Infinite-level Fock spaces, crystal bases, and tensor product of extremal weight modules of type $A_{+\infty}$

TL;DR

This paper explores the structure of extremal weight modules over an infinite-dimensional quantum group, revealing tensor category properties and describing socle filtrations via crystal bases and Fock space representations.

Contribution

It provides an explicit description of the socle filtration of tensor products of extremal weight modules using crystal bases and introduces the notion of saturated crystal valuation.

Findings

The category generated by extremal weight modules is a tensor category.

Explicit socle filtration descriptions are obtained for tensor products.

The study connects Fock space representations with crystal bases and dualities.

Abstract

We study the category $\mathcal{C}$ generated by extremal weight modules over $U_q(\mathfrak{gl}_{>0})$. We show that $\mathcal{C}$ is a tensor category, and give an explicit description of the socle filtration of tensor product of any two extremal weight modules. This follows from the study of Fock space $\mathcal{F}^\infty \otimes \mathcal{M}$ of infinite level, which has commuting actions of a parabolic $q$-boson algebra and $U_p(\mathfrak{gl}_{>0})$ with $p=-q^{-1}$. It contains a (semisimple) limit of the fermionic Fock space $\mathcal{F}^n$ of level $n$, which has a $q$-analogue of Howe duality often called level-rank duality. To describe the socle filtration of $\mathcal{F}^\infty \otimes \mathcal{M}$, we introduce the notion of a saturated crystal valuation, whose existence was observed for example in the embedding of an extremal weight module into a tensor product of fundamental weight modules of affine type due to Kashiwara and Beck-Nakajima.