] == 2, delta2[n, uuu, vvvs, X], + deltaMany[n, ttt, Drop[vvvs, 1], delta2[n, uuu, {vvvs[[1]], ttt}, X]]]]] + +psi2[{n1_, n2_}, {uuu_, vvv_}, www_, X0_] := Module[{X = Expand[X0]}, + Sum[Coefficient[Coefficient[X, uuu, i], vvv, + j]*(psiStarAB[n1, n2, i, j] /. {w -> www}) , {i, 0, n1}, {j, 0, n2}] + ] + +psiMany[ns_, uuus_, www_, X_] := Module[{ttt, vvvs, ms}, + If[Length[uuus] == 1, X /. {uuus[[1]] -> www}, + If[Length[uuus] == 2, psi2[ns, uuus, www, X], + ms = Join[{ns[[1]] + ns[[2]]}, Drop[ns, 2]]; + vvvs = Join[{ttt}, Drop[uuus, 2]]; + psiMany[ms, vvvs, www, psi2[Take[ns, 2], Take[uuus, 2], ttt, X]]]]] + + +(* === pushforward along the power map === *) + +(* compute the pushforward by composing the diagonal with the merging map *) + +Clear[slowOmegaAB] +slowOmegaAB[n_, d_, k_] := slowOmegaAB[n, d, k] = Module[ + {vars = Table[Subscript[ttt, i], {i, 1, d}], + dims = Table[n, {i, 1, d}]}, + Factor[psiMany[dims, vars, u, deltaMany[n, u, vars, u^k]]] + ] + +(* d-th power of u^k : P^n -> P^(n*d) *) +(* NOTE: this formula is valid for k>n too! + you can experimentally check this + by pre-normalizing and post-normalizing *) +Clear[omegaAB] +omegaAB[n_, d_, k_] := + Module[ + {idxs = Select[Table[i, {i, 0, n*d}], Not[Divisible[#, d]] &]} + , u^k*d^(n - k)*Product[u + (n*d - i)*\[Alpha] + i*\[Beta], {i, idxs}] + ] + +(* pushforward along d-th power : P^n -> P^(n*d) *) + +omega1[n_, d_, uuu_, www_, X0_] := Module[{X = Expand[X0]}, + Sum[Coefficient[X, uuu, i]*(omegaAB[n, d, i] /. {u -> www}) , {i, 0, n}] + ] + +(* P^n1 x P^n2 x P^n3 -> P^(d1*n1) x P^(d2*n2) x P^(d3*n3) -> P^(d1*n1 + \ +d2*n2 + d3*n3 *) +omegaLam[ns_, ds_, uuus_, www_, X0_] := Module[ + {X = Expand[X0], + m = Length[ns], + vars, Y, nds, ttt, i}, + vars = Table[Subscript[ttt, i], {i, 1, m}]; + nds = Table[ ns[[i]]*ds[[i]], {i, 1, m}]; + Y = X; + For[i = 1, i <= m, i++, + Y = omega1[ns[[i]], ds[[i]], uuus[[i]], vars[[i]], Y]]; + psiMany[nds, vars, www, Y] + ] + +(* === EQUIVARIANT CSM CLASS === *) + +uus[n_] := Table[uu[i], {i, 1, n}] +vvs[n_] := Table[vv[i], {i, 1, n}] + +(* total chern class of P^n *) +Clear[chernPnAB] +chernPnAB[n_] := chernPnAB[n] = normalizeVarAB[n, u, + Expand[Product[1 + u + wt[n, i], {i, 0, n}]]] + +chernPnABVar[n_, uuu_] := chernPnAB[n] /. {u -> uuu} + +EmptyPartQ[part_] := Length[part] == 0 + +DualPart[{}] := {} +DualPart[lam_] := + With[{m = lam[[1]]}, Table[Length[Select[lam, # >= i &]], {i, 1, m}]] + +toExpoForm[{}] := {} +toExpoForm[part_] := + Module[{k = Max[part]}, Table[Length[Select[part, # == j &]], {j, 1, k}]] + +posVectorQ[as_] := Map[# >= 0 &, as] /. {List -> And}; +kdeTriples[p_, ns_] := Module[ + {m = Length[ns], + posQ, oneK, A}, + oneK[k_] := + Table[{k, ns - es, es}, {es, Combinatorica`Compositions[p - k, m]}]; + A = Table [oneK[k], {k, 0, p - 1}]; + A = Select[Flatten[A, 1], posVectorQ[Snd[#]] &]; + A] + +Clear[csmXLam, csmDisj1, csmDisj, csmDisjSorted] + +csmXLam[{}] := csmXLam[{}] = 1 +csmXLam[{1}] := csmXLam[{1}] = chernPnAB[1] + +csmDisj1[0] := csmDisj1[0] = 1 +csmDisj1[1] := csmDisj1[1] = chernPnAB[1] + +csmDisj[{}] := csmDisj[{}] = 1 + +csmDisjSorted[{}] := csmDisjSorted[{}] = 1 + +(* equiv CSM of D(n) *) +csmDisj1[n_] := csmDisj1[n] = Module[ + {parts = Select[Combinatorica`Partitions[n], Length[#] < n &]}, + Expand[chernPnAB[n] - Sum[csmXLam[p], {p, parts}]] + ] + +(* equiv CSM of X_lambda *) +csmXLam[lambda_] := csmXLam[lambda] = Module[ + {es = toExpoForm[lambda], m, m1, ns, pairs, X}, + m = Length[es]; + ns = Range[m]; + pairs = Zip[ns, es]; (* i^e_ *) + + pairs = Select[pairs, Snd[#] > 0 &]; (* !!! *) + m1 = Length[pairs]; + ns = Map[Fst, pairs]; + es = Map[Snd, pairs]; + X = csmDisj[es]; + (* Print["xlam1 - ",pairs]; + Print["xlam2 - ",es," | ",ns," | ",uus[m1]," | ",X]; *) + + Expand[omegaLam[es, ns, uus[m1], u, X]] + ] + +(* equiv CSM of D(d1,d2,...) *) + +csmDisj[{n_}] := csmDisj[{n}] = csmDisj1[n] /. {u -> uu[1]} +csmDisj[ns0_] := csmDisj[ns0] = Module[ + {m = Length[ns0], + nis0, nis1, ns1, idxs, X, ttt}, + nis0 = Zip[ns0, Range[m]]; + nis1 = SortBy[nis0, -Fst[#] &]; + idxs = Map[Snd, nis1]; + ns1 = Select[Map[Fst, nis1], # > 0 &]; + (* Print["nis1 - ",nis0," | ",nis1]; + Print["nis2 - ",ns1]; *) + X = csmDisjSorted[ns1]; + X = X /. Table[uu[i] -> Subscript[ttt, i], {i, 1, m}]; + X /. Table[Subscript[ttt, i] -> uu[idxs[[i]]], {i, 1, m}] + ] + +(* a single term corresponding to a triple (k,ds,es) *) +Clear[singleKDE] +singleKDE[{k_, ds_, es_}] := singleKDE[{k, ds, es}] = Module[ + {A, B, + m, vars, dims, + pp, qq, rr, ss, + pps, qqs, rrs, sss + }, + m = Length[ds]; + pp[i_] := Subscript[pp$p, i]; + qq[i_] := Subscript[qq$q, i]; + rr[i_] := Subscript[rr$r, i]; + ss[i_] := Subscript[ss$s, i]; + pps = Table[pp[i], {i, 1, m}]; + qqs = Table[qq[i], {i, 1, m}]; + rrs = Table[rr[i], {i, 1, m}]; + sss = Table[ss[i], {i, 1, m}]; + vars = Join[{zzz}, pps, qqs]; + dims = Join[{k}, ds, es]; + A = csmDisj[dims] /. Table[uu[i] -> vars[[i]], {i, 1, 2 m + 1}]; + B = A; + For[i = 1, i <= m, i++, + B = Expand[delta2[es[[i]], qq[i], {rr[i], ss[i]}, B]]]; + For[i = 1, i <= m, i++, + B = Expand[psi2[{ds[[i]], es[[i]]}, {pp[i], ss[i]}, uu[i], B]]]; + B = psiMany[Join[{k}, es], Join[{zzz}, rrs], z, B]; + B + ] + +(* equiv CSM of D(d1,d2,...), but we require d1>=d2>=d3>=...>=dn>0 *) + +csmDisjSorted[{n_}] := csmDisjSorted[{n}] = csmDisj1[n] /. {u -> uu[1]} +csmDisjSorted[pns_] := csmDisjSorted[pns] = Module[ + {p = pns[[1]], + ns = Drop[pns, 1], + A, B, rest, + KDE + }, + KDE = kdeTriples[p, ns]; + (* Print["sorted1 - ",p," | ",ns]; + Print["sorted2 - ",KDE]; *) + A = (csmDisj1[p] /. {u -> z})*csmDisj[ns]; + rest = Sum[singleKDE[kde], {kde, KDE}]; + B = Expand[A - rest]; + B = B /. Table[uu[i] -> uu[i + 1], {i, 1, Length[ns]}]; + B = B /. {z -> uu[1]} + ] + + +hs2mat = {g -> u, a -> \[Alpha], b -> \[Beta]} + +csmXLam[{2, 2, 1, 1}] + +24 u^2 - 18 u^3 + 144 u \[Alpha] - 162 u^2 \[Alpha] + 180 \[Alpha]^2 - + 444 u \[Alpha]^2 - 24 u^2 \[Alpha]^2 + 18 u^3 \[Alpha]^2 - 360 \[Alpha]^3 - + 144 u \[Alpha]^3 + 162 u^2 \[Alpha]^3 - 180 \[Alpha]^4 + 444 u \[Alpha]^4 + + 360 \[Alpha]^5 + 144 u \[Beta] - 162 u^2 \[Beta] + 504 \[Alpha] \[Beta] - + 1056 u \[Alpha] \[Beta] + 48 u^2 \[Alpha] \[Beta] - + 36 u^3 \[Alpha] \[Beta] - 1584 \[Alpha]^2 \[Beta] + + 144 u \[Alpha]^2 \[Beta] - 162 u^2 \[Alpha]^2 \[Beta] - + 144 \[Alpha]^3 \[Beta] + 168 u \[Alpha]^3 \[Beta] + 864 \[Alpha]^4 \[Beta] + + 180 \[Beta]^2 - 444 u \[Beta]^2 - 24 u^2 \[Beta]^2 + 18 u^3 \[Beta]^2 - + 1584 \[Alpha] \[Beta]^2 + 144 u \[Alpha] \[Beta]^2 - + 162 u^2 \[Alpha] \[Beta]^2 + 648 \[Alpha]^2 \[Beta]^2 - + 1224 u \[Alpha]^2 \[Beta]^2 - 1224 \[Alpha]^3 \[Beta]^2 - 360 \[Beta]^3 - + 144 u \[Beta]^3 + 162 u^2 \[Beta]^3 - 144 \[Alpha] \[Beta]^3 + + 168 u \[Alpha] \[Beta]^3 - 1224 \[Alpha]^2 \[Beta]^3 - 180 \[Beta]^4 + + 444 u \[Beta]^4 + 864 \[Alpha] \[Beta]^4 + 360 \[Beta]^5 + +ref2211 = (180*b^2 - 360*b^3 - 180*b^4 + 360*b^5 + 504*a*b - 1584*a*b^2 - + 144*a*b^3 + 864*a*b^4 + 180*a^2 - 1584*a^2*b + 648*a^2*b^2 - + 1224*a^2*b^3 - 360*a^3 - 144*a^3*b - 1224*a^3*b^2 - 180*a^4 + 864*a^4*b + + 360*a^5 + 144*g^1*b - 444*g^1*b^2 - 144*g^1*b^3 + 444*g^1*b^4 + + 144*g^1*a - 1056*g^1*a*b + 144*g^1*a*b^2 + 168*g^1*a*b^3 - 444*g^1*a^2 + + 144*g^1*a^2*b - 1224*g^1*a^2*b^2 - 144*g^1*a^3 + 168*g^1*a^3*b + + 444*g^1*a^4 + 24*g^2 - 162*g^2*b - 24*g^2*b^2 + 162*g^2*b^3 - 162*g^2*a + + 48*g^2*a*b - 162*g^2*a*b^2 - 24*g^2*a^2 - 162*g^2*a^2*b + 162*g^2*a^3 - + 18*g^3 + 18*g^3*b^2 - 36*g^3*a*b + 18*g^3*a^2) /. hs2mat + +24 u^2 - 18 u^3 + 144 u \[Alpha] - 162 u^2 \[Alpha] + 180 \[Alpha]^2 - + 444 u \[Alpha]^2 - 24 u^2 \[Alpha]^2 + 18 u^3 \[Alpha]^2 - 360 \[Alpha]^3 - + 144 u \[Alpha]^3 + 162 u^2 \[Alpha]^3 - 180 \[Alpha]^4 + 444 u \[Alpha]^4 + + 360 \[Alpha]^5 + 144 u \[Beta] - 162 u^2 \[Beta] + 504 \[Alpha] \[Beta] - + 1056 u \[Alpha] \[Beta] + 48 u^2 \[Alpha] \[Beta] - + 36 u^3 \[Alpha] \[Beta] - 1584 \[Alpha]^2 \[Beta] + + 144 u \[Alpha]^2 \[Beta] - 162 u^2 \[Alpha]^2 \[Beta] - + 144 \[Alpha]^3 \[Beta] + 168 u \[Alpha]^3 \[Beta] + 864 \[Alpha]^4 \[Beta] + + 180 \[Beta]^2 - 444 u \[Beta]^2 - 24 u^2 \[Beta]^2 + 18 u^3 \[Beta]^2 - + 1584 \[Alpha] \[Beta]^2 + 144 u \[Alpha] \[Beta]^2 - + 162 u^2 \[Alpha] \[Beta]^2 + 648 \[Alpha]^2 \[Beta]^2 - + 1224 u \[Alpha]^2 \[Beta]^2 - 1224 \[Alpha]^3 \[Beta]^2 - 360 \[Beta]^3 - + 144 u \[Beta]^3 + 162 u^2 \[Beta]^3 - 144 \[Alpha] \[Beta]^3 + + 168 u \[Alpha] \[Beta]^3 - 1224 \[Alpha]^2 \[Beta]^3 - 180 \[Beta]^4 + + 444 u \[Beta]^4 + 864 \[Alpha] \[Beta]^4 + 360 \[Beta]^5 + +ref2211 - csmXLam[{2, 2, 1, 1}] diff --git a/slides/csm_slides_2017.pdf b/slides/csm_slides_2017.pdf new file mode 100644 Binary files /dev/null and b/slides/csm_slides_2017.pdf differ diff --git a/slides/motivic_slides_2018.pdf b/slides/motivic_slides_2018.pdf new file mode 100644 Binary files /dev/null and b/slides/motivic_slides_2018.pdf differ diff --git a/src/Math/RootLoci/Algebra.hs b/src/Math/RootLoci/Algebra.hs --- a/src/Math/RootLoci/Algebra.hs +++ b/src/Math/RootLoci/Algebra.hs @@ -5,19 +5,17 @@ -- to do -- -- > import Math.RootLoci.Algebra --- > import qualified Math.RootLoci.Algebra.FreeMod as ZMod +-- > import qualified Math.Algebra.Polynomial.FreeModule as ZMod -- module Math.RootLoci.Algebra ( ZMod , QMod , FreeMod - , module Math.RootLoci.Algebra.Polynomial +-- , module Math.RootLoci.Algebra.Polynomial , module Math.RootLoci.Algebra.SymmPoly --- , module ZMod -- apparently this does not work ) where -import Math.RootLoci.Algebra.FreeMod ( ZMod , QMod , FreeMod ) -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import Math.Algebra.Polynomial.FreeModule ( ZMod , QMod , FreeMod ) +import qualified Math.Algebra.Polynomial.FreeModule as ZMod -import Math.RootLoci.Algebra.Polynomial import Math.RootLoci.Algebra.SymmPoly diff --git a/src/Math/RootLoci/Algebra/FreeMod.hs b/src/Math/RootLoci/Algebra/FreeMod.hs deleted file mode 100644 --- a/src/Math/RootLoci/Algebra/FreeMod.hs +++ /dev/null @@ -1,214 +0,0 @@ - --- | Free modules. --- --- This module should be imported qualified - -{-# LANGUAGE BangPatterns, FlexibleInstances, TypeSynonymInstances #-} -module Math.RootLoci.Algebra.FreeMod where - --------------------------------------------------------------------------------- - -import Prelude hiding ( sum , product ) -import Data.List hiding ( sum , product ) - -import Data.Monoid -import Data.Ratio -import Data.Maybe - -import Math.Combinat.Sets ( choose ) - -import qualified Data.Map.Strict as Map -import Data.Map.Strict (Map) - --------------------------------------------------------------------------------- - --- | Free module over a coefficient ring with the given base. Internally a map --- storing the coefficients. We maintain the invariant that the coefficients --- are never zero. -newtype FreeMod coeff base = FreeMod { unFreeMod :: Map base coeff } deriving (Eq,Show) - --- | Free module with integer coefficients -type ZMod base = FreeMod Integer base - --- | Free module with rational coefficients -type QMod base = FreeMod Rational base - --------------------------------------------------------------------------------- - -instance (Monoid b, Ord b, Eq c, Num c) => Num (FreeMod c b) where - (+) = add - (-) = sub - negate = neg - (*) = mul - fromInteger = konst . fromInteger - abs = error "FreeMod/abs" - signum = error "FreeMod/signum" - --------------------------------------------------------------------------------- --- * Sanity checking - --- | Should be the identity function -normalize :: (Ord b, Eq