Crystal structures arising from representations of 𝐺𝐿(𝑚|𝑛)

Abstract

This paper provides results on the modular representation theory of the supergroup G L ( m | n ) . GL(m|n). Working over a field of arbitrary characteristic, we prove that the explicit combinatorics of certain crystal graphs describe the representation theory of a modular analogue of the Bernstein-Gelfand-Gelfand category O \mathcal {O} . In particular, we obtain a linkage principle and describe the effect of certain translation functors on irreducible supermodules. Furthermore, our approach accounts for the fact that G L ( m | n ) GL(m|n) has non-conjugate Borel subgroups and we show how Serganova’s odd reflections give rise to canonical crystal isomorphisms.