A conjecture on monomial realizations and polyhedral realizations for crystal bases

TL;DR

"The paper investigates a bridge between polyhedral realizations of crystal bases and monomial realizations. It fixes an adapted sequence $\iota$ and studies the union of monomial realizations $\mathcal{M}_{s,k,\iota}$ for the Langlands dual $\mathfrak{g}^L$, then compares the tropicalization to the polyhedral image ${\rm Im}(\Psi_{\iota})$, positing a central conjecture of equality. The authors prove the conjecture for several classes: finite types $A_n,B_n,C_n,D_n$, rank-2 Kac-Moody algebras, and classical affine types, providing explicit inequalities and monomial descriptions. This work clarifies how combinatorial models (Young tableaux, walls) encode the polyhedral inequalities and offers practical tools for computing crystal bases via both polyhedral and monomial frameworks."

Abstract

Crystal bases are powerful combinatorial tools in the representation theory of quantum groups $U_q(\mathfrak{g})$ for a symmetrizable Kac-Moody algebras $\mathfrak{g}$. The polyhedral realizations are combinatorial descriptions of the crystal base $B(\infty)$ for Verma modules in terms of the set of integer points of a polyhedral cone, which equals the string cone when $\mathfrak{g}$ is finite dimensional simple. It is a fundamental and natural problem to find explicit forms of the polyhedral cone. The monomial realization expresses crystal bases $B(λ)$ of integrable highest weight representations as Laurent monomials with double indexed variables. In this paper, we give a conjecture between explicit forms of the polyhedral cones and monomial realizations. We prove the conjecture is true when $\mathfrak{g}$ is a classical Lie algebra, a rank $2$ Kac-Moody algebra or a classical affine Lie algebra.

Theorems & Definitions (31)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Definition 2.6
• Theorem 3.1
• Lemma 3.2
• Example 3.3
• Example 3.4