| Safe Haskell | None |
|---|---|
| Language | Haskell2010 |
Math.RootLoci.CSM.Equivariant.Umbral
Description
The umbral formula for the open CSM classes.
The formula is the following:
A(mu) = 1 / aut(mu) * prod_i Theta(mu_i) Theta(p) = ( (1 + beta*s) (alpha+t)^p - (1 + alpha*s) (beta+t)^p ) / ( alpha - beta )
and the umbral subtitution resulting in the CSM class (at least for length(mu)>=3) is:
t^j -> P_j(m) s^k -> (n-3)(n-3-1)(...n-3-k+1) * Q(n-3-k)
Note that Theta(p) is actually a (symmetric) polynomial in alpha and beta; furthermore
it's linear in s and degree p in t.
Synopsis
- data ST = ST !Int !Int
- prettyMixedST :: forall b c. (Pretty b, Num c, Eq c, IsSigned c, Pretty c) => FreeMod (FreeMod c b) ST -> String
- theta :: ChernBase base => Int -> FreeMod (ZMod base) ST
- thetaQ :: ChernBase b => Int -> FreeMod (QMod b) ST
- integralUmbralFormula :: ChernBase base => Partition -> FreeMod (ZMod base) ST
- umbralFormula :: ChernBase base => Partition -> FreeMod (QMod base) ST
- affineWeights :: Int -> [ZMod AB]
- topChernClass :: ChernBase base => Int -> ZMod base
- umbralSubstPolyAff :: ChernBase base => Partition -> ST -> ZMod base
- umbralSubstitutionAff :: ChernBase base => Partition -> FreeMod (ZMod base) ST -> ZMod base
- umbralAffOpenCSM :: ChernBase base => Partition -> ZMod base
- affineZeroCSM :: ChernBase base => Int -> ZMod base
- umbralAffClosedCSM :: ChernBase base => Partition -> ZMod base
- umbralSubstPolyProj :: forall base. ChernBase base => Partition -> ST -> ZMod (Gam base)
- umbralSubstitutionProj :: ChernBase base => Partition -> FreeMod (ZMod base) ST -> ZMod (Gam base)
- umbralOpenCSM :: ChernBase base => Partition -> ZMod (Gam base)
- umbralClosedCSM :: ChernBase base => Partition -> ZMod (Gam base)
The umbral variables
A monomial s^k * t^j
prettyMixedST :: forall b c. (Pretty b, Num c, Eq c, IsSigned c, Pretty c) => FreeMod (FreeMod c b) ST -> String Source #
The umbral formula
theta :: ChernBase base => Int -> FreeMod (ZMod base) ST Source #
Theta(p) is defined by the formula
Theta(p) = ( (1 + beta*s) (alpha+t)^p - (1 + alpha*s) (beta+t)^p ) / ( alpha - beta )
This is actually a polynomial in alpha,beta,s,t, also symmetric in alpha and beta
thetaQ :: ChernBase b => Int -> FreeMod (QMod b) ST Source #
Same as theta but with rational coefficients
integralUmbralFormula :: ChernBase base => Partition -> FreeMod (ZMod base) ST Source #
This is just prod_i Theta_{mu_i}
umbralFormula :: ChernBase base => Partition -> FreeMod (QMod base) ST Source #
This is 1/aut(mu) * prod_i Theta_{mu_i}
The affine CSM
topChernClass :: ChernBase base => Int -> ZMod base Source #
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!
umbralSubstPolyAff :: ChernBase base => Partition -> ST -> ZMod base Source #
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
where n = length(mu) and m = weight(mu).
umbralSubstitutionAff :: ChernBase base => Partition -> FreeMod (ZMod base) ST -> ZMod base Source #
The (affine) umbral substitution
umbralAffOpenCSM :: ChernBase base => Partition -> ZMod base Source #
CSM of the open stratums from the umbral the formula
affineZeroCSM :: ChernBase base => Int -> ZMod base Source #
CSM class of the zero orbit (which is just the top Chern class)
umbralAffClosedCSM :: ChernBase base => Partition -> ZMod base Source #
Sum over the strata in the closure (including the zero orbit!)
The projective CSM
umbralSubstPolyProj :: forall base. ChernBase base => Partition -> ST -> ZMod (Gam base) Source #
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
where n = length(mu) and m = weight(mu).
umbralSubstitutionProj :: ChernBase base => Partition -> FreeMod (ZMod base) ST -> ZMod (Gam base) Source #
The (projective) umbral substitution