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# GKMVariety -- the class of all GKM varieties

## Description

A GKMVariety $X$ is a MutableHashTable representing a GKM variety $X$ with an action of a torus $T$. Its keys include:

- points, whose value is a list representing the torus-fixed points of $X$
- characterRing, whose value is a ring representing the character ring of $T$
- momentGraph, whose value is the MomentGraph of $X$
- charts, whose value is a HashTable representing the (negatives of) characters of the torus action on each torus-invariant affine chart around a torus-fixed point. The keys of X.charts are X.points and the values are lists consisting of lists of integers.

Every GKMVariety created by methods in this package has at least the two keys points and characterRing. The following example is the projective space $\mathbb P^2$ as a GKMVariety.

    i1 : PP2 = projectiveSpace 2

    o1 = a "GKM variety" with an action of a 3-dimensional torus

    o1 : GKMVariety

    i2 : peek PP2

    o2 = GKMVariety{cache => CacheTable{...1...}                               }
                    characterRing => ZZ[T ..T ]
                                         0   2
                    charts => HashTable{set {0} => {{-1, 1, 0}, {-1, 0, 1}}}
                                        set {1} => {{1, -1, 0}, {0, -1, 1}}
                                        set {2} => {{1, 0, -1}, {0, 1, -1}}
                    momentGraph => a "moment graph" on 3 vertices with 3 edges
                    points => {set {0}, set {1}, set {2}}

## See also

- makeGKMVariety -- constructs a GKM variety
- Example: generalized flag varieties
- Example: smooth toric varieties

## Functions and methods returning a GKM variety:

- `generalizedFlagVariety(String,ZZ,List)` -- see generalizedFlagVariety -- makes a generalized flag variety as a GKM variety
- `generalizedFlagVariety(String,ZZ,List,Ring)` -- see generalizedFlagVariety -- makes a generalized flag variety as a GKM variety
- `generalizedSchubertVariety(GKMVariety,Thing)` -- see generalizedSchubertVariety -- create a generalized Schubert variety
- GKMVariety ** GKMVariety -- product of GKM varieties
- `makeGKMVariety(KClass)` -- see makeGKMVariety -- constructs a GKM variety
- `makeGKMVariety(List,List,Ring)` -- see makeGKMVariety -- constructs a GKM variety
- `makeGKMVariety(List,Ring)` -- see makeGKMVariety -- constructs a GKM variety
- `makeGKMVariety(MomentGraph)` -- see makeGKMVariety -- constructs a GKM variety
- `makeGKMVariety(MomentGraph,Ring)` -- see makeGKMVariety -- constructs a GKM variety
- `makeGKMVariety(NormalToricVariety)` -- see makeGKMVariety -- constructs a GKM variety
- `makeGKMVariety(NormalToricVariety,Ring)` -- see makeGKMVariety -- constructs a GKM variety
- `projectiveSpace(ZZ)` -- see projectiveSpace -- constructs projective space as a GKM variety
- `projectiveSpace(ZZ,Ring)` -- see projectiveSpace -- constructs projective space as a GKM variety

## Methods that use a GKM variety:

- `ampleKClass(GKMVariety)` -- see ampleKClass -- the class of an ample line bundle
- `ampleKClass(GKMVariety,KClass)` -- see ampleKClass -- the class of an ample line bundle
- `bruhatOrder(GKMVariety)` -- see bruhatOrder -- computes the Bruhat order on a generalized flag variety
- `cellOrder(GKMVariety)` -- see cellOrder -- the poset of a stratification of a GKM variety
- `charts(GKMVariety)` -- see charts -- outputs the torus-invariant affine charts of a GKM variety
- `charts(GKMVariety,List)` -- see charts -- outputs the torus-invariant affine charts of a GKM variety
- `diagonalMap(GKMVariety)` -- see diagonalMap -- constructs the diagonal morphism
- `flagMap(GKMVariety,GKMVariety)` -- see flagMap -- creates equivariant maps between generalized flag varieties
- `lieType(GKMVariety)` -- see lieType -- outputs the Lie type of a generalized flag variety
- `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
- map(GKMVariety,GKMVariety,List) -- creates a EquivariantMap
- momentGraph(GKMVariety) -- view the moment graph of a GKM variety
- momentGraph(GKMVariety,MomentGraph) -- define a moment graph for a GKM variety
- normalToricVariety(GKMVariety) -- converts a GKM variety back into a toric variety
- `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

## For the programmer

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GKMVarieties/Documentation_GKMVarieties.m2:373:0.