coincident-root-loci-0.2: Equivariant CSM classes of coincident root loci

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

# The umbral variables

data ST Source #

A monomial s^k * t^j

Constructors

 ST !Int !Int

Instances

 Source # Methods(==) :: ST -> ST -> Bool #(/=) :: ST -> ST -> Bool # Source # Methodscompare :: ST -> ST -> Ordering #(<) :: ST -> ST -> Bool #(<=) :: ST -> ST -> Bool #(>) :: ST -> ST -> Bool #(>=) :: ST -> ST -> Bool #max :: ST -> ST -> ST #min :: ST -> ST -> ST # Source # MethodsshowsPrec :: Int -> ST -> ShowS #show :: ST -> String #showList :: [ST] -> ShowS # Source # Methodsmappend :: ST -> ST -> ST #mconcat :: [ST] -> ST # Source # Methods

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

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

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

Sum over the strata in the closure

# 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

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

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

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

Sum over the strata in the closure