Math.RootLoci.CSM.Equivariant.Umbral

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.

The umbral variables

data ST

A monomial s^k * t^j

Constructors

ST !Int !Int

Instances

Eq ST

Defined in Math.RootLoci.CSM.Equivariant.Umbral

Methods

(==) :: ST -> ST -> Bool

(/=) :: ST -> ST -> Bool

Ord ST

Defined in Math.RootLoci.CSM.Equivariant.Umbral

Methods

compare :: ST -> ST -> Ordering

(<) :: ST -> ST -> Bool

(<=) :: ST -> ST -> Bool

(>) :: ST -> ST -> Bool

(>=) :: ST -> ST -> Bool

max :: ST -> ST -> ST

min :: ST -> ST -> ST

Show ST

Defined in Math.RootLoci.CSM.Equivariant.Umbral

Methods

showsPrec :: Int -> ST -> ShowS

show :: ST -> String

showList :: [ST] -> ShowS

Semigroup ST

Defined in Math.RootLoci.CSM.Equivariant.Umbral

Methods

(<>) :: ST -> ST -> ST

sconcat :: NonEmpty ST -> ST

stimes :: Integral b => b -> ST -> ST

Monoid ST

Defined in Math.RootLoci.CSM.Equivariant.Umbral

Methods

mempty :: ST

mappend :: ST -> ST -> ST

mconcat :: [ST] -> ST

Pretty ST

Defined in Math.RootLoci.CSM.Equivariant.Umbral

Methods

pretty :: ST -> String

prettyInParens :: ST -> String

prettyMixedST :: forall b c. (Pretty b, Num c, Eq c, IsSigned c, Pretty c) => FreeMod (FreeMod c b) ST -> String

The umbral formula

theta :: ChernBase base => Int -> FreeMod (ZMod base) ST

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

Same as theta but with rational coefficients

integralUmbralFormula :: ChernBase base => Partition -> FreeMod (ZMod base) ST

This is just prod_i Theta_{mu_i}

umbralFormula :: ChernBase base => Partition -> FreeMod (QMod base) ST

This is 1/aut(mu) * prod_i Theta_{mu_i}

The affine CSM

affineWeights :: Int -> [ZMod AB]

Weights of the representation Sym^m C^2

topChernClass :: ChernBase base => Int -> ZMod base

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

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

The (affine) umbral substitution

umbralAffOpenCSM :: ChernBase base => Partition -> ZMod base

CSM of the open stratums from the umbral the formula

affineZeroCSM :: ChernBase base => Int -> ZMod base

CSM class of the zero orbit (which is just the top Chern class)

umbralAffClosedCSM :: ChernBase base => Partition -> ZMod base

Sum over the strata in the closure (including the zero orbit!)

The projective CSM

umbralSubstPolyProj :: forall base. ChernBase base => Partition -> ST -> ZMod (Gam base)

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)

The (projective) umbral substitution

umbralOpenCSM :: ChernBase base => Partition -> ZMod (Gam base)

CSM of the open stratums from the umbral the formula (for length(mu) >= 3)

umbralClosedCSM :: ChernBase base => Partition -> ZMod (Gam base)

Sum over the strata in the closure