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

- 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

## See also

- bases(FlagMatroid) -- compute the bases of a flag matroid
- euler(KClass) -- computes the equivariant Euler characteristic of an equivariant K-class

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GKMVarieties/Documentation_GKMVarieties.m2:1957:0.