# generalizedFlagVariety -- makes a generalized flag variety as a GKM variety

## Synopsis

• Usage:
X = generalizedFlagVariety(LT,d,L)
X = generalizedFlagVariety(LT,d,L,R)
• Inputs:
• LT, , one of "A", "B", "C", or "D"
• d, an integer, the dimension of the root system
• L, a list, of integers strictly between 1 and d (inclusive)
• R, a ring, the character ring of the torus acting on the generalized flag variety
• Outputs:
• X, , representing the corresponding generalized flag variety

## Description

Let $G$ be the Lie group corresponding to $LT_d$, and let $w = a_1w_1 + \cdots + a_dw_d$ be a nonnegative $\mathbb Z$-linear combination of fundamental weights in the root system of type $LT_d$, where $a_i$ is the number of times $i$ appears in the list $L$. (See Example: generalized flag varieties for conventions regarding classical Lie groups and their root systems). This method outputs the GKM variety representing the generalized flag variety $G/P$ embedded in the irreducible representation of $G$ with the highest weight $w$.

The following example features the Lagrangian Grassmannian $LGr(2,4)$ of 2-dimensional subspaces in $\mathbb C^4$ that are isotropic under the standard alternating form. Its MomentGraph is a complete graph on 4 vertices.

 i1 : LGr24 = generalizedFlagVariety("C",2,{2}) o1 = a GKM variety with an action of a 2-dimensional torus o1 : GKMVariety i2 : peek LGr24 o2 = GKMVariety{cache => CacheTable{...2...} } characterRing => ZZ[T ..T ] 0 1 charts => HashTable{{set {0*, 1*}} => {{1, 1}, {2, 0}, {0, 2}} } {set {0*, 1}} => {{2, 0}, {1, -1}, {0, -2}} {set {0, 1*}} => {{0, 2}, {-1, 1}, {-2, 0}} {set {0, 1}} => {{0, -2}, {-2, 0}, {-1, -1}} momentGraph => a moment graph on 4 vertices with 6 edges points => {{set {0, 1}}, {set {0, 1*}}, {set {0*, 1}}, {set {0*, 1*}}} i3 : momentGraph LGr24 o3 = a moment graph on 4 vertices with 6 edges o3 : MomentGraph i4 : euler ampleKClass LGr24 -1 -1 -1 -1 o4 = T T + T T + 1 + T T + T T 0 1 0 1 0 1 0 1 o4 : ZZ[T ..T ] 0 1

## Caveat

Spin groups have not been implemented.