Extremal tensor products of Demazure crystals

TL;DR

This paper characterizes when tensor products of Demazure crystals are extremal and demonstrates that such tensor products decompose into Demazure crystals if and only if they are extremal, providing a clear criterion for their structure.

Contribution

It introduces a local criterion for extremality of tensor products of Demazure crystals and proves the equivalence between extremality and decomposition into Demazure crystals.

Findings

Tensor products of Demazure crystals decompose into Demazure crystals iff they are extremal.

A local characterization for extremality of tensor products of Demazure crystals.

The primary component of the tensor square of any Demazure crystal is always Demazure.

Abstract

Demazure crystals are subcrystals of highest weight irreducible $\mathfrak{g}$-crystals. In this article, we study tensor products of a larger class of subcrystals, called extremal, and give a local characterization for exactly when the tensor product of Demazure crystals is extremal. We then show that tensor products of Demazure crystals decompose into direct sums of Demazure crystals if and only if the tensor product is extremal, thus providing a sufficient and necessary local criterion for when the tensor product of Demazure crystals is itself Demazure. As an application, we show that the primary component in the tensor square of any Demazure crystal is always Demazure.