Positive level, negative level and level zero

TL;DR

This survey explores the combinatorics and geometry of integrable quantum affine Lie algebra representations at various levels, emphasizing crystal structures, Weyl group actions, and the alcove walk method.

Contribution

It provides a comprehensive overview of the structure and combinatorial models of extremal weight modules at different levels, including new insights into the alcove walk approach.

Findings

Detailed descriptions of affine Weyl group orbits and crystal graphs.

Illustrations of Macdonald polynomials in the context of level representations.

Survey of the alcove walk method for affine flag varieties.

Abstract

This is a survey on the combinatorics and geometry of integrable representations of quantum affine Lie algebras with a particular focus on level 0. Pictures and examples are included to illustrate the affine Weyl group orbits, crystal graphs and Macdonald polynomials that provide detailed understanding of the structure of the extremal weight modules and their characters. The final section surveys the alcove walk method of working with the positive level, negative level and level zero affine flag varieties and describes the corresponding actions of the affine Hecke algebra.