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

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