c, Num c) => FreeMod c b -> FreeMod c b -normalize = FreeMod . Map.filter (/=0) . unFreeMod - --- | Safe equality testing (should be identical to @==@) -safeEq :: (Ord b, Eq b, Eq c, Num c) => FreeMod c b -> FreeMod c b -> Bool -safeEq x y = normalize x == normalize y - --------------------------------------------------------------------------------- --- * Constructing and deconstructing - --- | The additive unit -zero :: FreeMod c b -zero = FreeMod $ Map.empty - --- | A module generator -generator :: Num c => b -> FreeMod c b -generator x = FreeMod $ Map.singleton x 1 - --- | A single generator with a coefficient -singleton :: (Ord b) => b -> c -> FreeMod c b -singleton b c = FreeMod $ Map.singleton b c - --- | Conversion from list. --- Note that we assume here that each generator appears at most once! -fromList :: (Eq c, Num c, Ord b) => [(b,c)] -> FreeMod c b -fromList = FreeMod . Map.fromList . filter cond where - cond (b,x) = (x/=0) - --- | Conversion to list -toList :: FreeMod c b -> [(b,c)] -toList = Map.toList . unFreeMod - --- | Extract the coefficient of a generator -coeffOf :: (Ord b, Num c) => b -> FreeMod c b -> c -coeffOf b (FreeMod x) = case Map.lookup b x of - Just c -> c - Nothing -> 0 - --- | Finds the term with the largest generator (in the natural ordering of the generators) -findMaxTerm :: (Ord b) => FreeMod c b -> Maybe (b,c) -findMaxTerm (FreeMod m) = if Map.null m - then Nothing - else Just (Map.findMax m) - --- | Finds the term with the smallest generator (in the natural ordering of the generators) -findMinTerm :: (Ord b) => FreeMod c b -> Maybe (b,c) -findMinTerm (FreeMod m) = if Map.null m - then Nothing - else Just (Map.findMin m) - --------------------------------------------------------------------------------- --- * Basic operations - --- | Negation -neg :: Num c => FreeMod c b -> FreeMod c b -neg (FreeMod m) = FreeMod (Map.map negate m) - --- | Additions -add :: (Ord b, Eq c, Num c) => FreeMod c b -> FreeMod c b -> FreeMod c b -add (FreeMod m1) (FreeMod m2) = FreeMod (Map.mergeWithKey f id id m1 m2) where - f _ x y = case x+y of { 0 -> Nothing ; z -> Just z } - --- | Subtraction -sub :: (Ord b, Eq c, Num c) => FreeMod c b -> FreeMod c b -> FreeMod c b -sub (FreeMod m1) (FreeMod m2) = FreeMod (Map.mergeWithKey f id (Map.map negate) m1 m2) where - f _ x y = case x-y of { 0 -> Nothing ; z -> Just z } - --- | Scaling by a number -scale :: (Ord b, Eq c, Num c) => c -> FreeMod c b -> FreeMod c b -scale 0 _ = zero -scale x (FreeMod m) = FreeMod (Map.mapMaybe f m) where - f y = case x*y of { 0 -> Nothing ; z -> Just z } - --- | Dividing by a number (assuming that the coefficient ring is integral, and each coefficient --- is divisible by the given number) -invScale :: (Ord b, Eq c, Integral c, Show c) => c -> FreeMod c b -> FreeMod c b -invScale d (FreeMod m) = FreeMod (Map.mapMaybe f m) where - f a = case divMod a d of - (b,0) -> case b of { 0 -> Nothing ; z -> Just z } - _ -> error $ "FreeMod/invScale: not divisible by " ++ show d - --------------------------------------------------------------------------------- - --- | Summation -sum :: (Ord b, Eq c, Num c) => [FreeMod c b] -> FreeMod c b -sum [] = zero -sum zms = (foldl1' add) zms - --- | Linear combination -linComb :: (Ord b, Eq c, Num c) => [(c, FreeMod c b)] -> FreeMod c b -linComb = sumWith where - - sumWith :: (Ord b, Eq c, Num c) => [(c, FreeMod c b)] -> FreeMod c b - sumWith [] = zero - sumWith zms = sum [ scale c zm | (c,zm) <- zms ] - --- | Expand each generator into a term in another module and then sum the results -flatMap :: (Ord b1, Ord b2, Eq c, Num c) => (b1 -> FreeMod c b2) -> FreeMod c b1 -> FreeMod c b2 -flatMap f = sum . map g . Map.assocs . unFreeMod where - g (x,c) = scale c (f x) - -flatMap' :: (Ord b1, Ord b2, Eq c2, Num c2) => (c1 -> c2) -> (b1 -> FreeMod c2 b2) -> FreeMod c1 b1 -> FreeMod c2 b2 -flatMap' embed f = sum . map g . Map.assocs . unFreeMod where - g (x,c) = scale (embed c) (f x) - --- | The histogram of a multiset of generators is naturally an element of the given Z-module. -{-# SPECIALIZE histogram :: Ord b => [b] -> ZMod b #-} -histogram :: (Ord b, Num c) => [b] -> FreeMod c b -histogram xs = FreeMod $ foldl' f Map.empty xs where - f old x = Map.insertWith (+) x 1 old - --------------------------------------------------------------------------------- --- * Rings - --- | The multiplicative unit -one :: (Monoid b, Num c) => FreeMod c b -one = konst 1 - --- | A constant -konst :: (Monoid b) => c -> FreeMod c b -konst c = FreeMod (Map.singleton mempty c) - --- | Multiplying two ring elements -mul :: (Ord b, Monoid b, Eq c, Num c) => FreeMod c b -> FreeMod c b -> FreeMod c b -mul xx yy = sum [ (f x c) | (x,c) <- toList xx ] where - f x c = FreeMod $ Map.fromList [ (x<>y, cd) | (y,d) <- ylist , let cd = c*d , cd /= 0 ] - ylist = toList yy - --- | Product -product :: (Ord b, Monoid b, Eq c, Num c) => [FreeMod c b] -> FreeMod c b -product [] = generator mempty -product xs = foldl1' mul xs - --- | Multiplies by a monomial -mulMonom :: (Ord b, Monoid b) => b -> FreeMod c b -> FreeMod c b -mulMonom monom = FreeMod . Map.mapKeys (mappend monom) . unFreeMod - --------------------------------------------------------------------------------- --- * Misc - --- | A symmetric polynomial of some generators -symPoly :: (Ord a, Monoid a) => Int -> [a] -> ZMod a -symPoly k xs = fromList $ map (\x -> (x,1)) $ (map mconcat $ choose k xs) - --- | Changing the base set -mapBase :: (Ord a, Ord b) => (a -> b) -> FreeMod c a -> FreeMod c b -mapBase f = onFreeMod (Map.mapKeys f) - --- | Changing the coefficient ring -mapCoeff :: (Ord b) => (c1 -> c2) -> FreeMod c1 b -> FreeMod c2 b -mapCoeff f = onFreeMod' (Map.map f) - --- | Extract a subset of terms -filterBase :: (Ord a, Ord b) => (a -> Maybe b) -> FreeMod c a -> FreeMod c b -filterBase f = onFreeMod (Map.fromList . mapMaybe g . Map.toList) where - g (k,x) = case f k of { Just k' -> Just (k',x) ; Nothing -> Nothing } - -onFreeMod :: (Ord a, Ord b) => (Map a c -> Map b c) -> FreeMod c a -> FreeMod c b -onFreeMod f = FreeMod . f . unFreeMod - -onFreeMod' :: (Ord a, Ord b) => (Map a c -> Map b d) -> FreeMod c a -> FreeMod d b -onFreeMod' f = FreeMod . f . unFreeMod - --------------------------------------------------------------------------------- diff --git a/src/Math/RootLoci/Algebra/Polynomial.hs b/src/Math/RootLoci/Algebra/Polynomial.hs deleted file mode 100644 --- a/src/Math/RootLoci/Algebra/Polynomial.hs +++ /dev/null @@ -1,102 +0,0 @@ - --- | Univariate polynomials - -{-# LANGUAGE GeneralizedNewtypeDeriving #-} -module Math.RootLoci.Algebra.Polynomial where - --------------------------------------------------------------------------------- - -import Data.Array ( assocs ) - -import Math.Combinat.Numbers - -import Math.RootLoci.Misc - -import qualified Math.RootLoci.Algebra.FreeMod as ZMod -import Math.RootLoci.Algebra.FreeMod ( FreeMod , ZMod , QMod ) - --------------------------------------------------------------------------------- --- * Polynomials - --- | Standard univariate polynomials -newtype Poly coeff = Poly { fromPoly :: FreeMod coeff X } deriving (Eq,Num,Show) - --- | Univariate polynomials using /rising factorials/ as a basis function -newtype RisingPoly coeff = RisingPoly { fromRisingPoly :: FreeMod coeff RisingF } deriving (Eq,Show) - --- | Univariate polynomials using /falling factorials/ as a basis function -newtype FallingPoly coeff = FallingPoly { fromFallingPoly :: FreeMod coeff FallingF } deriving (Eq,Show) - -instance (Num c, Show c, Eq c, IsSigned c) => Pretty (Poly c) where pretty (Poly p) = pretty p -instance (Num c, Show c, Eq c, IsSigned c) => Pretty (RisingPoly c) where pretty (RisingPoly p) = pretty p -instance (Num c, Show c, Eq c, IsSigned c) => Pretty (FallingPoly c) where pretty (FallingPoly p) = pretty p - --------------------------------------------------------------------------------- --- * Monomials - --- | A power of @x@ (that is, a monomial of the form @x^i@) -newtype X = X Int deriving (Eq,Ord,Show) - -instance Monoid X where - mempty = X 0 - mappend (X e) (X f) = X (e+f) - -instance Pretty X where - pretty (X e) = case e of - 0 -> "1" - 1 -> "x" - _ -> "x^" ++ show e - --------------------------------------------------------------------------------- --- * Rising and falling factorials - --- | Rising factorial @x^(k) = x(x+1)(x+2)...(x+k-1)@ -newtype RisingF = RF Int deriving (Eq,Ord,Show) - --- | Falling factorial @x_(k) = x(x-1)(x-2)...(x-k+1)@ -newtype FallingF = FF Int deriving (Eq,Ord,Show) - -instance Pretty RisingF where - pretty (RF k) = case k of - 0 -> "1" - 1 -> "x" - _ -> "x^(" ++ show k ++ ")" - -instance Pretty FallingF where - pretty (FF k) = case k of - 0 -> "1" - 1 -> "x" - _ -> "x_(" ++ show k ++ ")" - -risingPoly :: RisingF -> Poly Integer -risingPoly (RF k) = Poly $ ZMod.fromList - [ (X p, abs c) | (p,c) <- assocs (signedStirling1stArray k) ] - -fallingPoly :: FallingF -> Poly Integer -fallingPoly (FF k) = Poly $ ZMod.fromList - [ (X p, c) | (p,c) <- assocs (signedStirling1stArray k) ] - --------------------------------------------------------------------------------- --- * Lagrange interpolation - -lagrangeInterp :: [(Rational,Rational)] -> Poly Rational -lagrangeInterp = Poly . lagrangeInterp' - -lagrangeInterp' :: [(Rational,Rational)] -> QMod X -lagrangeInterp' xys = final where - final = ZMod.sum [ ZMod.scale (ys!!j) (lagrangePoly' xs j) | j<-[0..m-1] ] where - m = length xys - (xs,ys) = unzip xys - -lagrangePoly' :: [Rational] -> Int -> QMod X -lagrangePoly' xs j = ZMod.scale (1/denom) numer where - numer = ZMod.product [ term i | i<-[0..m-1] , i /= j ] - denom = product [ x j - x i | i<-[0..m-1] , i /= j ] - m = length xs - x i = xs !! i - term i = ZMod.fromList - [ (X 1 , 1 ) - , (X 0 , - x i ) - ] - --------------------------------------------------------------------------------- diff --git a/src/Math/RootLoci/Algebra/SymmPoly.hs b/src/Math/RootLoci/Algebra/SymmPoly.hs --- a/src/Math/RootLoci/Algebra/SymmPoly.hs +++ b/src/Math/RootLoci/Algebra/SymmPoly.hs @@ -30,28 +30,43 @@ -- -{-# LANGUAGE DataKinds, TypeFamilies, Rank2Types, GADTs, StandaloneDeriving #-} +{-# LANGUAGE BangPatterns, DataKinds, TypeFamilies, Rank2Types, GADTs, StandaloneDeriving #-} module Math.RootLoci.Algebra.SymmPoly where -------------------------------------------------------------------------------- -import Data.Proxy +import Control.Monad +import Data.List +-- Semigroup became a superclass of Monoid +#if MIN_VERSION_base(4,11,0) +import Data.Foldable +import Data.Semigroup +#endif + +import System.Random -- testing only + import Math.Combinat.Sign import Math.Combinat.Numbers +import Math.Combinat.Sets ( choose ) import qualified Data.Map.Strict as Map -import Control.Monad -import System.Random +import Data.Array ( Array ) +import Data.Array.IArray -import Math.RootLoci.Algebra.FreeMod (ZMod) -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import Math.Algebra.Polynomial.FreeModule (ZMod) +import qualified Math.Algebra.Polynomial.FreeModule as ZMod -import Math.RootLoci.Misc.Pretty +import Math.RootLoci.Misc.Common +import Math.Algebra.Polynomial.Pretty -import Unsafe.Coerce as Unsafe +-------------------------------------------------------------------------------- +-- | An elementary symmetric polynomial of some generators +symPoly :: (Ord a, Monoid a) => Int -> [a] -> ZMod a +symPoly k xs = ZMod.fromList $ map (\x -> (x,1)) $ (map mconcat $ choose k