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On combinatorics of string polytopes in types $B$ and $C$

Yunhyung Cho, Naoki Fujita, Eunjeong Lee

TL;DR

This work develops explicit descriptions of string cones for types $B_n$ and $C_n$ by folding Gleizer--Postnikov's type $A$ construction, and uses this to classify when string polytopes are unimodularly equivalent to Gelfand--Tsetlin polytopes. It shows that in type $C_n$ the unimodular equivalence $ riangle_{m i}^{(C)}( ho) cong GT_{C_n}( ho)$ unless ${m i}={m i}_{C}^{(n)}$, while demonstrating that ${m j}_{C}^{(n)}$ yields a non-integral vertex and hence is not equivalent. The authors also classify simplicial string cones in types $B_n$ and $C_n$ and describe the intricate folding relations among string cones and their path-based inequalities using symplectic wiring diagrams and canonical paths. Overall, the paper advances a unified combinatorial framework for string polytopes across classical types and sharpens the unimodular equivalence landscape with Gelfand--Tsetlin polytopes.

Abstract

A string polytope is a rational convex polytope whose lattice points parametrize a highest weight crystal basis, which is obtained from a string cone by explicit affine inequalities depending on a highest weight. It also inherits geometric information of a flag variety such as toric degenerations, Newton-Okounkov bodies, mirror symmetry, Schubert calculus, and so on. In this paper, we study combinatorial properties of string polytopes in types $B$ and $C$ by giving an explicit description of string cones in these types which is analogous to Gleizer-Postnikov's description of string cones in type $A$. As an application, we characterize string polytopes in type $C$ which are unimodularly equivalent to the Gelfand-Tsetlin polytope in type $C$ for a specific highest weight.

On combinatorics of string polytopes in types $B$ and $C$

TL;DR

This work develops explicit descriptions of string cones for types and by folding Gleizer--Postnikov's type construction, and uses this to classify when string polytopes are unimodularly equivalent to Gelfand--Tsetlin polytopes. It shows that in type the unimodular equivalence unless , while demonstrating that yields a non-integral vertex and hence is not equivalent. The authors also classify simplicial string cones in types and and describe the intricate folding relations among string cones and their path-based inequalities using symplectic wiring diagrams and canonical paths. Overall, the paper advances a unified combinatorial framework for string polytopes across classical types and sharpens the unimodular equivalence landscape with Gelfand--Tsetlin polytopes.

Abstract

A string polytope is a rational convex polytope whose lattice points parametrize a highest weight crystal basis, which is obtained from a string cone by explicit affine inequalities depending on a highest weight. It also inherits geometric information of a flag variety such as toric degenerations, Newton-Okounkov bodies, mirror symmetry, Schubert calculus, and so on. In this paper, we study combinatorial properties of string polytopes in types and by giving an explicit description of string cones in these types which is analogous to Gleizer-Postnikov's description of string cones in type . As an application, we characterize string polytopes in type which are unimodularly equivalent to the Gelfand-Tsetlin polytope in type for a specific highest weight.
Paper Structure (7 sections, 26 theorems, 92 equations, 22 figures, 1 table)

This paper contains 7 sections, 26 theorems, 92 equations, 22 figures, 1 table.

Table of Contents

  1. Introduction
  2. Gleizer--Postnikov description of string cones
  3. Wiring diagrams and rigorous paths
  4. String cone inequalities
  5. Folding procedure for string cones
  6. Folding procedure for rigorous paths
  7. Gelfand--Tsetlin type string polytopes in type $C$

Key Result

Theorem 1.1

Let $\mathfrak{g}$ be a simple Lie algebra of type $B_n$ or $C_n$ with $n \geq 2$. Then, for ${\bm i} \in R(w_0)$, the following are equivalent.

Figures (22)

  • Figure 1: Wiring diagrams for ${\bm i} = (1,2,1,3,2,1)$ and ${\bm i}' = (1,3,2,1,3,2)$.
  • Figure 2: Forbidden fragments.
  • Figure 3: Oriented wiring diagrams for ${\bm i} = (1,2,1,3,2,1)$ and ${\bm i}' = (1,3,2,1,3,2)$.
  • Figure 4: Chambers $\mathscr C_j$ for $\bm i = (1,3,2,1,3,2)$.
  • Figure 5: Symplectic wiring diagram for $\bm i = (1,2,3,1,2,3,1,2,3)$.
  • ...and 17 more figures

Theorems & Definitions (65)

  • Theorem 1.1: see Theorem \ref{['thm_simplicial_string_cones_BC']} and Remark \ref{['r:second_main_type_B']}
  • Theorem 1.2: Theorem \ref{['thm_GT_string_polytopes_C']}
  • Definition 2.1: GleizerPostnikov
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4: GleizerPostnikov
  • Example 2.5
  • Lemma 2.6: CKLP21_GC
  • Example 2.7
  • Proposition 2.8: see CKLP21_GC
  • ...and 55 more