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

## Synopsis

• Function: latticePoints
• Usage:
P = latticePoints(FM)
• Inputs:
• FM, ,
• 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 : List

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