xs) + -------------------------------------------------------------------------------- -- * Base monomials @@ -185,8 +200,31 @@ -------------------------------------------------------------------------------- +-- Semigroup became a superclass of Monoid +#if MIN_VERSION_base(4,11,0) + +instance Semigroup AB where + (AB a1 b1) <> (AB a2 b2) = AB (a1+a2) (b1+b2) + +instance Semigroup Chern where + (Chern e1 f1) <> (Chern e2 f2) = Chern (e1+e2) (f1+f2) + +instance Semigroup Schur where + (<>) = error "Schur/mappend: not a monoid" + instance Monoid AB where mempty = AB 0 0 + +instance Monoid Chern where + mempty = Chern 0 0 + +instance Monoid Schur where + mempty = Schur 0 0 + +#else + +instance Monoid AB where + mempty = AB 0 0 (AB a1 b1) `mappend` (AB a2 b2) = AB (a1+a2) (b1+b2) instance Monoid Chern where @@ -197,6 +235,8 @@ mempty = Schur 0 0 mappend = error "Schur/mappend: not a monoid" +#endif + -------------------------------------------------------------------------------- instance Pretty AB where @@ -227,9 +267,20 @@ instance Graded Chern where grade (Chern e f) = e + 2*f instance Graded Schur where grade (Schur i j) = i + j +-- | Filters out the given grade filterGrade :: (Ord b, Graded b) => Int -> ZMod b -> ZMod b filterGrade g = ZMod.onFreeMod filt where filt = Map.filterWithKey $ \x _ -> (grade x == g) + +-- | Separates the different grades +separateGradedParts :: (Ord b, Graded b) => ZMod b -> Array Int (ZMod b) +separateGradedParts input = arr where + table = foldl' worker Map.empty (ZMod.toList input) + worker !old (base,coeff) = insertMap (:[]) (:) (grade base) (base,coeff) old + size = if Map.null table then 0 else fst (Map.findMax table) + arr = listArray (0,size) + [ ZMod.fromList (Map.findWithDefault [] d table) + | d <- [0..size] ] -------------------------------------------------------------------------------- -- * Conversions diff --git a/src/Math/RootLoci/CSM/Aluffi.hs b/src/Math/RootLoci/CSM/Aluffi.hs --- a/src/Math/RootLoci/CSM/Aluffi.hs +++ b/src/Math/RootLoci/CSM/Aluffi.hs @@ -30,14 +30,14 @@ import Math.RootLoci.Classic import Math.RootLoci.Misc -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import qualified Math.Algebra.Polynomial.FreeModule as ZMod -------------------------------------------------------------------------------- -- * CSM computation -- | Paolo Aluffi's explicit formula for the (non-equivariant) CSM of open coincident root loci aluffiOpenCSM :: Partition -> ZMod G -aluffiOpenCSM part@(Partition ps) = ZMod.invScale (aut part) xsum where +aluffiOpenCSM part@(Partition ps) = ZMod.divideByConst (aut part) xsum where n = sum ps d = length ps xsum = ZMod.fromList [ ( G (n-d+k) , coeff k ) | k<-[0..d] ] diff --git a/src/Math/RootLoci/CSM/Equivariant/Direct.hs b/src/Math/RootLoci/CSM/Equivariant/Direct.hs --- a/src/Math/RootLoci/CSM/Equivariant/Direct.hs +++ b/src/Math/RootLoci/CSM/Equivariant/Direct.hs @@ -29,7 +29,7 @@ import Math.RootLoci.Geometry import Math.RootLoci.Misc -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import qualified Math.Algebra.Polynomial.FreeModule as ZMod import Math.RootLoci.CSM.Equivariant.PushForward import qualified Math.RootLoci.CSM.Equivariant.Ordered as Ordered @@ -47,14 +47,14 @@ directCalcOpenCSM :: ChernBase base => Partition -> ZMod (Gam base) directCalcOpenCSM part@(Partition xs) = result where m = partitionWeight part - result = ZMod.invScale (aut part) $ pi_star m middle + result = ZMod.divideByConst (aut part) $ pi_star m middle middle = delta_star_ part distinct distinct = Ordered.formulaDistinctCSM (length xs) -------------------------------------------------------------------------------- -- | To compute the CSM of the closed loci, we just some over the open strata --- in the closure. +-- in the closure directClosedCSM :: ChernBase base => Partition -> ZMod (Gam base) directClosedCSM = polyCache2 calc where @@ -62,4 +62,3 @@ calc part = ZMod.sum [ directOpenCSM q | q <- Set.toList (closureSet part) ] -------------------------------------------------------------------------------- - diff --git a/src/Math/RootLoci/CSM/Equivariant/Ordered.hs b/src/Math/RootLoci/CSM/Equivariant/Ordered.hs --- a/src/Math/RootLoci/CSM/Equivariant/Ordered.hs +++ b/src/Math/RootLoci/CSM/Equivariant/Ordered.hs @@ -43,6 +43,12 @@ -------------------------------------------------------------------------------- +-- Semigroup became a superclass of Monoid +#if MIN_VERSION_base(4,11,0) +import Data.Foldable +import Data.Semigroup +#endif + import Math.Combinat.Classes import Math.Combinat.Numbers import Math.Combinat.Sign @@ -56,7 +62,7 @@ import qualified Data.Set as Set ; import Data.Set (Set) -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import qualified Math.Algebra.Polynomial.FreeModule as ZMod import Math.RootLoci.CSM.Equivariant.PushForward @@ -181,10 +187,22 @@ -- | A formal monomial @q^k@ newtype QPow = QPow Int deriving (Eq,Ord,Show) +#if MIN_VERSION_base(4,11,0) + +instance Semigroup QPow where + (<>) (QPow e) (QPow f) = QPow (e+f) + instance Monoid QPow where mempty = QPow 0 + +#else + +instance Monoid QPow where + mempty = QPow 0 mappend (QPow e) (QPow f) = QPow (e+f) +#endif + instance Pretty QPow where pretty (QPow k) = showVarPower "q" k @@ -217,7 +235,7 @@ , (monom [1,2] (-3) , 2) ] | n >= 3 = ZMod.sum - [ ZMod.scale coeff $ (ZMod.symPoly (n-3-k) us) * (ZMod.generator $ monom [] k) + [ ZMod.scale coeff $ (symPoly (n-3-k) us) * (ZMod.generator $ monom [] k) | k<-[0..n-3] , let coeff = negateIfOdd (n-3+k) (factorial (n-3) `div` factorial k) ] @@ -270,7 +288,7 @@ {- smaller = ZMod.sum [ ZMod.scale coeff $ - (ZMod.symPoly (n-k) us) * (embed $ computeQPolys k) + (symPoly (n-k) us) * (embed $ computeQPolys k) | k<-[0..n-1] , let coeff = negateIfOdd (n+k) (factorial n `div` factorial k) ] diff --git a/src/Math/RootLoci/CSM/Equivariant/PushForward.hs b/src/Math/RootLoci/CSM/Equivariant/PushForward.hs --- a/src/Math/RootLoci/CSM/Equivariant/PushForward.hs +++ b/src/Math/RootLoci/CSM/Equivariant/PushForward.hs @@ -47,7 +47,7 @@ import Math.RootLoci.Geometry import Math.RootLoci.Misc -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import qualified Math.Algebra.Polynomial.FreeModule as ZMod -------------------------------------------------------------------------------- -- * The function tau @@ -94,7 +94,7 @@ full = ZMod.generator (Eta ks mempty) -- == sigma_n(eta) rest = ZMod.sum [ sigma (n-i) * tauEta (i-2) * ab | i<-[2..n] ] - sigma k = ZMod.symPoly k [ Eta [k] mempty | k<-ks ] + sigma k = symPoly k [ Eta [k] mempty | k<-ks ] -- | a group generator on the left is a subset (=product) of U-s, which -- we map to a linear combinaton of H-s @@ -224,7 +224,7 @@ g :: Integer -> ZMod (Gam Chern) -> ZMod (Gam Chern) -> ZMod (Gam Chern) g k prev1 prev2 - = ZMod.invScale (mm-k) + = ZMod.divideByConst (mm-k) $ mulGam prev1 + ZMod.scale k (mulInjMonom c1 prev1) + ZMod.scale k (mulInjMonom c2 prev2) @@ -245,9 +245,9 @@ g :: Integer -> ZMod Chern -> ZMod Chern -> ZMod Chern g k prev1 prev2 - = ZMod.invScale (mm-k) - $ ZMod.scale ( k) - $ (ZMod.mulMonom c1 prev1 + ZMod.mulMonom c2 prev2) + = ZMod.divideByConst (mm-k) + $ ZMod.scale ( k) + $ (ZMod.mulByMonom c1 prev1 + ZMod.mulByMonom c2 prev2) mm = fromIntegral m :: Integer diff --git a/src/Math/RootLoci/CSM/Equivariant/Recursive.hs b/src/Math/RootLoci/CSM/Equivariant/Recursive.hs --- a/src/Math/RootLoci/CSM/Equivariant/Recursive.hs +++ b/src/Math/RootLoci/CSM/Equivariant/Recursive.hs @@ -31,7 +31,7 @@ import Math.RootLoci.Geometry import Math.RootLoci.Misc -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import qualified Math.Algebra.Polynomial.FreeModule as ZMod -------------------------------------------------------------------------------- -- * CSM calculation @@ -66,7 +66,7 @@ openCSM = polyCache2 calcOpenCSM where calcOpenCSM :: ChernBase base => Partition -> ZMod (Gam base) - calcOpenCSM part = ZMod.invScale thisCoeff (pushdown `ZMod.sub` smaller) where + calcOpenCSM part = ZMod.divideByConst thisCoeff (pushdown `ZMod.sub` smaller) where n = partitionWeight part pushdown = lowerClass part smaller = ZMod.linComb [ (c , openCSM q) | (q,c) <- Map.assocs theClosure ] diff --git a/src/Math/RootLoci/CSM/Equivariant/Umbral.hs b/src/Math/RootLoci/CSM/Equivariant/Umbral.hs --- a/src/Math/RootLoci/CSM/Equivariant/Umbral.hs +++ b/src/Math/RootLoci/CSM/Equivariant/Umbral.hs @@ -19,6 +19,12 @@ -------------------------------------------------------------------------------- +-- Semigroup became a superclass of Monoid +#if MIN_VERSION_base(4,11,0) +import Data.Foldable +import Data.Semigroup +#endif + import Math.Combinat.Classes import Math.Combinat.Numbers import Math.Combinat.Partitions.Integer @@ -31,8 +37,11 @@ import Math.RootLoci.Geometry import Math.RootLoci.Misc -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import Math.Algebra.Polynomial.Misc ( IsSigned(..) ) +import Math.Algebra.Polynomial.Pretty +import qualified Math.Algebra.Polynomial.FreeModule as ZMod + import Math.RootLoci.CSM.Equivariant.PushForward ( tau , piStarTableAff , piStarTableProj ) import Math.RootLoci.CSM.Equivariant.Ordered ( formulaQPoly ) @@ -46,10 +55,23 @@ = ST !Int !Int deriving (Eq,Ord,Show) +-- Semigroup became a superclass of Monoid +#if MIN_VERSION_base(4,11,0) + +instance Semigroup ST where + (ST s1 t1) <> (ST s2 t2) = ST (s1+s2) (t1+t2) + instance Monoid ST where mempty = ST 0 0 + +#else + +instance Monoid ST where + mempty = ST 0 0 (ST s1 t1) `mappend` (ST s2 t2) = ST (s1+s2) (t1+t2) +#endif + instance Pretty ST where pretty st = case st of ST 0 0 -> "" @@ -57,7 +79,7 @@ ST 0 f -> showVarPower "t" f ST e f -> showVarPower "s" e ++ "*" ++ showVarPower "t" f -prettyMixedST :: forall b c. (Pretty b, Num c, Eq c, IsSigned c, Show c) => FreeMod (FreeMod c b) ST -> String +prettyMixedST :: forall b c. (Pretty b, Num c, Eq c, IsSigned c, Pretty c) => FreeMod (FreeMod c b) ST -> String prettyMixedST = prettyFreeMod'' prettyInner pretty where prettyInner :: FreeMod c b -> String @@ -78,8 +100,8 @@ | otherwise = error "theta: non-positive input" where - term0 = [ (ST 0 i , ZMod.scale (binomial p i) ( tau (p-i-1)) ) | i<-[0..p-1] ] - term1 = [ (ST 1 i , ZMod.scale (binomial p i) (ZMod.mulMonom c2_monom $ tau (p-i-2)) ) | i<-[0..p-2] ] + term0 = [ (ST 0 i , ZMod.scale (binomial p i) ( tau (p-i-1)) ) | i<-[0..p-1] ] + term1 = [ (ST 1 i , ZMod.scale (binomial p i) (ZMod.mulByMonom c2_monom $ tau (p-i-2)) ) | i<-[0..p-2] ] ++ [ (ST 1 p , ZMod.konst (-1) ) ] c2_monom = select0 (alphaBeta,c2) @@ -107,6 +129,25 @@ -------------------------------------------------------------------------------- -- * The affine CSM +-- | Weights of the representation @Sym^m C^2@ +affineWeights :: Int -> [ZMod AB] +affineWeights m = + [ ZMod.fromList [ ( AB 1 0 , fi (m-j) ) , ( AB 0 1 , fi j ) ] + | j <- [0..m] + ] + where + fi :: Int -> Integer + fi = fromIntegral + +-- | The top Chern class of the representation is just the product of weights. +-- This represents the zero orbit, and we need to add this to the closure in the +-- affine case! +topChernClass :: ChernBase base => Int -> ZMod base +topChernClass m = select1 (total , abToChern total) where + total = product [ w | w <- affineWeights m ] + +-------------------------------------------------------------------------------- + -- | The polynomial to be substituted in the place of @s^k*t^j@: -- -- > s^k*t^j -> P_j(m) * Q_k(n-3-k) * (n-3)_k @@ -144,18 +185,25 @@ -- the current umbral formula only works for @n >= 3@ ?? calc