Tensor decomposition of Demazure crystals for symmetrizable Kac-Moody Lie algebras

TL;DR

This work advances the crystal-theoretic understanding of Demazure modules for symmetrizable Kac–Moody algebras by establishing necessary and sufficient conditions for when tensor products of Demazure crystals decompose into disjoint unions of Demazure crystals. The main criterion hinges on the minimal coset representative $v_{\min}^{\lambda}$ belonging to a specific subgroup, equivalently rendering the tensor product an extremal subset of $B(\lambda)\otimes B(\mu)$. The authors give explicit decompositions in terms of $u(b,v)$ and demonstrate that Demazure characters arising from these decompositions admit nonnegative expressions via Demazure operators, with direct implications for the positivity of key polynomials in type $A$. They further extend these results to $B_v(\lambda)\otimes B_w(\infty)$ and provide analogous decompositions, enriching the toolkit for studying Demazure characters, extremal subsets, and related combinatorics in the Kac–Moody setting.

Abstract

We study the tensor product of Demazure crystals for symmetrizable Kac-Moody Lie algebras. It is not necessary that the tensor product of Demazure crystals is isomorphic to a disjoint union of Demazure crystals. In this paper, we provide necessary and sufficient conditions for the decomposition of the tensor product of Demazure crystals as a disjoint union of Demazure crystals. Our results are the generalization of the results proved by Anthony Joseph and Takafumi Kouno. As an application, we obtain a sufficient condition when the product of Demazure characters is a linear combination of Demazure characters with nonnegative integer coefficients. In particular, we obtain a partial solution for the key positivity problem.

Theorems & Definitions (41)

• Theorem 1.0.1
• Theorem 1.0.2
• Theorem 1.0.3
• Theorem 1.0.4
• Theorem 1.0.5
• Lemma 2.1.1
• Definition 2.2.1
• Definition 2.3.1
• Remark 2.4.1
• Definition 2.4.2