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GKMVarieties :: MomentGraph

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 implemenented before (see MG: moment graph computations), have not been imported into this package yet.

See also

Functions and methods returning a moment graph :

Methods that use a moment graph :

For the programmer

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