mu | n < 3 = forgetGamma (Direct.directOpenCSM mu) - | otherwise = ZMod.invScale (aut mu) + | otherwise = ZMod.divideByConst (aut mu) $ umbralSubstitutionAff mu $ integralUmbralFormula mu where n = numberOfParts mu --- | Sum over the strata in the closure +-------------------------------------------------------------------------------- + +-- | CSM class of the zero orbit (which is just the top Chern class) +affineZeroCSM :: ChernBase base => Int -> ZMod base +affineZeroCSM m = topChernClass m + +-- | Sum over the strata in the closure (including the zero orbit!) umbralAffClosedCSM :: ChernBase base => Partition -> ZMod base umbralAffClosedCSM = polyCache1 calc where calc :: ChernBase base => Partition -> ZMod base - calc part = ZMod.sum [ umbralAffOpenCSM q | q <- Set.toList (closureSet part) ] + calc part = affineZeroCSM (weight part) + + ZMod.sum [ umbralAffOpenCSM q | q <- Set.toList (closureSet part) ] -------------------------------------------------------------------------------- -- * The projective CSM @@ -198,7 +246,7 @@ -- the current umbral formula only works for @n >= 3@ ?? calc mu | n < 3 = Direct.directOpenCSM mu - | otherwise = ZMod.invScale (aut mu) + | otherwise = ZMod.divideByConst (aut mu) $ umbralSubstitutionProj mu $ integralUmbralFormula mu where diff --git a/src/Math/RootLoci/CSM/Projective.hs b/src/Math/RootLoci/CSM/Projective.hs --- a/src/Math/RootLoci/CSM/Projective.hs +++ b/src/Math/RootLoci/CSM/Projective.hs @@ -39,7 +39,7 @@ import Math.RootLoci.Geometry import Math.RootLoci.Misc -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import qualified Math.Algebra.Polynomial.FreeModule as ZMod -------------------------------------------------------------------------------- @@ -171,7 +171,7 @@ -- | we know that (pi_* upperCSM) = sum (chi * openCSM) calcOpenCSM :: Partition -> ZMod G - calcOpenCSM part = ZMod.invScale thisCoeff (pushdown - smaller) where + calcOpenCSM part = ZMod.divideByConst thisCoeff (pushdown - smaller) where n = partitionWeight part pushdown = lowerCSM part -- pi_star n (upperCSM part) smaller = ZMod.linComb [ (c , openCSM q) | (q,c) <- Map.assocs theClosure ] diff --git a/src/Math/RootLoci/Classic.hs b/src/Math/RootLoci/Classic.hs --- a/src/Math/RootLoci/Classic.hs +++ b/src/Math/RootLoci/Classic.hs @@ -70,21 +70,67 @@ -- check_hilbert2 = and [ hilbert p == hilbert2 p | n<-[0..20] , p<-partitions n ] -------------------------------------------------------------------------------- --- * Schubert +-- * Enumerative geometry --- | Number of 4-tangent lines to a generic degree @d@ surface -quadTangentLines :: Int -> Integer -quadTangentLines d0 +-- | The degree of the dual curve is @d(d-1)@ +degreeOfDualCurve :: Int -> Integer +degreeOfDualCurve d0 + | d < 2 = 0 + | otherwise = d*(d-1) + where + d = fromIntegral d0 :: Integer + +-- | Number of flex lines to a generic degree @d@ plane curve +numberOfCurveFlexes :: Int -> Integer +numberOfCurveFlexes d0 + | d < 3 = 0 + | otherwise = 3*d*(d-2) + where + d = fromIntegral d0 :: Integer + +-- | Number of bitangent lines to a generic degree @d@ plane curve +numberOfCurveBiTangents :: Int -> Integer +numberOfCurveBiTangents d0 + | d < 3 = 0 + | otherwise = div ((-3 + d)* (-2 + d)* d* (3 + d)) 2 + where + d = fromIntegral d0 :: Integer + +-- | Number of 4-tangent lines to a generic degree @d@ surface (Schubert) +numberOfSurface4xTangents :: Int -> Integer +numberOfSurface4xTangents d0 | d < 8 = 0 | otherwise = d * (d - 4) * (d - 5) * (d - 6) * (d - 7) * (d^3 + 6*d^2 + 7*d - 30) where d = fromIntegral d0 :: Integer -- | Number of lines meeting a generic degree @d@ surface at point with 5x multiplicity -quintFlexLines :: Int -> Integer -quintFlexLines d0 +numberOfSurface5xHyperflexes :: Int -> Integer +numberOfSurface5xHyperflexes d0 | d < 5 = 0 - | otherwise = error "quintFlexLines" + | otherwise = (35*d^3 - 200*d^2 + 240*d) + where + d = fromIntegral d0 :: Integer + +-- | Bidegree of bitangent locus of a generic hypersurface +-- +-- (See: Kathlen Kohn, Bernt Ivar Utstol Nodland, Paolo Tripoli: Secants, bitangents, and their congruences) +-- +bidegreeOfSurfaceBiTangents :: Int -> (Integer,Integer) +bidegreeOfSurfaceBiTangents d0 + | d < 4 = ( 0 , 0 ) + | otherwise = ( div (d*(d-1)*(d-2)*(d-3)) 2 , div (d*(d-2)*(d-3)*(d+3)) 2 ) + where + d = fromIntegral d0 :: Integer + +-- | Bidegree of the flex locus of a generic hypersurface +-- +-- (See: Kathlen Kohn, Bernt Ivar Utstol Nodland, Paolo Tripoli: Secants, bitangents, and their congruences) +-- +bidegreeOfSurfaceFlexes :: Int -> (Integer,Integer) +bidegreeOfSurfaceFlexes d0 + | d < 4 = ( 0 , 0 ) + | otherwise = ( d*(d-1)*(d-3) , 3*d*(d-2) ) where d = fromIntegral d0 :: Integer diff --git a/src/Math/RootLoci/Dual/Localization.hs b/src/Math/RootLoci/Dual/Localization.hs --- a/src/Math/RootLoci/Dual/Localization.hs +++ b/src/Math/RootLoci/Dual/Localization.hs @@ -11,6 +11,7 @@ -- out the result (we know a priori that it is a homogenenous polynomial -- in @alpha@ and @beta@). +{-# LANGUAGE DataKinds #-} module Math.RootLoci.Dual.Localization where -------------------------------------------------------------------------------- @@ -27,13 +28,23 @@ import qualified Data.Map as Map -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import qualified Math.Algebra.Polynomial.FreeModule as ZMod +import Math.Algebra.Polynomial.Univariate +import Math.Algebra.Polynomial.Univariate.Lagrange + import Math.RootLoci.Algebra import Math.RootLoci.Classic -------------------------------------------------------------------------------- +type X = U "x" + +mkX :: Int -> X +mkX = U + +-------------------------------------------------------------------------------- + -- | The localization formula as a string which Mathematica can parse localizeMathematica :: Partition -> String localizeMathematica (Partition xs) = formula where @@ -86,7 +97,7 @@ localizeDual :: Partition -> ZMod AB localizeDual part = ZMod.mapBase worker $ localizeInterpolatedZ part where c = codim part - worker (X i) = AB (c-i) i + worker (U i) = AB (c-i) i -- | We can use Lagrange interpolation to express the dual class from the -- localization formula (since we know a priori that the result is a homogeneous @@ -97,7 +108,7 @@ codim = sum xs - length xs bs = map fromIntegral [ 2..codim+2 ] :: [Rational] qs = [ localizeEval part 1 b | b<-bs ] :: [Rational] - final = lagrangeInterp' (zip bs qs) + final = unUni $ lagrangeInterp (zip bs qs) localizeInterpolatedZ :: Partition -> ZMod X localizeInterpolatedZ = (ZMod.mapCoeff f . localizeInterpolatedQ) where diff --git a/src/Math/RootLoci/Dual/Restriction.hs b/src/Math/RootLoci/Dual/Restriction.hs --- a/src/Math/RootLoci/Dual/Restriction.hs +++ b/src/Math/RootLoci/Dual/Restriction.hs @@ -27,7 +27,7 @@ import qualified Data.Set as Set ; import Data.Set (Set) -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import qualified Math.Algebra.Polynomial.FreeModule as ZMod import Math.RootLoci.Algebra import Math.RootLoci.Classic @@ -144,7 +144,7 @@ -- | The dual class of the closure agress with the lowest degree part of the CSM class. dualClassFromProjCSM :: forall base. ChernBase base => ZMod (Gam base) -> ZMod base -dualClassFromProjCSM csm = dualClassFromAffCSM (ZMod.filterBase nogamma csm) where +dualClassFromProjCSM csm = dualClassFromAffCSM (ZMod.mapMaybeBase nogamma csm) where nogamma :: Gam base -> Maybe base nogamma (Gam k ab) = if k==0 then Just ab else Nothing diff --git a/src/Math/RootLoci/Geometry/Cohomology.hs b/src/Math/RootLoci/Geometry/Cohomology.hs --- a/src/Math/RootLoci/Geometry/Cohomology.hs +++ b/src/Math/RootLoci/Geometry/Cohomology.hs @@ -23,16 +23,22 @@ import Data.List import Data.Monoid +-- Semigroup became a superclass of Monoid +#if MIN_VERSION_base(4,11,0) +import Data.Foldable +import Data.Semigroup +#endif + import Math.Combinat.Numbers import qualified Data.Map as Map import qualified Data.Set as Set -import qualified Math.RootLoci.Algebra.FreeMod as ZMod -import Math.RootLoci.Algebra.FreeMod ( ZMod , FreeMod(..) , unFreeMod ) +import qualified Math.Algebra.Polynomial.FreeModule as ZMod +import Math.Algebra.Polynomial.FreeModule ( ZMod , FreeMod(..) , unFreeMod ) import Math.RootLoci.Algebra.SymmPoly -import Math.RootLoci.Misc.Pretty +import Math.Algebra.Polynomial.Pretty -------------------------------------------------------------------------------- -- * The non-equivariant case @@ -54,8 +60,41 @@ -------------------------------------------------------------------------------- +-- Semigroup became a superclass of Monoid +#if MIN_VERSION_base(4,11,0) + +instance Semigroup US where + (US us1) <> (US us2) = + if nub us3 == us3 + then US us3 + else error "[U]/monoid: duplicate indices" + where + us3 = sort (us1 ++ us2) + +instance Semigroup HS where + (HS hs1) <> (HS hs2) = + if nub hs3 == hs3 + then HS hs3 + else error "[H]/monoid: duplicate indices" + where + hs3 = sort (hs1 ++ hs2) + +instance Semigroup G where + (G e) <> (G f) = G (e+f) + instance Monoid US where mempty = US [] + +instance Monoid HS where + mempty = HS [] + +instance Monoid G where + mempty = G 0 + +#else + +instance Monoid US where + mempty = US [] (US us1) `mappend` (US us2) = if nub us3 == us3 then US us3 @@ -75,6 +114,8 @@ instance Monoid G where mempty = G 0 (G e) `mappend` (G f) = G (e+f) + +#endif -------------------------------------------------------------------------------- @@ -149,13 +190,13 @@ injectZMod = ZMod.mapBase injectMonom forgetGamma :: Ord base => ZMod (Gam base) -> ZMod base -forgetGamma = ZMod.filterBase f where +forgetGamma = ZMod.mapMaybeBase f where f (Gam k ab) = case k of 0 -> Just ab _ -> Nothing forgetEquiv :: ChernBase base => ZMod (Gam base) -> ZMod G -forgetEquiv = ZMod.filterBase f where +forgetEquiv = ZMod.mapMaybeBase f where f (Gam k ab) = if (ab == mempty) then Just (G k) else Nothing @@ -211,18 +252,50 @@ -------------------------------------------------------------------------------- -- | This is a hack to reuse the same pushforward code -unsafeEtaToOmega :: Ord ab => FreeMod coeff (Eta ab) -> FreeMod coeff (Omega ab) +unsafeEtaToOmega :: (Eq coeff, Num coeff, Ord ab) => FreeMod coeff (Eta ab) -> FreeMod coeff (Omega ab) unsafeEtaToOmega = ZMod.mapBase f where f (Eta js ab) = Omega js ab -unsafeOmegaToEta :: Ord ab => FreeMod coeff (Omega ab) -> FreeMod coeff (Eta ab) +unsafeOmegaToEta :: (Eq coeff, Num coeff, Ord ab) => FreeMod coeff (Omega ab) -> FreeMod coeff (Eta ab) unsafeOmegaToEta = ZMod.mapBase f where f (Omega js ab) = Eta js ab -------------------------------------------------------------------------------- +#if MIN_VERSION_base(4,11,0) + +instance Semigroup ab => Semigroup (Omega ab) where + (Omega as ab1) <> (Omega bs ab2) = + if nub cs == cs + then Omega cs (ab1 <> ab2) + else error "Omega/monoid: duplicate indices" + where + cs = sort (as ++ bs) + +instance Semigroup ab => Semigroup (Eta ab) where + (Eta fs ab1) <> (Eta gs ab2) = + if nub hs == hs + then Eta hs (ab1 <> ab2) + else error "Eta/monoid: duplicate indices" + where + hs = sort (fs ++ gs) + +instance Semigroup ab => Semigroup (Gam ab) where + (Gam e ab1) <> (Gam f ab2) = Gam (e+f) (ab1 <> ab2) + instance Monoid ab => Monoid (Omega ab) where mempty = Omega [] mempty + +instance Monoid ab => Monoid (Eta ab) where + mempty = Eta [] mempty + +instance Monoid ab => Monoid (Gam ab) where + mempty = Gam 0 mempty + +#else + +instance Monoid ab => Monoid (Omega ab) where + mempty = Omega [] mempty (Omega as ab1) `mappend` (Omega