momentGraph -- creates a moment graph

Synopsis

Usage:

G = momentGraph(L,E,H)
Inputs:
- L, a list, of vertices
- E, a hash table, whose keys are lists of two vertices representing edges and values are characters of corresponding 1-dimensional orbits
- H, a ring, a polynomial ring representing the equivariant cohomology ring of a point
Outputs:
- G, a moment graph,

Description

This method creates a MomentGraph from the data of vertices, edges and their associated characters, and a ring representing the equivariant cohomology ring of a point (with trivial torus-action). The following example is the moment graph of the projective 2-space $\mathbb P^2$.

i1 : V = {set{0}, set{1}, set{2}};

i2 : E = hashTable {({set{0},set{1}},{-1,1,0}), ({set{0},set{2}},{-1,0,1}), ({set{1},set{2}},{0,-1,1})}

o2 = HashTable{{set {0}, set {1}} => {-1, 1, 0}}
               {set {0}, set {2}} => {-1, 0, 1}
               {set {1}, set {2}} => {0, -1, 1}

o2 : HashTable

i3 : t = symbol t; H = QQ[t_0..t_2]

o4 = H

o4 : PolynomialRing

i5 : G = momentGraph(V,E,H)

o5 = a "moment graph" on 3 vertices with 3 edges 

o5 : MomentGraph

i6 : peek G

o6 = MomentGraph{cache => CacheTable{}                               }
                 edges => HashTable{{set {0}, set {1}} => {-1, 1, 0}}
                                    {set {0}, set {2}} => {-1, 0, 1}
                                    {set {1}, set {2}} => {0, -1, 1}
                 HTpt => H
                 vertices => {set {0}, set {1}, set {2}}

i7 : underlyingGraph G

o7 = Graph{set {0} => {set {1}, set {2}}}
           set {1} => {set {0}, set {2}}
           set {2} => {set {0}, set {1}}

o7 : Graph

Ways to use momentGraph :

momentGraph(List,HashTable,Ring)
momentGraph(GKMVariety) -- view the moment graph of a GKM variety
momentGraph(GKMVariety,MomentGraph) -- define a moment graph for a GKM variety

For the programmer

The object momentGraph is a method function.

momentGraph -- creates a moment graph

Synopsis

Description

See also

Ways to use momentGraph :

For the programmer