Title: A conjecture on monomial realizations and polyhedral realizations for crystal bases Show Chinese title

Title: 关于晶体基的单项式实现和多面体实现的一个猜想

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 $B(\lambda)$ 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. Show Chinese abstract

Abstract: 晶体基是量子群表示论中的强大组合工具，适用于对称化Kac-Moody代数$U_q(\mathfrak{g})$的情形$\mathfrak{g}$。 多面体实现是以多面锥的整点集合形式描述Verma模的晶体基$B(\infty)$，当$\mathfrak{g}$是有限维单李代数时，它等于弦锥。 找到多面锥的显式形式是一个基本且自然的问题。 单项式实现将可积最高权表示的晶体基$B(\lambda)$表达为具有双指标变量的Laurent单项式。 本文提出了关于多面锥的显式形式与单项式实现之间的一个猜想，并证明了当$\mathfrak{g}$是经典李代数、秩为$2$的Kac-Moody代数或经典仿射李代数时该猜想成立。