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, a string, 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, a GKM variety, 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*}} => {{0, 2}, {-1, 1}, {-2, 0}} }
                                    {set {0, 1}} => {{0, -2}, {-2, 0}, {-1, -1}}
                                    {set {1*, 0*}} => {{1, 1}, {2, 0}, {0, 2}}
                                    {set {1, 0*}} => {{2, 0}, {1, -1}, {0, -2}}
                momentGraph => a "moment graph" on 4 vertices with 6 edges 
                points => {{set {0, 1}}, {set {0, 1*}}, {set {1, 0*}}, {set {1*, 0*}}}

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.

Ways to use generalizedFlagVariety :

generalizedFlagVariety(String,ZZ,List)
generalizedFlagVariety(String,ZZ,List,Ring)

For the programmer