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}}

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.

GKMVariety -- the class of all GKM varieties

Description

See also

Functions and methods returning a GKM variety :

Methods that use a GKM variety :

For the programmer