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GKMVarieties :: generalizedFlagVariety

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

Synopsis

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.

See also

Ways to use generalizedFlagVariety :

For the programmer

The object generalizedFlagVariety is a method function.