Level-rank dualities and moving vectors

TL;DR

This paper develops a uniform combinatorial framework for classical affine Lie algebra dualities, extending previous character-based methods and linking defect calculations to moving vectors.

Contribution

It constructs combinatorial models for all classical affine types, extending Uglov maps and characterizing dualities without relying on character calculations.

Findings

Established combinatorial models (Maya diagrams and abaci) for all classical affine types.

Extended Uglov map to all classical affine types.

Linked defect of cyclotomic KLR algebra to moving vectors.

Abstract

Duality relations between Lie algebras are a significant phenomenon in Lie algebra representation theory, with level-rank duality as a famous example. Level-rank dualities for affine Lie algebras of type $A^{(1)}$ were first discovered by Frenkel in 1982, and later extended to all classical non-twisted affine types by Hasegawa in 1989 through elaborate character calculations. In this paper, for all classical affine Lie algebras, we construct appropriate Fock spaces in a uniform way and establish corresponding combinatorial models (Maya diagrams and abaci), extending Uglov map to all classcial affine types. Through the action of the Virasoro algebra, we completely characterize the joint highest weight vectors in the Fock space, thereby obtaining the corresponding level-rank duality theory. Our method no longer relies on character calculations. Using this new level-rank duality theory, the defect of the cyclotomic KLR algebra $\mathscr{R}^{\Lambda}_{\beta}$ of classical affine type can be interpreted as the sum of the components of the correponding moving vector.