bs ab2) = if nub cs == cs then Omega cs (ab1 <> ab2) @@ -242,6 +315,8 @@ instance Monoid ab => Monoid (Gam ab) where mempty = Gam 0 mempty (Gam e ab1) `mappend` (Gam f ab2) = Gam (e+f) (ab1 <> ab2) + +#endif -------------------------------------------------------------------------------- diff --git a/src/Math/RootLoci/Geometry/Forget.hs b/src/Math/RootLoci/Geometry/Forget.hs --- a/src/Math/RootLoci/Geometry/Forget.hs +++ b/src/Math/RootLoci/Geometry/Forget.hs @@ -62,10 +62,10 @@ countDirectCoarsenings :: Partition -> Map Partition Integer countDirectCoarsenings part = Map.fromListWith (+) list where list = - [ ( fromExponentialFrom ((i1+i2,1):(i1,e1-1):(i2,e2-1):rest) , fromIntegral (e1*e2) ) + [ ( fromExponentialForm ((i1+i2,1):(i1,e1-1):(i2,e2-1):rest) , fromIntegral (e1*e2) ) | ( (i1,e1):(i2,e2):[] , rest ) <- choose' 2 ies ] ++ - [ ( fromExponentialFrom ((2*i,1):(i,e-2):rest) , binomial e 2 ) + [ ( fromExponentialForm ((2*i,1):(i,e-2):rest) , binomial e 2 ) | ( (i,e):[] , rest ) <- choose' 1 ies , e >= 2 ] diff --git a/src/Math/RootLoci/Geometry/Mobius.hs b/src/Math/RootLoci/Geometry/Mobius.hs --- a/src/Math/RootLoci/Geometry/Mobius.hs +++ b/src/Math/RootLoci/Geometry/Mobius.hs @@ -31,7 +31,7 @@ import Math.Combinat.Partitions.Set import Math.Combinat.Sets -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import qualified Math.Algebra.Polynomial.FreeModule as ZMod import Math.RootLoci.Algebra import Math.RootLoci.Misc diff --git a/src/Math/RootLoci/Misc.hs b/src/Math/RootLoci/Misc.hs --- a/src/Math/RootLoci/Misc.hs +++ b/src/Math/RootLoci/Misc.hs @@ -4,10 +4,10 @@ module Math.RootLoci.Misc ( module Math.RootLoci.Misc.Common , module Math.RootLoci.Misc.PTable - , module Math.RootLoci.Misc.Pretty + , module Math.Algebra.Polynomial.Pretty ) where import Math.RootLoci.Misc.Common import Math.RootLoci.Misc.PTable -import Math.RootLoci.Misc.Pretty +import Math.Algebra.Polynomial.Pretty diff --git a/src/Math/RootLoci/Misc/Common.hs b/src/Math/RootLoci/Misc/Common.hs --- a/src/Math/RootLoci/Misc/Common.hs +++ b/src/Math/RootLoci/Misc/Common.hs @@ -1,7 +1,7 @@ -- | Some auxilary functions -{-# LANGUAGE BangPatterns, TypeSynonymInstances, FlexibleInstances, DeriveFunctor #-} +{-# LANGUAGE CPP, BangPatterns, TypeSynonymInstances, FlexibleInstances, DeriveFunctor #-} module Math.RootLoci.Misc.Common where -------------------------------------------------------------------------------- @@ -9,6 +9,7 @@ import Data.List import Data.Monoid import Data.Ratio +import Data.Ord import Control.Monad @@ -21,10 +22,6 @@ import qualified Data.Map.Strict as Map import Data.Map (Map) --- import qualified Math.RootLoci.Algebra.FreeMod as ZMod --- import Math.RootLoci.Algebra.SymmPoly --- import Math.RootLoci.Geometry.Cohomology - -------------------------------------------------------------------------------- -- * Pairs @@ -35,6 +32,10 @@ -------------------------------------------------------------------------------- -- * Lists +{-# SPECIALIZE sum' :: [Int] -> Int #-} +sum' :: Num a => [a] -> a +sum' = foldl' (+) 0 + {-# SPECIALIZE unique :: [Partition] -> [Partition] #-} unique :: Ord a => [a] -> [a] unique = map head . group . sort @@ -48,6 +49,34 @@ histogram xs = foldl' f Map.empty xs where f old x = Map.insertWith (+) x 1 old +#if MIN_VERSION_base(4,8,0) +-- sortOn already in base, nothing to do +#else +-- sortOn not yet in base, let's define it +sortOn :: Ord b => (a -> b) -> [a] -> [a] +sortOn f = sortBy (comparing f) +#endif + +longZipWith :: (a -> c) -> (b -> c) -> (a -> b -> c) -> [a] -> [b] -> [c] +longZipWith f g h = go where + go (x:xs) (y:ys) = h x y : go xs ys + go xs [] = map f xs + go [] ys = map g ys + +evens :: [a] -> [a] +evens (x:xs) = x : odds xs +evens [] = [] + +odds :: [a] -> [a] +odds (_:xs) = evens xs +odds [] = [] + +interleave :: [a] -> [a] -> [a] +interleave = go where + go (x:xs) (y:ys) = x : y : go xs ys + go [] [] = [] + go _ _ = error "interleave: input lists do not have the same length" + -------------------------------------------------------------------------------- -- * Maps @@ -60,6 +89,21 @@ Just y -> y Nothing -> error "unsafeDeleteLookup: key not found" +-- | Example usage: @insertMap (:[]) (:) ...@ +insertMap :: Ord k => (b -> a) -> (b -> a -> a) -> k -> b -> Map k a -> Map k a +insertMap f g k y = Map.alter h k where + h mb = case mb of + Nothing -> Just (f y) + Just x -> Just (g y x) + +-- | Example usage: @buildMap (:[]) (:) ...@ +buildMap :: Ord k => (b -> a) -> (b -> a -> a) -> [(k,b)] -> Map k a +buildMap f g xs = foldl' worker Map.empty xs where + worker !old (k,y) = Map.alter h k old where + h mb = case mb of + Nothing -> Just (f y) + Just x -> Just (g y x) + -------------------------------------------------------------------------------- -- * Partitions @@ -72,6 +116,14 @@ aut part = product $ map factorial es where es = map snd $ toExponentialForm part +-- | TODO: move this into combinat +exponentVector :: Partition -> [Int] +exponentVector p = go 1 (toExponentialForm p) where + go _ [] = [] + go !i ef@((j,e):rest) = if i Maybe Sign @@ -101,6 +154,7 @@ instance IsSigned Int where signOf = signOfNum instance IsSigned Integer where signOf = signOfNum instance IsSigned Rational where signOf = signOfNum +-} -------------------------------------------------------------------------------- -- * Numbers @@ -130,3 +184,48 @@ prod' = foldl' (*) 1 -------------------------------------------------------------------------------- +-- * Utility + +-- | Put into parentheses +paren :: String -> String +paren s = '(' : s ++ ")" + +-------------------------------------------------------------------------------- + +-- | Exponential form of a partition +expFormString :: Partition -> String +expFormString p = "(" ++ intercalate "," (map f ies) ++ ")" where + ies = toExponentialForm p + f (i,e) = show i ++ "^" ++ show e + +extendStringL :: Int -> String -> String +extendStringL k s = s ++ replicate (k - length s) ' ' + +extendStringR :: Int -> String -> String +extendStringR k s = replicate (k - length s) ' ' ++ s + +-------------------------------------------------------------------------------- +-- * Mathematica-formatted output + +class Mathematica a where + mathematica :: a -> String + +instance Mathematica Int where + mathematica = show + +instance Mathematica Integer where + mathematica = show + +instance Mathematica String where + mathematica = show + +instance Mathematica Partition where + mathematica (Partition ps) = "{" ++ intercalate "," (map show ps) ++ "}" + +data Indexed a = Indexed String a + +instance Mathematica a => Mathematica (Indexed a) where + mathematica (Indexed x sub) = "Subscript[" ++ x ++ "," ++ mathematica sub ++ "]" + +-------------------------------------------------------------------------------- + diff --git a/src/Math/RootLoci/Misc/Pretty.hs b/src/Math/RootLoci/Misc/Pretty.hs deleted file mode 100644 --- a/src/Math/RootLoci/Misc/Pretty.hs +++ /dev/null @@ -1,137 +0,0 @@ - -{-# LANGUAGE FlexibleInstances #-} - --- | Pretty-printing - -module Math.RootLoci.Misc.Pretty where - --------------------------------------------------------------------------------- - -import Data.List - -import Math.Combinat.Sign -import Math.Combinat.Partitions.Integer - -import qualified Data.Map.Strict as Map -import Data.Map.Strict (Map) - -import Math.RootLoci.Algebra.FreeMod ( FreeMod, ZMod, QMod ) -import qualified Math.RootLoci.Algebra.FreeMod as ZMod - -import Math.RootLoci.Misc.Common - --------------------------------------------------------------------------------- - -class Pretty a where - pretty :: a -> String - --- instance Pretty a => Pretty (ZMod a) where --- pretty = prettyZMod pretty - -instance (Num c, Eq c, Show c, IsSigned c, Pretty b) => Pretty (FreeMod c b) where - pretty = prettyFreeMod' True pretty - --------------------------------------------------------------------------------- --- * Pretty printing elements of free modules - --- | Example: @showVarPower "x" 5 == "x^5"@ -showVarPower :: String -> Int -> String -showVarPower name expo = case expo of - 0 -> "1" - 1 -> name - _ -> name ++ "^" ++ show expo - --------------------------------------------------------------------------------- - --- | no multiplication sign (ok for mathematica and humans) -prettyZMod_ :: (b -> String) -> ZMod b -> String -prettyZMod_ = prettyFreeMod' False - --- | multiplication sign (ok for maple etc) -prettyZMod :: (b -> String) -> ZMod b -> String -prettyZMod = prettyFreeMod' True - --------------------------------------------------------------------------------- - -prettyFreeMod' - :: (Num c, Eq c, Show c, IsSigned c) - => Bool -- ^ use star for multiplication (@False@ means just concatenation) - -> (b -> String) -- ^ show base - -> FreeMod c b - -> String -prettyFreeMod' star showBase what = final where - final = if take 3 stuff == " + " then drop 3 stuff else drop 1 stuff - stuff = concatMap f (ZMod.toList what) - f (g, 1) = plus ++ showBase' g - f (g, -1) = minus ++ showBase' g - f (g, c) = sgn c ++ {- extendStringR 3 -} (show $ abs c) ++ starStr ++ showBase' g - -- cond (_,c) = (c/=0) - starStr = if star then "*" else " " - showBase' g = case showBase g of - "" -> "1" -- "(1)" - s -> s - sgn c = case signOf c of - Just Minus -> minus - _ -> plus - plus = " + " - minus = " - " - -prettyFreeMod'' - :: (c -> String) -- ^ show coefficient - -> (b -> String) -- ^ show base - -> FreeMod c b - -> String -prettyFreeMod'' showCoeff showBase what = result where - result = intercalate " + " (map f $ ZMod.toList what) - f (g, c) = showCoeff c ++ starStr ++ showBase' g - starStr = "*" -- if star then "*" else " " - showBase' g = case showBase g of - "" -> "1" -- "(1)" - s -> s - --------------------------------------------------------------------------------- --- * Utility - --- | Put into parentheses -paren :: String -> String -paren s = '(' : s ++ ")" - --------------------------------------------------------------------------------- - --- | Exponential form of a partition -expFormString :: Partition -> String -expFormString p = "(" ++ intercalate "," (map f ies) ++ ")" where - ies = toExponentialForm p - f (i,e) = show i ++ "^" ++ show e - -extendStringL :: Int -> String -> String -extendStringL k s = s ++ replicate (k - length s) ' ' - -extendStringR :: Int -> String -> String -extendStringR k s = replicate (k - length s) ' ' ++ s - --------------------------------------------------------------------------------- --- * Mathematica-formatted output - -class Mathematica a where - mathematica :: a -> String - -instance Mathematica Int where - mathematica = show - -instance Mathematica Integer where - mathematica = show - -instance Mathematica String where - mathematica = show - -instance Mathematica Partition where - mathematica (Partition ps) = "{" ++ intercalate "," (map show ps) ++ "}" - -data Indexed a = Indexed String a - -instance Mathematica a => Mathematica (Indexed a) where - mathematica (Indexed x sub) = "Subscript[" ++ x ++ "," ++ mathematica sub ++ "]" - --------------------------------------------------------------------------------- - diff --git a/src/Math/RootLoci/Motivic/Abstract.hs b/src/Math/RootLoci/Motivic/Abstract.hs new file mode 100644 --- /dev/null +++ b/src/Math/RootLoci/Motivic/Abstract.hs @@ -0,0 +1,430 @@ + +-- | The abstract motivic algorithm +-- +-- See: B. Komuves: Motivic characteristic classes of discriminant strata +-- +-- TODO: caching of results (otherwise it is very slow) + +{-# LANGUAGE