Crystal duality and Littlewood-Richardson rule of extremal weight crystals

TL;DR

This paper explores the structure of a category of $ ext{gl}_$-crystals, establishing its monoidal nature, describing its Grothendieck ring, and providing an explicit Littlewood-Richardson rule for extremal weight crystals.

Contribution

It introduces a new categorical framework for extremal weight crystals and connects it to algebraic structures via an Ore extension, also deriving an explicit Littlewood-Richardson rule.

Findings

The category of extremal weight crystals is monoidal.

The Grothendieck ring is anti-isomorphic to an Ore extension of the character ring.

An explicit Littlewood-Richardson rule for extremal weight crystals is provided.

Abstract

We consider a category of $\gl_\infty$-crystals, whose objects are disjoint unions of extremal weight crystals of non-negative level with certain finite conditions on the multiplicity of connected components. We show that it is a monoidal category under tensor product of crystals and the associated Grothendieck ring is anti-isomorphic to an Ore extension of the character ring of integrable lowest weight $\gl_\infty$-modules with respect to derivations shifting the characters of fundamental modules. A Littlewood-Richardson rule of extremal weight crystals with non-negative level is described explicitly in terms of classical Littlewood-Richardson coefficients.