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GKMVarieties :: GKMVariety

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

Functions and methods returning a GKM variety :

Methods that use a GKM variety :

For the programmer

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