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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.

See also

Functions and methods returning an equivariant K-class:

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.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/GKMVarieties/Documentation_GKMVarieties.m2:555:0.