Decomposition of tensor products of Demazure crystals

TL;DR

This paper investigates the structure of tensor products of Demazure crystals, providing conditions for their components to be Demazure crystals, with applications to key polynomial positivity and Demazure operator rules.

Contribution

It offers a necessary and sufficient condition for tensor product components to be Demazure crystals and derives explicit formulas for these components.

Findings

Connected components of tensor products are sometimes Demazure crystals.

Explicit formulas for connected components are provided.

Applications include positivity of key polynomial structure constants.

Abstract

A Demazure crystal is the basis at $q=0$ of a Demazure module. Demazure crystals play an important role in Schubert calculus because the character of a Demazure crystal in type A is identical to a key polynomial, which is closely related to Schubert polynomials. In this paper, we study tensor products of Demazure crystals. Each connected component of a tensor product of Demazure crystals need not be isomorphic to some Demazure crystal. We provide a necessary and sufficient condition for every connected component of a tensor product to be isomorphic to some Demazure crystal. Also, we obtain the explicit formula for connected components. As applications, we study the positivity for structure constants of products of key polynomials, and we obtain an equation of crystals, which is an analog of the Leibniz rule for Demazure operators.