Alternating sign matrices and Verma modules

TL;DR

This paper establishes an isomorphism between the poset of alternating sign matrices under Bruhat order and a poset of submodules of a Verma module for rak{sl}_n, linking combinatorics and representation theory.

Contribution

It introduces a novel connection between alternating sign matrices and Verma modules, enriching the understanding of their algebraic and combinatorial structures.

Findings

Poset of alternating sign matrices is isomorphic to a submodule poset of a Verma module.

The submodule poset can be characterized via intersections of Verma submodules.

The poset relates to Kazhdan-Lusztig cells, bridging combinatorics and Lie algebra representations.

Abstract

We show that the poset of alternating sign matrices, with Bruhat order, is isomorphic to the poset of certain submodules of the dominant Verma module for the special linear Lie algebra $\frak{sl}_n$. The latter poset consists of the intersections of Verma submodules and can also be defined in terms of a Kazhdan-Lusztig cell.