CPP, BangPatterns, FlexibleInstances, TypeSynonymInstances, + MultiParamTypeClasses, FunctionalDependencies, GeneralizedNewtypeDeriving, + TypeFamilies + #-} +module Math.RootLoci.Motivic.Abstract where + +-------------------------------------------------------------------------------- + +import Data.Char +import Data.List +import Data.Ord +import Data.Maybe + +import Data.Map.Strict (Map) +import qualified Data.Map.Strict as Map + +import qualified Math.Algebra.Polynomial.FreeModule as ZMod +import Math.Algebra.Polynomial.FreeModule (ZMod,QMod,FreeMod) +import Math.Algebra.Polynomial.Pretty + +import Math.Combinat.Classes hiding (empty) +import Math.Combinat.Tuples +import Math.Combinat.Partitions +import Math.Combinat.Permutations hiding (permute) + +-- import Debug.Trace +-- debug s x y = trace (">>> " ++ s ++ " -> " ++ show x) y + +import Math.RootLoci.Motivic.Classes +import Math.RootLoci.Misc.Common + +-------------------------------------------------------------------------------- +-- * The abstract algorithm + +-- | The (abstract) class of @Sym^n(X)@ +symn :: Num c => Dim -> FreeMod c SingleLam +symn dim = ZMod.generator $ SingleLam (Bindings [dim]) (Single [(DeBruijn 0, 1)]) + +-- | The open stratum X(1,1,...,1) +open :: Dim -> ZMod SingleLam +open d@(Dim n) = symn d `ZMod.sub` rest where + rest = ZMod.sum [ xlam p | p <- partitions n , width p < n ] + +zeros :: Int -> ZMod MultiLam +zeros k = ZMod.generator + $ MultiLam (Bindings []) (Multi $ replicate k (Single [])) + +-- | The open stratum X(lambda) +xlam :: Partition -> ZMod SingleLam +xlam p = + if height p == 1 + then open (Dim $ weight p) + else normalize $ psi $ omega123 $ dvec $ dimVector p + +-- | The open stratum D(n1,n2,...) +dvec :: [Dim] -> ZMod MultiLam +dvec dims0 = permute invperm sorted where + invperm = inversePermutation perm + perm = sortingPermutationDesc dims0 + dims1 = permuteList perm dims0 + (dims2,dzeros) = span (>0) dims1 -- separate zero dimensions + sorted = cross (dvecSorted dims2) (zeros $ length dzeros) + +-- | The open stratum D(n1,n2,...), assuming @n1 >= n2 >= n3 >= ...@ +dvecSorted :: [Dim] -> ZMod MultiLam +dvecSorted [] = error "dvec: empty dimension vector shouldn't appear in the algorithm" +dvecSorted [n] = singleToMulti (open n) +dvecSorted (p:ns) = normalize (ZMod.sub big rest) where + big = cross (singleToMulti $ open p) (dvecSorted ns) + rest = ZMod.sum + [ theta (dvec (k : interleave ds es)) + | ds <- dimTuples ns + , let es = zipWith (-) ns ds + , let l = sum es + , l > 0 + , let k = p - l + , k >= 0 + ] + +-------------------------------------------------------------------------------- +-- * Data types and instances + +-- | A variable, implemented as a /de Bruijn level/ (indexing starts from 0) +newtype Var + = DeBruijn Int + deriving (Eq,Ord,Show) + +-------------------------------------------------------------------------------- + +-- | We use de Bruijn levels to index the bound variables, and ecah bound variables has a dimension +newtype Bindings + = Bindings [Dim] + deriving (Eq,Ord,Show) + +numberOfBoundVariables :: Bindings -> Int +numberOfBoundVariables (Bindings ds) = length ds + +dimensionTable :: Bindings -> Map Var Dim +dimensionTable (Bindings dims) = Map.fromList $ zip (map DeBruijn [0..]) dims + +-------------------------------------------------------------------------------- + +-- | An expression living on @Sym^n(X)@, with free variables +newtype Single + = Single [(Var,Int)] + deriving (Eq,Ord,Show) + +unSingle :: Single -> [(Var,Int)] +unSingle (Single ves) = ves + +-- | An expression living on @Sym^{n_1}(X) x ... x Sym^{n_r}(X)@, with free variables +newtype Multi + = Multi [Single] + deriving (Eq,Ord,Show) + +-- | A lambda expression living on @Sym^n(X)@, with variables bound to @Sym^d(X)@ with different dimensions +data SingleLam + = SingleLam !Bindings !Single + deriving (Eq,Ord,Show) + +-- | A lambda expression living on @Sym^{n_1}(X) x ... x Sym^{n_r}(X)@, with variables bound to @Sym^d(X)@ with different dimensions +data MultiLam + = MultiLam !Bindings !Multi + deriving (Eq,Ord,Show) + +-------------------------------------------------------------------------------- + +instance Pretty Bindings where + pretty (Bindings dims) = "\\" ++ concat (zipWith f vars dims) where + vars = map DeBruijn [0..] + f v d = "(" ++ pretty v ++ ":S" ++ show d ++ ")" + +instance Pretty Var where + pretty (DeBruijn i) = chr (97 + i) : [] + +instance Pretty (Var,Int) where + pretty (v,0) = "1" + pretty (v,1) = pretty v + pretty (v,e) = pretty v ++ "^" ++ show e + +instance Pretty Single where + pretty (Single ves) = intercalate "*" $ map pretty ves + +instance Pretty Multi where + pretty (Multi ts) = "[" ++ (intercalate "," $ map pretty ts) ++ "]" + +instance Pretty SingleLam where + pretty (SingleLam binds body) = "{" ++ pretty binds ++ "->" ++ pretty body ++ "}" + +instance Pretty MultiLam where + pretty (MultiLam binds body) = "{" ++ pretty binds ++ "->" ++ pretty body ++ "}" + +-------------------------------------------------------------------------------- + +instance Degree SingleLam where + type MultiDegree SingleLam = Int + multiDegree (SingleLam binds (Single ves)) = sum [ (unDim $ (Map.!) dimTable v) * e | (v,e) <- ves ] where + dimTable = dimensionTable binds + totalDegree = multiDegree + +instance Degree MultiLam where + type MultiDegree MultiLam = [Int] + multiDegree (MultiLam binds (Multi bodies)) = map totalDegree [ (SingleLam binds b) | b <- bodies ] + totalDegree = sum . multiDegree + +-------------------------------------------------------------------------------- + +instance Empty Bindings where + empty = Bindings [] + +instance Empty Single where + empty = Single [] + +instance Empty Multi where + empty = Multi [] + +instance Empty SingleLam where + empty = SingleLam empty empty + +instance Empty MultiLam where + empty = MultiLam empty empty + +-------------------------------------------------------------------------------- + +-- | Shift de Bruijn levels +class Shift a where + shift :: Int -> a -> a + +instance Shift Var where + shift k (DeBruijn l) = DeBruijn (k+l) + +instance Shift Single where + shift k (Single ves) = Single [ (shift k v, e) | (v,e) <- ves ] + +instance Shift Multi where + shift k (Multi terms) = Multi $ map (shift k) terms + +-------------------------------------------------------------------------------- + +-- | Rename variables +class Rename a where + rename :: (Var -> Var) -> a -> a + +instance Rename Var where + rename f v = f v + +instance Rename (Var,Int) where + rename f (v,e) = (f v, e) + +instance Rename Single where + rename f (Single ves) = Single $ map (rename f) ves + +instance Rename Multi where + rename f (Multi ts) = Multi $ map (rename f) ts + +-------------------------------------------------------------------------------- + +-- | Extract the exponent of a given variable +exponentOf :: Var -> Single -> Int +exponentOf u (Single ves) = sum [ e | (v,e) <- ves , v==u ] + +-- | Extract the exponent vector of a given variable +exponentVectorOf :: Var -> Multi -> [Int] +exponentVectorOf v (Multi ts) = map (exponentOf v) ts + +-------------------------------------------------------------------------------- + +instance Normalize Single where + normalize (Single ves) = Single $ Map.toList $ Map.fromListWith (+) ves + +instance Normalize Multi where + normalize (Multi terms) = Multi (map normalize terms) + +normalizeWithExpo :: (Rename term, Normalize term, Ord expo) => (expo -> Bool) -> (Var -> term -> expo) -> (Bindings,term) -> (Bindings,term) +normalizeWithExpo cond expo (binds,body) = (binds',body') where + Bindings dims = binds + vars = map DeBruijn [0..] + vby = [ (v,(d,es)) | (v,d) <- zip vars dims , let es = expo v body , cond es ] + sorted = sortOn snd vby + dims' = map (fst . snd) sorted + binds' = Bindings dims' + f v = DeBruijn $ fromJust $ findIndex (\pair -> fst pair == v) sorted + body' = normalize $ rename f $ body + +instance Normalize SingleLam where + normalize (SingleLam binds body) = SingleLam binds' body' where + (binds',body') = normalizeWithExpo (>0) exponentOf (binds,body) + +instance Normalize MultiLam where + normalize (MultiLam binds body) = MultiLam binds' body' where + cond ds = any (>0) ds + (binds',body') = normalizeWithExpo cond f (binds,body) + f v = {- reverse . -} exponentVectorOf v + +instance (Eq c, Num c) => Normalize (FreeMod c SingleLam) where + normalize = ZMod.mapBase normalize + +instance (Eq c, Num c) => Normalize (FreeMod c MultiLam) where + normalize = ZMod.mapBase normalize + +-------------------------------------------------------------------------------- + +instance SuperNormalize Multi where + superNormalize (Multi ts) = Multi $ reverse $ dropWhile isempty $ reverse $ map normalize $ ts where + isempty (Single xs) = null xs + +instance SuperNormalize MultiLam where + superNormalize mlam = MultiLam binds (superNormalize body) where + (MultiLam binds body) = normalize mlam + +instance (Eq c, Num c) => SuperNormalize (FreeMod c MultiLam) where + superNormalize = ZMod.mapBase superNormalize + +-------------------------------------------------------------------------------- + +instance Cross Bindings where + cross (Bindings ds) (Bindings es) = Bindings (ds++es) + crossInterleave = error "Bindings/crossInterleave: undefined" + +instance Cross Multi where + cross (Multi xs) (Multi ys) = Multi (xs++ys) + crossInterleave (Multi xs) (Multi ys) = Multi (interleave xs ys) + +instance Cross MultiLam where + cross (MultiLam binds1 bodies1) (MultiLam binds2 bodies2) = normalize $ MultiLam binds3 bodies3 where + n1 = numberOfBoundVariables binds1 + binds3 = binds1 `cross` binds2 + bodies3 = bodies1 `cross` (shift n1 bodies2) + + crossInterleave (MultiLam binds1 bodies1) (MultiLam binds2 bodies2) = normalize $ MultiLam binds3 bodies3 where + n1 = numberOfBoundVariables binds1 + binds3 = binds1 `cross` binds2 + bodies3 = bodies1 `crossInterleave` (shift n1 bodies2) + +instance (Eq c, Num c) => Cross (FreeMod c MultiLam) where + cross x y = normalize $ ZMod.mulWith cross x y + crossInterleave x y = normalize $ ZMod.mulWith crossInterleave x y + +-------------------------------------------------------------------------------- + +instance SingleToMulti Single Multi where + singleToMulti = Multi . (:[]) + +instance SingleToMulti SingleLam MultiLam where + singleToMulti (SingleLam binds single) = MultiLam binds (Multi [single]) + +instance (Eq c, Num c) => SingleToMulti (FreeMod c SingleLam) (FreeMod c MultiLam) where + singleToMulti = ZMod.mapBase singleToMulti + +-------------------------------------------------------------------------------- + +instance Omega (Var,Int) where + omega 0 _ = omegaZeroError + omega k (v,d) = (v,d*k) + +instance Omega Single where + omega 0 _ = Single [] -- omegaZeroError + omega k (Single ves) = Single $ map (omega k) ves + +instance Omega Multi where + omega 0 _ = omegaZeroError + omega k (Multi ts) = Multi $ map (omega k) ts + +instance Omega SingleLam where + omega 0 _ = omegaZeroError + omega k (SingleLam binds body) = SingleLam binds (omega k body) + +instance Omega MultiLam where + omega 0 _ = omegaZeroError + omega k (MultiLam binds body) = MultiLam binds (omega k body) + +instance (Eq c, Num c) => Omega (FreeMod c SingleLam) where + omega 0 = omegaZeroError + omega k = ZMod.mapBase (omega k) + +instance (Eq c, Num c) => Omega (FreeMod c MultiLam) where + omega 0 = omegaZeroError + omega k = normalize . ZMod.mapBase (omega