latticePoints(FlagMatroid) -- lattice points of a base polytope of a flag matroid

Synopsis

Function: latticePoints
Usage:

P = latticePoints(FM)
Inputs:
- FM, a flag matroid,
Outputs:
- P, a list, of lists of integers representing the lattice points

Description

For a basis $B= \{B_1, \ldots, B_k\}$ of a flag matroid $M$ (see bases(FlagMatroid)), let $e_B$ be the sum over $i = 1, \ldots, k$ of the indicator vectors of $B_i$. The base polytope of a flag matroid $M$ is the convex hull of $e_B$ as $B$ ranges over all bases of $M$. This method computes the lattice points of the base polytope of a flag matroid, exploiting the strong normality property as proven in [CDMS18].

i1 : FM = flagMatroid {uniformMatroid(1,4),uniformMatroid(2,4)}

o1 = a "flag matroid" with rank sequence {1, 2} on 4 elements 

o1 : FlagMatroid

i2 : P = latticePoints FM

o2 = ({2, 1, 0, 0}, {2, 0, 1, 0}, {1, 1, 1, 0}, {2, 0, 0, 1}, {1, 1, 0, 1},
     ------------------------------------------------------------------------
     {1, 0, 1, 1}, {1, 2, 0, 0}, {0, 2, 1, 0}, {0, 2, 0, 1}, {0, 1, 1, 1},
     ------------------------------------------------------------------------
     {1, 0, 2, 0}, {0, 1, 2, 0}, {0, 0, 2, 1}, {1, 0, 0, 2}, {0, 1, 0, 2},
     ------------------------------------------------------------------------
     {0, 0, 1, 2})

o2 : Sequence

In terms of equivariant K-theory, the lattice points of the base polytope of a flag matroid is equal to the integer-point transform of the equivariant Euler characteristic (see euler(KClass)) of the KClass defined by the flag matroid shifted by the $O(1)$ bundle on the (partial) flag variety.

i3 : X = generalizedFlagVariety("A",3,{1,2})

o3 = a "GKM variety" with an action of a 4-dimensional torus

o3 : GKMVariety

i4 : FM = flagMatroid {uniformMatroid(1,4),uniformMatroid(2,4)}

o4 = a "flag matroid" with rank sequence {1, 2} on 4 elements 

o4 : FlagMatroid

i5 : C = makeKClass(X,FM)

o5 = an "equivariant K-class" on a GKM variety 

o5 : KClass

i6 : chiCO1 = euler(C * ampleKClass X)

      2      2      2        2                        2               2  
o6 = T T  + T T  + T T  + T T  + T T T  + T T T  + T T  + T T T  + T T  +
      0 1    0 2    0 3    0 1    0 1 2    0 1 3    0 2    0 2 3    0 3  
     ------------------------------------------------------------------------
      2      2        2               2    2        2
     T T  + T T  + T T  + T T T  + T T  + T T  + T T
      1 2    1 3    1 2    1 2 3    1 3    2 3    2 3

o6 : ZZ[T ..T ]
         0   3

i7 : set P === set exponents chiCO1

o7 = true

Ways to use this method: