[Paper Review] Vertices of Gelfand-Tsetlin Polytopes

Jesús A. De Loera, Tyrrell B. McAllister|ArXiv.org|Sep 19, 2003

Advanced Combinatorial Mathematics10 references4 citations

TL;DR

This paper provides a combinatorial characterization of vertices in Gelfand-Tsetlin polytopes, introducing a tiling-based method to compute vertex coordinates and face dimensions. It disproves a conjecture by Berenstein and Kirillov by constructing an infinite family of non-integral vertices for $ n \geq 5 $, with denominators growing arbitrarily with $ n $, while showing the Ehrhart polynomial remains valid—providing the first such family of non-integral polytopes with polynomial Ehrhart functions.

ABSTRACT

This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns arising in the representation theory $\mathfrak{gl}_n \C$ and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowest-dimensional face containing a given Gelfand-Tsetlin pattern. As an application, we disprove a conjecture of Berenstein and Kirillov about the integrality of all vertices of the Gelfand-Tsetlin polytopes. We can construct for each $n\geq5$ a counterexample, with arbitrarily increasing denominators as $n$ grows, of a non-integral vertex. This is the first infinite family of non-integral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the non-integral vertices when $n$ is fixed.

Motivation & Objective

• To provide a combinatorial characterization of the vertices of Gelfand-Tsetlin polytopes.
• To develop a method for computing the dimension of the lowest-dimensional face containing a given GT-pattern.
• To disprove the conjecture by Berenstein and Kirillov that all vertices of GT-polytopes are integral.
• To establish a bound on the denominators of non-integral vertices when $ n $ is fixed.
• To prove that the Ehrhart counting function of GT-polytopes remains a polynomial despite the existence of non-integral vertices.

Proposed method

• The authors define a tiling of a GT-pattern by grouping equal and adjacent entries into tiles, forming a partition of the pattern's positions.
• They associate a tiling matrix $ A_{\mathscr{P}} $ to each GT-pattern, where entries count how many times each free tile appears in each row.
• Vertices are characterized by the condition that every $ s \times s $ submatrix of the tiling matrix $ A_{\mathscr{P}} $ has non-zero determinant, where $ s $ is the number of free tiles.
• The dimension of the minimal face containing a GT-pattern is computed as $ s - \text{rank}(A_{\mathscr{P}}) $, where $ s $ is the number of free tiles.
• A bound on denominators of non-integral vertices is derived using the spectral radius of $ A_{\mathscr{P}} $, leading to the bound $ (n-1)^{\binom{n+1}{2} - n - 1} $.
• The Ehrhart polynomial is proven to be a univariate polynomial by showing that for large $ m $, the vectors in Kostant's formula remain in the same chamber, ensuring polynomial behavior.

Experimental results

Research questions

• RQ1 Are all vertices of Gelfand-Tsetlin polytopes integral, as conjectured by Berenstein and Kirillov?
• RQ2 Can a combinatorial characterization of vertices be given using tiling structures of GT-patterns?
• RQ3 What is the maximal denominator that can appear in a non-integral vertex of a GT-polytope for fixed $ n $?
• RQ4 Does the Ehrhart counting function of a GT-polytope remain a polynomial even when the polytope has non-integral vertices?
• RQ5 How does the tiling matrix $ A_{\mathscr{P}} $ relate to the face dimension and vertex structure of GT-polytopes?

Key findings

• The paper constructs, for each $ n \geq 5 $, an infinite family of GT-polytopes with non-integral vertices, where the denominators of the vertices can be arbitrarily large as $ n $ increases.
• The authors disprove the conjecture of Berenstein and Kirillov by exhibiting explicit counterexamples with non-integral vertices for $ n \geq 5 $.
• A bound on the denominators of non-integral vertices is established as $ (n-1)^{\binom{n+1}{2} - n - 1} $, valid for fixed $ n $.
• The dimension of the minimal face containing a GT-pattern is given by $ s - \text{rank}(A_{\mathscr{P}}) $, where $ s $ is the number of free tiles and $ A_{\mathscr{P}} $ is the tiling matrix.
• The Ehrhart counting function $ f(m) = \#(GT(m\lambda, m\mu) \cap \mathbb{Z}^{n+1 \choose 2}) $ is proven to be a univariate polynomial, despite the existence of non-integral vertices.
• The proof of polynomiality relies on the fact that for large $ m $, the vectors in Kostant’s formula remain in the same chamber of the root system $ A_n $, ensuring a single polynomial expression.