Level-rank dualities and 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.