Decomposition of tensor products of Demazure crystals

Abstract

In general, a tensor product of Demazure crystals does not decompose into a disjoint union of Demazure crystals. However, under a certain condition, a tensor product decomposes into a disjoint union of Demazure crystals. In this paper, we introduce a necessary and sufficient condition for every connected component of a tensor product of two Demazure crystals to be isomorphic to some Demazure crystal. Moreover, we consider a recursive formula describing connected components of tensor products of arbitrary Demazure crystals. As an application, we discuss the key positivity problem, which is the problem whether a product of key polynomials is a linear combination of key polynomials with nonnegative integer coefficients or not. Also, we obtain a crystal-theoretic analog of the Leibniz rule for Demazure operators.