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# MomentGraph -- the class of all moment graphs

## Description

The moment graph of a GKM variety $X$ with an action of a torus $T$ has vertices corresponding to the $T$-fixed points $X^T$ and edges corresponding to the one-dimensional $T$-orbits. If $\{v_1,v_2\}$ is an edge and the corresponding one-dimensional $T$-orbit closure is $\mathbb P^1$ where $v_1 = 0$ and $v_2 = \infty$, then denote $m(v_1,v_2)$ to be the negative of the character of the action of $T$ on $\mathbb A^1 \subset \mathbb P^1$ (where $v_1 \in \mathbb A^1$).

A MomentGraph is a HashTable with three keys:

- vertices, whose values represent the vertices of the moment graph
- edges, whose value is a HashTable; its keys are pairs {a,b} of elements in vertices representing the edges of the moment graph, and the values are the characters $m(a,b)$
- HTpt, whose value is a ring representing the equivariant cohomology ring of a point

## Caveat

Functionalities concerning intersection cohomology of sheaves on moment graphs, which had been implemented before (see MG: moment graph computations), have not been imported into this package yet.

## See also

- momentGraph -- creates a moment graph
- makeGKMVariety -- constructs a GKM variety
- GKMVariety -- the class of all GKM varieties

## Functions and methods returning a moment graph:

- `momentGraph(List,HashTable,Ring)` -- see momentGraph -- creates a moment graph
- MomentGraph ** MomentGraph -- the product of two moment graphs
- momentGraph(GKMVariety) -- view the moment graph of a GKM variety

## Methods that use a moment graph:

- `cellOrder(MomentGraph)` -- see cellOrder -- the poset of a stratification of a GKM variety
- cellOrder(MomentGraph,Poset) -- define a cell order on a moment graph
- `makeGKMVariety(MomentGraph)` -- see makeGKMVariety -- constructs a GKM variety
- `makeGKMVariety(MomentGraph,Ring)` -- see makeGKMVariety -- constructs a GKM variety
- momentGraph(GKMVariety,MomentGraph) -- define a moment graph for a GKM variety
- underlyingGraph(MomentGraph) -- the underlying (undirected) graph of a moment graph

## For the programmer

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GKMVarieties/Documentation_GKMVarieties.m2:1198:0.