k) + +-------------------------------------------------------------------------------- + +instance Omega123 Multi where + omega123 (Multi ts) = Multi $ zipWith omega [1..] ts + +instance Omega123 MultiLam where + omega123 (MultiLam binds body) = MultiLam binds (omega123 body) + +instance (Eq c, Num c) => Omega123 (FreeMod c MultiLam) where + omega123 = normalize . ZMod.mapBase omega123 + +-------------------------------------------------------------------------------- + +instance Psi Multi Single where + psi (Multi ts) = normalize $ Single $ concat $ map unSingle ts + +instance Psi MultiLam SingleLam where + psi (MultiLam binds body) = SingleLam binds (psi body) + +instance (Eq c, Num c) => Psi (FreeMod c MultiLam) (FreeMod c SingleLam) where + psi = normalize . ZMod.mapBase psi + +instance (Eq c, Num c) => Psi [FreeMod c SingleLam] (FreeMod c SingleLam) where + psi = normalize . psi . crossMany . map singleToMulti + +-------------------------------------------------------------------------------- + +instance PsiEvenOdd Multi where + psiEvenOdd (Multi ts) = normalize $ Multi $ zipWith f (evens ts) (odds ts) where + f (Single xs) (Single ys) = Single (xs++ys) + +instance PsiEvenOdd MultiLam where + psiEvenOdd (MultiLam binds body) = MultiLam binds (psiEvenOdd body) + +instance PsiEvenOdd (ZMod MultiLam) where + psiEvenOdd = normalize . ZMod.mapBase psiEvenOdd + +-------------------------------------------------------------------------------- + +instance Pontrjagin SingleLam where + pontrjaginOne = empty + pontrjaginMul a b = psi $ cross (singleToMulti a) (singleToMulti b) + +instance Pontrjagin MultiLam where + pontrjaginOne = empty + pontrjaginMul a b = psiEvenOdd $ crossInterleave a' b' where + (a',b') = extendToCommonSize (a,b) + +-------------------------------------------------------------------------------- + +instance ExtendToCommonSize Multi where + extendToCommonSize (Multi xs, Multi ys) = (Multi xs', Multi ys') where + (xs',ys') = extendToCommonSize (xs,ys) + +instance ExtendToCommonSize MultiLam where + extendToCommonSize (MultiLam as xs, MultiLam bs ys) = (MultiLam as xs', MultiLam bs ys') where + (xs',ys') = extendToCommonSize (xs,ys) + +-------------------------------------------------------------------------------- + +instance Permute Multi where + permute p (Multi ts) = Multi (permuteList p ts) + +instance Permute MultiLam where + permute p (MultiLam binds multi) = MultiLam binds (permute p multi) + +instance Permute (ZMod MultiLam) where + permute p = ZMod.mapBase (permute p) + +-------------------------------------------------------------------------------- + +instance Theta Multi where + theta (Multi (u:us)) = Multi (a:bs) where + a = psi $ Multi (u : odds us) + bs = zipWith f (evens us) (odds us) + f (Single u) (Single v) = normalize $ Single (u ++ v) + +instance Theta MultiLam where + theta (MultiLam binds body) = MultiLam binds (theta body) + +instance Theta (ZMod MultiLam) where + theta = normalize . ZMod.mapBase theta + +-------------------------------------------------------------------------------- diff --git a/src/Math/RootLoci/Motivic/Classes.hs b/src/Math/RootLoci/Motivic/Classes.hs new file mode 100644 --- /dev/null +++ b/src/Math/RootLoci/Motivic/Classes.hs @@ -0,0 +1,182 @@ + +{-# LANGUAGE FlexibleInstances, TypeSynonymInstances, + MultiParamTypeClasses, FunctionalDependencies, + TypeFamilies, DataKinds, GeneralizedNewtypeDeriving + #-} +module Math.RootLoci.Motivic.Classes where + +-------------------------------------------------------------------------------- + +import Data.Char +import Data.List +import Data.Ord +import Data.Maybe + +import GHC.TypeLits + +import Data.Map.Strict (Map) +import qualified Data.Map.Strict as Map + +import qualified Math.Algebra.Polynomial.FreeModule as ZMod +import Math.Algebra.Polynomial.FreeModule (ZMod,QMod,FreeMod) +import Math.Algebra.Polynomial.Pretty + +import Math.Combinat.Classes hiding (empty) +import Math.Combinat.Tuples +import Math.Combinat.Partitions +import Math.Combinat.Permutations hiding (permute) + +import Math.Algebra.Polynomial.Class +import Math.Algebra.Polynomial.Monomial.Indexed + +import Math.RootLoci.Misc.Common + +-------------------------------------------------------------------------------- +-- * Dimensions + +-- | A dimension (@d@ in @Sym^d(X)@) +newtype Dim + = Dim Int + deriving (Eq,Ord,Show,Num) + +unDim :: Dim -> Int +unDim (Dim d) = d + +dimVector :: Partition -> [Dim] +dimVector = map Dim . exponentVector + +dimTuples :: [Dim] -> [[Dim]] +dimTuples + = (map . map) Dim + . tuples' + . map unDim + +-------------------------------------------------------------------------------- +-- * Classes + +-- | Degree of something +class Degree a where + type MultiDegree a :: * + totalDegree :: a -> Int + multiDegree :: a -> MultiDegree a + +instance (KnownNat n) => Degree (XS v n) where + type MultiDegree (XS v n) = [Int] + totalDegree = totalDegXS + multiDegree = xsToExponents + +-------------------------------------------------------------------------------- + +class Empty a where + empty :: a + +instance Empty [a] where + empty = [] + +instance Empty (Maybe a) where + empty = Nothing + +instance Empty Int where + empty = 0 + +instance KnownNat n => Empty (XS v n) where + empty = emptyXS + +-------------------------------------------------------------------------------- + +-- | Normalize terms and lambdas +class Normalize a where + normalize :: a -> a + +-- | This is a hack because there is some issue when this is included in normalize that i don't want to debug right now +class SuperNormalize a where + superNormalize :: a -> a + +-------------------------------------------------------------------------------- + +-- | Exterior (or cross) product +class Cross a where + cross :: a -> a -> a + crossMany :: [a] -> a + crossMany = foldl1' cross + crossInterleave :: a -> a -> a -- ^ interleaved cross product of vectors + +instance Cross [a] where + cross = (++) + crossMany = concat + crossInterleave xs ys = interleave xs ys + +------------------------------------------------------------------------------- + +-- | Conversion from scalar to vector +class SingleToMulti s t | s->t, t->s where + singleToMulti :: s -> t + +-------------------------------------------------------------------------------- + +omegaZeroError :: a +omegaZeroError = error "Omega^0 should not appear in the algorithm" + +-- | replicating points (power map) +class Omega a where + omega :: Int -> a -> a + +-------------------------------------------------------------------------------- + +-- | @Omega^{1,2,3,...}@ +class Omega123 a where + omega123 :: a -> a + +-------------------------------------------------------------------------------- + +-- | The merging (or multiplication) map +class Psi t s | t->s where + psi :: t -> s + +-------------------------------------------------------------------------------- + +-- | The interleaved pairwise merging map +class PsiEvenOdd t where + psiEvenOdd :: t -> t + +-------------------------------------------------------------------------------- + +-- | Pontrjagin ring +class Pontrjagin a where + pontrjaginOne :: a + pontrjaginMul :: a -> a -> a + +-------------------------------------------------------------------------------- + +class ExtendToCommonSize a where + extendToCommonSize :: (a,a) -> (a,a) + +instance Empty a => ExtendToCommonSize [a] where + extendToCommonSize (xs,ys) = (xs',ys') where + a = length xs + b = length ys + n = max a b + xs' = xs ++ replicate (n-a) empty + ys' = ys ++ replicate (n-b) empty + +-------------------------------------------------------------------------------- + +-- | Applying permutations +class Permute a where + permute :: Permutation -> a -> a + +instance Permute [a] where + permute = permuteList + +-------------------------------------------------------------------------------- + +-- | The custom pusforward @Theta@ appearing in the algorithm +-- +-- we subdivide the input as @[z;x1,y1,x2,y2,x3,y3...]@ +-- and then duplicate each of @y1,y2,y3...@, then combine the left copies of @y_i@ with +-- @z@, and the right copies of @y_i@ with the corresponding @x_i@-s, resulting in +-- @[z*y1*y2*...;x1*y1,x2*y2,...]@ +class Theta a where + theta :: a -> a --mypf :: a -> a + +-------------------------------------------------------------------------------- diff --git a/src/Math/RootLoci/Motivic/Homology.hs b/src/Math/RootLoci/Motivic/Homology.hs new file mode 100644 --- /dev/null +++ b/src/Math/RootLoci/Motivic/Homology.hs @@ -0,0 +1,173 @@ + +-- | Motivic classes in homology + +{-# LANGUAGE + DataKinds, KindSignatures, TypeOperators, ScopedTypeVariables, MultiParamTypeClasses, + TypeFamilies, FlexibleContexts, TypeSynonymInstances, FlexibleInstances + #-} +module Math.RootLoci.Motivic.Homology where + +-------------------------------------------------------------------------------- + +import Data.Array + +import Data.Proxy +import GHC.TypeLits + +import Unsafe.Coerce as Unsafe + +import Math.Combinat.Classes +import Math.Combinat.Numbers +import Math.Combinat.Partitions + +import qualified Math.Algebra.Polynomial.FreeModule as ZMod + +import qualified Math.Algebra.Polynomial.Monomial.Infinite as XInf +import qualified Math.Algebra.Polynomial.Monomial.Indexed as XS + +import Math.Algebra.Polynomial.Multivariate.Infinite as XInf +import Math.Algebra.Polynomial.Multivariate.Indexed as XS + +import Math.Algebra.Polynomial.Class +import Math.Algebra.Polynomial.Univariate +import Math.Algebra.Polynomial.Pretty + +import Math.RootLoci.Geometry.Cohomology ( G(..) ) + +import Math.RootLoci.Misc.Common +import Math.RootLoci.Motivic.Abstract as Abstract +import Math.RootLoci.Motivic.Classes + +import Math.RootLoci.CSM.Aluffi + +-------------------------------------------------------------------------------- + +interpretSingleLam :: (Dim -> KRing Integer) -> SingleLam -> KRing Integer +interpretSingleLam symFun (SingleLam (Bindings bindings) (Single body)) = result where + syms = map symFun bindings + result = psiAny $ crossKs $ map f body + f (DeBruijn i, e) = omegaH e (syms!!i) + +csmPn :: Dim -> KRing Integer +csmPn (Dim d) = Uni $ ZMod.fromList [ (U k , binomial (d+1) (k+1)) | k<-[0..d] ] + +-- | CSM class in homology +csm_xlam_P1 :: Partition -> KRing Integer +csm_xlam_P1 part = Uni $ ZMod.flatMap f (Abstract.xlam part) where + f x = unUni (interpretSingleLam csmPn x) + +-- | CSM class in cohomology (via Poincare duality) +csm_xlam_P1_cohom :: Partition -> ZMod.ZMod G +csm_xlam_P1_cohom part = ZMod.mapBase f $ unUni $ csm_xlam_P1 part where + n = weight part + f (U k) = G (n-k) + +-- | Compares Aluffi's CSM formula to the motivic algorithm (up to partitions of size @n@) +test_motivic_csm_vs_aluffi :: Int -> Bool +test_motivic_csm_vs_aluffi n = and + [ csm_xlam_P1_cohom part == aluffiOpenCSM part + | k<-[1..n] , part <- partitions k + ] + +-------------------------------------------------------------------------------- + +instance SingleToMulti (KRing c) (GRing c) where + singleToMulti = embedInf + +instance Ring c => Psi (GRing c) (KRing c) where + psi = psiAny + +instance Ring c => Omega (KRing c) where + omega = omegaH + +-------------------------------------------------------------------------------- + +type KRing c = Univariate c "u" -- ^ @lim_n H_*(Sym^n(P1))@ +type GRing c = XInf.Poly c "u" -- ^ @lim_{n1,n2,...