Crystal isomorphisms for irreducible highest weight U_{v}{\hat{sl}}_{e})-modules of higher level

TL;DR

This paper provides a simplified description of crystal graph isomorphisms for higher-level irreducible modules of quantum affine sl_e, using embeddings into infinite-level crystal structures.

Contribution

It generalizes previous work to higher levels and explicitly constructs embeddings of level l crystals into infinite-level crystals for better understanding.

Findings

Explicit bijections between multipartitions and crystal graphs.

Embedding of level l crystals into $U_{v}( ilde{sl}_{ ext{infinity}})$-crystals.

Simplified description of crystal isomorphisms for higher-level modules.

Abstract

We study the crystal graphs of irreducible $U_{v}(\hat{sl}}_{e})$-modules of higher level l. Generalizing works of the first author, we obtain a simple description of the bijections between the classes of multipartitions which naturally label these graphs: the Uglov multipartitions. This is achieved by expliciting an embedding of the $U_{v}(\hat{sl}}_{e})$-crystals of level l into $U_{v}(\hat{sl}_{\infty})$-crystals associated to highest weight modules.