Embeddings among quantum affine $\mathfrak{sl}_n$

TL;DR

This paper constructs an explicit embedding of quantum affine rak{sl}_n into rak{sl}_{n+1}, unifying previous embeddings and enabling applications in canonical bases, modular representation theory, and categorical representations.

Contribution

It introduces a new explicit embedding of quantum affine rak{sl}_n into rak{sl}_{n+1}, generalizing prior embeddings and connecting various areas of representation theory.

Findings

Unified embedding of quantum affine rak{sl}_n into rak{sl}_{n+1}

Compatibility with quantum affine Schur algebras and idempotent versions

Applications to canonical bases and modular representation theory

Abstract

We establish an explicit embedding of a quantum affine $\mathfrak{sl}_n$ into a quantum affine $\mathfrak{sl}_{n+1}$. This embedding serves as a common generalization of two natural, but seemingly unrelated, embeddings, one on the quantum affine Schur algebra level and the other on the non-quantum level. The embedding on the quantum affine Schur algebras is used extensively in the analysis of canonical bases of quantum affine $\mathfrak{sl}_n$ and $\mathfrak{gl}_n$. The embedding on the non-quantum level is used crucially in a work of Riche and Williamson on the study of modular representation theory of general linear groups over a finite field. The same embedding is also used in a work of Maksimau on the categorical representations of affine general linear algebras. We further provide a more natural compatibility statement of the embedding on the idempotent version with that on the quantum affine Schur algebra level. A $\mathfrak{gl}_n$-variant of the embedding is also established.