} H_*(Sym^n1(P1) x Sym^n2(P1) x ... )@ + +-- fuck Haskell's type level naturals, they are completely unusable +-- type NRing c k = XS.Poly c "u" k -- ^ @lim_{n1,...,nk} H_*(Sym^n1(P1) x ... x Sym^nk(P1) )@ + +embedInf :: KRing c -> GRing c +embedInf = XInf.Poly . ZMod.unsafeMapBase f . unUni where + f (U k) = if k > 0 then XInf [k] else XInf [] + +project1 :: GRing c -> KRing c +project1 = Uni . ZMod.unsafeMapBase f . XInf.unPoly where + f (XInf ns) = U $ head ns + +delta2 :: Ring c => KRing c -> GRing c +delta2 = XInf.Poly . ZMod.flatMap f . unUni where + f (U k) = ZMod.sum [ ZMod.generator (XInf [i,k-i]) | i<-[0..k] ] + +deltaN :: Ring c => Int -> KRing c -> GRing c +deltaN n input + | n <= 0 = error "deltaN: n <= 0" + | n == 1 = embedInf input + | n == 2 = delta2 input + | otherwise = unify1st2nd + $ mapCoeffP delta2 + $ separate1st (deltaN (n-1) input) + where + mapCoeffP f = XInf.Poly . ZMod.mapCoeff f . XInf.unPoly + +psi2 :: Ring c => GRing c -> KRing c +psi2 = Uni . ZMod.mapMaybeBaseCoeff f . XInf.unPoly where + f (XInf xs) = let [i,j] = take 2 (xs ++ [0,0]) + in Just ( U (i+j) , fromInteger (binomial (i+j) i) ) + +psiNaive :: (Ring c) => Int -> GRing c -> KRing c +psiNaive n input + | n <= 0 = error "psiN: n <= 0" + | n == 1 = project1 input + | n == 2 = psi2 input + | otherwise = psi2 $ kkToG2 $ psiNaive (n-1) $ separate1st input + +psiAny :: Ring c => GRing c -> KRing c +psiAny = Uni . ZMod.mapMaybeBaseCoeff f . XInf.unPoly where + f (XInf is) = Just (U (sum is) , fromInteger (multinomial is)) + +omegaNaive :: Ring c => Int -> KRing c -> KRing c +omegaNaive n = psiAny . deltaN n + +omegaH :: Ring c => Int -> KRing c -> KRing c +omegaH d = Uni . ZMod.mapMaybeBaseCoeff f . unUni where + f (U k) = Just (U k, fromIntegral d ^ k) + +separate1st :: forall c n. (Ring c) => GRing c -> GRing (KRing c) +separate1st = XInf.Poly . ZMod.mapMaybeBaseCoeff g . ZMod.mapCoeff f . XInf.unPoly where + f c = scalarP c :: KRing c + g (XInf (k:ns)) = Just (XInf ns, c) where + c = monomP (U k) + +unify1st :: forall c n. (Ring c) => GRing (KRing c) -> GRing c +unify1st = XInf.Poly . ZMod.fromList . concatMap f . ZMod.toList . XInf.unPoly where + f (XInf xs , Uni poly) = [ (XInf (k:xs) , c) | (U k, c) <- ZMod.toList poly ] + +unify1st2nd :: forall c n. (Ring c) => GRing (GRing c) -> GRing c +unify1st2nd = XInf.Poly . ZMod.fromList . concatMap f . ZMod.toList . XInf.unPoly where + f (XInf xs , XInf.Poly poly) = [ (XInf (kl++xs) , c) | (XInf kl0, c) <- ZMod.toList poly , let kl = take 2 (kl0++[0,0]) ] + +crossKs :: Ring c => [KRing c] -> GRing c +crossKs = XInf.Poly . ZMod.productWith empty cross . map (ZMod.mapBase sing) . map unUni where + sing (U k) = XInf [k] + cross (XInf as) (XInf bs) = XInf (as++bs) + empty = XInf [] + +kkToG2 :: Ring c => KRing (KRing c) -> GRing c +kkToG2 = XInf.Poly . ZMod.fromList . concatMap f . ZMod.toList . unUni where + f (U k , Uni poly) = [ (XInf [k,l] , c) | (U l, c) <- ZMod.toList poly ] + +unifyKK :: Ring c => KRing (KRing c) -> KRing c +unifyKK = Uni . ZMod.fromList . concatMap f . ZMod.toList . unUni where + f (U k , Uni poly) = [ (U (k+l) , c) | (U l, c) <- ZMod.toList poly ] + +-------------------------------------------------------------------------------- + +{- +deltaN :: Num c => Int -> KRing c -> GRing c +deltaN 0 = error "deltaN: 0" +deltaN 1 = embed +deltaN 2 = delta2 +deltaN +-} diff --git a/src/Math/RootLoci/Segre/Equivariant.hs b/src/Math/RootLoci/Segre/Equivariant.hs new file mode 100644 --- /dev/null +++ b/src/Math/RootLoci/Segre/Equivariant.hs @@ -0,0 +1,137 @@ + +-- | The equivariant Segre-Schwartz-MacPherson classes +-- +-- We can recover the Segre-SM classes by dividing the CSM class +-- by the total Chern class of the tangent bundle of the (smooth) +-- ambient variety. +-- +-- The Segre-SM class is useful because it behaves well wrt. pullback. +-- + +{-# LANGUAGE ScopedTypeVariables, BangPatterns #-} +module Math.RootLoci.Segre.Equivariant where + +-------------------------------------------------------------------------------- + +import Math.Combinat.Classes +import Math.Combinat.Numbers +import Math.Combinat.Sign +import Math.Combinat.Compositions +import Math.Combinat.Partitions.Integer +import Math.Combinat.Numbers.Series + +import Data.Array (Array) +import Data.Array.IArray + +import Math.RootLoci.Algebra +import Math.RootLoci.Geometry +import Math.RootLoci.Misc + +import qualified Math.Algebra.Polynomial.FreeModule as ZMod + +import Math.RootLoci.CSM.Equivariant.Umbral -- this is the fastest one + +-------------------------------------------------------------------------------- +-- * The total Chern class + +-- | Total Chern class of the representation @Sym^m C^2@ +-- +-- > c(Sym^m C^2) = \prod_{i=0}^m (1 + i*a + (m-i)*b) +-- +affTotalChernClass :: ChernBase base => Int -> ZMod base +affTotalChernClass m = select1 (total , abToChern total) where + total = product [ 1 + w | w <- affineWeights m ] + +-- | Parts of the total Chern class, separated by degree +affTotalChernClassByDegree :: ChernBase base => Int -> [ZMod base] +affTotalChernClassByDegree = elems . separateGradedParts . affTotalChernClass + +-------------------------------------------------------------------------------- +-- * Inverse of the total Chern class + +-- | Infinite power series expansion (by degree) of the multiplicative +-- inverse of the total Chern class of the representation @Sym^m C^2@ +-- +-- This is just the sum of all complete symmetric polynomials of the sums. +-- +recipTotalChernClass :: forall base. ChernBase base => Int -> [ZMod base] +recipTotalChernClass m = pseries' coeffs where + + coeffs = zip (map ZMod.neg prodWeights) [1..] + prodWeights = tail (affTotalChernClassByDegree m) + +-- | Another implementation of 'recipTotalChernClass' +recipTotalChernClass2 :: forall base. ChernBase base => Int -> [ZMod base] +recipTotalChernClass2 m = integralReciprocalSeries (affTotalChernClassByDegree m) where + +-- | A third, very slow implementation of 'recipTotalChernClass' +recipTotalChernClassSlow :: forall base. ChernBase base => Int -> [ZMod base] +recipTotalChernClassSlow m = select2 (list , map abToChern list) where + + weights = affineWeights m + list = [ grade d | d <- [0..] ] + + grade :: Int -> ZMod AB + grade d = negateIfOdd d + $ ZMod.sum (map mkProduct $ compositions (m+1) d) + + mkProduct es = ZMod.product [ (weightPowers!i) !! e | (i,e) <- zip [0..m] es ] + + -- much faster to cache to powers of the weights! + weightPowers :: Array Int [ZMod AB] + weightPowers = listArray (0,m) [ wtPowList (weights !! i) | i <- [0..m] ] + + wtPowList :: ZMod AB -> [ZMod AB] + wtPowList w = go 1 where { go !x = x : go (x*w) } + + +-------------------------------------------------------------------------------- + +-- | Divides a polynomial with the total chern class. As the result is an +-- infinite power series, we return it's homogeneous parts as an infinite list. +-- +-- Equivalent (but should be faster than) to: +-- +-- > separeteGradedParts what `mulSeries` (recipTotalChernClass m) +-- +divideByTotalChernClass :: ChernBase base => Int -> ZMod base -> [ZMod base] +divideByTotalChernClass m what = convolveWithPSeries' coeffs numerList where + + numerArr = separateGradedParts what + numerList = elems numerArr + + coeffs = zip (map ZMod.neg prodWeights) [1..] + prodWeights = tail (affTotalChernClassByDegree m) + +-- | Another, very slow implementation of 'divideByTotalChernClass' +divideByTotalChernClassSlow :: ChernBase base => Int -> ZMod base -> [ZMod base] +divideByTotalChernClassSlow m what = final where + (0,n) = bounds numerArr + numerArr = separateGradedParts what + denomList = recipTotalChernClassSlow m + final = [ part d | d <- [0..] ] + part deg = ZMod.sum + [ (numerArr ! i) * (denomList !! j) + | j <- [ max 0 (deg-n) .. deg ] + , let i = deg - j + ] + +-------------------------------------------------------------------------------- +-- * Affine Segre-SM classes + +-- | Affine equivariant Segre-SM class of the open strata +affineOpenSegreSM :: ChernBase base => Partition -> [ZMod base] +affineOpenSegreSM part = divideByTotalChernClass m (umbralAffOpenCSM part) where + m = weight part + +-- | Affine equivariant Segre-SM class of the zero orbit +affineZeroSegreSM :: ChernBase base => Int -> [ZMod base] +affineZeroSegreSM m = divideByTotalChernClass m (affineZeroCSM m) + +-- | Affine equivariant Segre-SM class of the closure of the strata (including the zero orbit!) +affineClosedSegreSM :: ChernBase base => Partition -> [ZMod base] +affineClosedSegreSM part = divideByTotalChernClass m (umbralAffClosedCSM part) where + m = weight part + +-------------------------------------------------------------------------------- + diff --git a/test/Tests/CSM/Equivariant.hs b/test/Tests/CSM/Equivariant.hs --- a/test/Tests/CSM/Equivariant.hs +++ b/test/Tests/CSM/Equivariant.hs @@ -1,8 +1,7 @@ -- | Tests for the equivariant CSM class - -{-# LANGUAGE Rank2Types, GADTs, TypeFamilies #-} +{-# LANGUAGE Rank2Types, GADTs, TypeFamilies, PackageImports #-} module Tests.CSM.Equivariant where -------------------------------------------------------------------------------- @@ -13,7 +12,7 @@ import Math.Combinat.Partitions.Integer import Math.Combinat.Partitions.Set -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import qualified "polynomial-algebra" Math.Algebra.Polynomial.FreeModule as ZMod import Math.RootLoci.Algebra import Math.RootLoci.Geometry diff --git a/test/Tests/CSM/Projective.hs b/test/Tests/CSM/Projective.hs --- a/test/Tests/CSM/Projective.hs +++ b/test/Tests/CSM/Projective.hs @@ -1,7 +1,7 @@ -- | Tests for the non-equivarant CSM classes -{-# LANGUAGE Rank2Types, GADTs, TypeFamilies #-} +{-# LANGUAGE Rank2Types, GADTs, TypeFamilies, PackageImports #-} module Tests.CSM.Projective where -------------------------------------------------------------------------------- @@ -12,7 +12,7 @@ import Math.Combinat.Partitions.Integer import Math.Combinat.Partitions.Set -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import qualified "polynomial-algebra" Math.Algebra.Polynomial.FreeModule as ZMod import Math.RootLoci.Algebra import Math.RootLoci.Geometry diff --git a/test/Tests/Pushforward.hs b/test/Tests/Pushforward.hs --- a/test/Tests/Pushforward.hs +++ b/test/Tests/Pushforward.hs @@ -2,7 +2,7 @@ -- | Tests for the push-forward -{-# LANGUAGE Rank2Types, GADTs, TypeFamilies #-} +{-# LANGUAGE Rank2Types, GADTs, TypeFamilies, PackageImports #-} module Tests.Pushforward where -------------------------------------------------------------------------------- @@ -12,7 +12,7 @@ import Math.Combinat.Classes import Math.Combinat.Partitions -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import qualified "polynomial-algebra" Math.Algebra.Polynomial.FreeModule as ZMod import Math.RootLoci.Algebra import Math.RootLoci.Geometry diff --git a/test/Tests/RootVsClass/Check.hs b/test/Tests/RootVsClass/Check.hs --- a/test/Tests/RootVsClass/Check.hs +++ b/test/Tests/RootVsClass/Check.hs @@ -1,7 +1,7 @@ -- | Checking polymorphic functions -{-# LANGUAGE Rank2Types, GADTs, TypeFamilies, ScopedTypeVariables #-} +{-# LANGUAGE Rank2Types, GADTs, TypeFamilies, ScopedTypeVariables, PackageImports #-} module Tests.RootVsClass.Check where -------------------------------------------------------------------------------- @@ -15,7 +15,7 @@ import Math.RootLoci.Misc import Math.RootLoci.Geometry.Cohomology -import qualified Math.RootLoci.Algebra.FreeMod as ZMod +import qualified "polynomial-algebra" Math.Algebra.Polynomial.FreeModule as ZMod import Math.RootLoci.CSM.Equivariant.Umbral ( ST )