Semi-Modules and Crystal Bases via Affine Deligne-Lusztig Varieties

TL;DR

This paper establishes an explicit correspondence between semi-modules and crystal bases, two combinatorial parameterizations of affine Deligne-Lusztig varieties for GL_n, enhancing understanding of their structure.

Contribution

It introduces a direct link between semi-modules and crystal bases, connecting two different combinatorial frameworks for affine Deligne-Lusztig varieties.

Findings

Explicit correspondence between semi-modules and crystal bases

Enhanced understanding of affine Deligne-Lusztig varieties

Bridging combinatorial parameterizations

Abstract

There are two combinatorial ways of parameterizing the $J_b$-orbits of the irreducible components of affine Deligne-Lusztig varieties for $GL_n$ and superbasic $b$. One way is to use the extended semi-modules introduced by Viehmann. The other way is to use the crystal bases introduced by Kashiwara and Lusztig. In this paper, we give an explicit correspondence between them using the crystal structure.