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

Methods

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

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

Ord ST

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

Methods

showsPrec :: Int -> ST -> ShowS

show :: ST -> String

showList :: [ST] -> ShowS

Monoid ST

Methods

mempty :: ST

mappend :: ST -> ST -> ST

mconcat :: [ST] -> ST

Pretty ST

Methods

pretty :: ST -> String

prettyMixedST :: forall b c. (Pretty b, Num c, Eq c, IsSigned c, Show 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

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

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

Sum over the strata in the closure

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