KClass -- the class of all equivariant K-classes

Description

For $X$ a GKM variety with an action of a torus $T$ whose character ring is $R$, a $T$-equivariant $K$-class $C \in K_T^0(X)$ of is encoded by its image in $K_T^0(X^T) = \prod_{x\in X^T} R$, under the injective restriction map $K_T^0(X) \to K_T^0(X^T)$. See [Corollary 5.12; VV03] or [Corollary A.5; RK03] for details.

A KClass C is a HashTable consisting of two keys:

variety, whose value is a GKMVariety of which C is a K-class of
KPolynomials, whose value is a HashTable; its keys are X.points and the values are Laurent polynomials in the character ring representing the values of the K-class under the restriction map.

Functions and methods returning an equivariant K-class :

ampleKClass(GKMVariety) -- see ampleKClass -- the class of an ample line bundle
makeKClass(GKMVariety,List) -- see makeKClass -- constructs an equivariant K-class
makeKClass(GKMVariety,FlagMatroid) -- the equivariant K-class of a flag matroid
makeKClass(GKMVariety,ToricDivisor) -- create the KClass from a ToricDivisor
orbitClosure(GKMVariety,Matrix) -- see orbitClosure -- computes the equivariant K-class of a torus orbit closure of a point in a generalized flag variety
trivialKClass(GKMVariety) -- see trivialKClass -- the equivariant K-class of the structure sheaf

Methods that use an equivariant K-class :

ampleKClass(GKMVariety,KClass) -- see ampleKClass -- the class of an ample line bundle
euler(KClass) -- computes the equivariant Euler characteristic of an equivariant K-class
isWellDefined(KClass) -- whether the input is a well-defined equivariant K-class
KClass * KClass -- computes the product of two equivariant K-classes
KClass + KClass -- computes the sum of two equivariant K-classes
KClass ^ ZZ -- computes powers of an equivariant K-classes
makeGKMVariety(KClass) -- see makeGKMVariety -- constructs a GKM variety

For the programmer

The object KClass is a type, with ancestor classes HashTable < Thing.

KClass -- the class of all equivariant K-classes

Description

See also

Functions and methods returning an equivariant K-class :

Methods that use an equivariant K-class :

For the programmer