cellOrder -- the poset of a stratification of a GKM variety

Synopsis

Usage:

P = cellOrder(G)

P = cellOrder(X)
Inputs:
- G, a moment graph, on which a cell order has been defined already
- X, a GKM variety, whose moment graph has a cell order defined already
Outputs:
- P, an instance of the type Poset,

Description

If a moment graph $G$ arises from a (possibly singular) GKM variety $X$ with an equivariant stratification, with each strata having a unique torus-fixed point, the vertices of $G$ (which correspond to the torus-fixed point of $X$) form a poset where $v_1 \leq v_2$ if the closure of the stratum corresponding to $v_1$ contains that of $v_2$. The following example features the Schubert variety of projective lines in $\mathbb P^3$ meeting a distinguished line. The poset of its stratification by smaller Schubert cells is a subposet of the Bruhat poset.

i1 : Gr24 = generalizedFlagVariety("A",3,{2})

o1 = a "GKM variety" with an action of a 4-dimensional torus

o1 : GKMVariety

i2 : X = generalizedSchubertVariety(Gr24, {set{0,2}})

o2 = a "GKM variety" with an action of a 4-dimensional torus

o2 : GKMVariety

i3 : cellOrder X

o3 = Relation Matrix: | 1 1 1 1 1 |
                      | 0 1 0 1 1 |
                      | 0 0 1 1 1 |
                      | 0 0 0 1 1 |
                      | 0 0 0 0 1 |

o3 : Poset

Ways to use cellOrder :

cellOrder(GKMVariety)
cellOrder(MomentGraph)
cellOrder(MomentGraph,Poset) -- define a cell order on a moment graph

For the programmer

The object cellOrder is a method function.

cellOrder -- the poset of a stratification of a GKM variety

Synopsis

Description

See also

Ways to use cellOrder :

For the programmer