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 |