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

TL;DR

This paper investigates the tensor product structure of Demazure crystals in symmetrizable Kac-Moody Lie algebras, providing conditions for their decomposition and implications for Demazure character positivity.

Contribution

It establishes necessary and sufficient conditions for decomposing tensor products of Demazure crystals into disjoint unions, generalizing prior results and addressing key positivity issues.

Findings

Conditions for Demazure crystal tensor product decomposition

Criteria for Demazure character linear combinations

Partial solution to key positivity problem

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.