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# gtPolytope -- the polytope defined by the inequalities and equations appearing in the definition of Gelfand-Tsetlin patterns

- Usage:
    `gtPolytope("A",lambda)`
- Inputs:
  - lambda, a list,
- Outputs:
  - P, a convex polyhedron,

## Description

Currently only supported for $\mathfrak{g} = sl_n$.

Let $\lambda$ be a partition with $n$ parts, written in nonincreasing order. A Gelfand-Tsetlin pattern of shape $\lambda$ is a triangular array of the following form:

$\begin{array}{ccccccccc} x_{n,1} & & x_{n,2} & & x_{n,3} & & \cdots && x_{n,n} \\ &x_{n-1,1} & & x_{n-1,2} & & \cdots & x_{n-1,n-1} & \\ && \ddots \\ && & x_{2,1} && x_{2,2} \\ &&&& x_{1,1}\end{array}$

Each entry $x_{i,j}$ is a nonnegative integer, the top row $x_{n,i}$ corresponds to $\lambda$, and the entries satisfy the inequalities $x_{k,i} \geq x_{k-1,i} \geq x_{k,i+1}$.

This function outputs the polytope defined by these inequalities and equations.

    i1 : P=gtPolytope("A",{2,0,0})

    o1 = P

    o1 : Polyhedron

    i2 : dim P

    o2 = 2

    i3 : halfspaces(P)

    o3 = (| -1 0 0 1  0 0  |, 0)
          | 0  0 0 0  0 -1 |
          | 0  0 0 -1 0 1  |

    o3 : Sequence

    i4 : hyperplanes(P)

    o4 = (| -1 0 0 0 0 0 |, | -2 |)
          | 0  1 0 0 0 0 |  | 0  |
          | 0  0 1 0 0 0 |  | 0  |
          | 0  0 0 0 1 0 |  | 0  |

    o4 : Sequence

    i5 : vertices(P)

    o5 = | 2 2 2 |
         | 0 0 0 |
         | 0 0 0 |
         | 0 2 2 |
         | 0 0 0 |
         | 0 0 2 |

                  6       3
    o5 : Matrix QQ  <-- QQ

## Ways to use gtPolytope:

- `gtPolytope(String,List)`

## For the programmer

The object gtPolytope is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/LieAlgebraRepresentations/documentation.m2:2279:0.