Direct.hs

-- | We compute the open CSM classes directly, generalizing Aluffi's argument
-- to the equivariant case:
--
-- First we compute the CSM of set of the distinct /ordered/ points, then
-- push that forward first with @delta_*@ then with @pi_*@ to get the
-- CSM of the distinct unordered points with given multiplicities.
--
-- After that, we can get the closed CSM classes by summing over the
-- strata in the closure.
--
-- This is faster, especially since we have a (recursive) formula for the 
-- CSM of the distinct ordered points.

{-# LANGUAGE BangPatterns, TypeSynonymInstances, FlexibleInstances #-}
module Math.RootLoci.CSM.Equivariant.Direct 
  ( directOpenCSM
  , directClosedCSM
  )
  where

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import Math.Combinat.Partitions.Integer

import qualified Data.Set as Set

import Math.RootLoci.Algebra
import Math.RootLoci.Geometry
import Math.RootLoci.Misc

import qualified Math.RootLoci.Algebra.FreeMod as ZMod

import Math.RootLoci.CSM.Equivariant.PushForward
import qualified Math.RootLoci.CSM.Equivariant.Ordered as Ordered

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-- | CSM class of the open strata.
--  
-- We just push-forward first with Delta then down with Pi the conjectured 
-- (recursive) formula for the CSM of the set of distinct ordered points
-- 
directOpenCSM :: ChernBase base => Partition -> ZMod (Gam base)
directOpenCSM = polyCache2 directCalcOpenCSM where

  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
    middle   = delta_star_ part distinct
    distinct = Ordered.formulaDistinctCSM (length xs)

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-- | To compute the CSM of the closed loci, we just some over the open strata
-- in the closure.
directClosedCSM :: ChernBase base => Partition -> ZMod (Gam base)
directClosedCSM = polyCache2 calc where
  
  calc :: ChernBase base => Partition -> ZMod (Gam base)
  calc part = ZMod.sum [ directOpenCSM q | q <- Set.toList (closureSet part) ] 

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