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

Synopsis

• Usage:
P = cellOrder(G)
P = cellOrder(X)
• Inputs:
• G, , on which a cell order has been defined already
• X, , 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