isWellDefined(KClass) -- whether the input is a well-defined equivariant K-class

Synopsis

Function: isWellDefined
Usage:

isWellDefined C
Inputs:
- C, an equivariant K-class,
Outputs:
- a Boolean value, whether or not a list of Laurent polynomials satisfies edge compatibility condition

Description

If $\{f_x \mid x\in X^T\}$ is a collection of Laurent polynomials in the character ring $\mathbb Z[T_0, \ldots, T_n]$ of the torus $T$ acting on a GKMVariety $X$, one per each torus-fixed point, representing an element $C$ of $K_T^0(X^T)$, then $C$ is in the image of $K_T^0(X)$ under the injective restriction map $K_T^0(X)\to K_T^0(X^T)$ if and only if it satisfies the following "edge compatibility condition":

For each one-dimensional $T$-orbit-closure in $X$ with boundary points $x$ and $x'$, one has $$f_x \equiv f_{x'} \ \mod \ 1 - T^{\lambda(x,x')}$$ where $\lambda(x,x')$ is the character of the action of $T$ on the one-dimensional orbit. See [Corollary 5.12; VV03] or [Corollary A.5; RK03] for details.

i1 : PP3 = projectiveSpace 3

o1 = a "GKM variety" with an action of a 4-dimensional torus

o1 : GKMVariety

i2 : isWellDefined ampleKClass PP3 --the O(1) class on PP3 is well-defined

o2 = true

i3 : badC = makeKClass(PP3, reverse gens PP3.characterRing) --reverse the order of Laurent polynomials defining the O(1) class

o3 = an "equivariant K-class" on a GKM variety 

o3 : KClass

i4 : isWellDefined badC --no longer well-defined
incompatible edges {{set {2}, set {3}}, {set {0}, set {2}}, {set {1}, set {3}}, {set {0}, set {1}}}
o4 = false

Caveat

A MomentGraph must be defined on the GKMVariety on which the KClass is a $K$-class of.

Ways to use this method: