CRYSTAL STRUCTURES ARISING FROM REPRESENTATIONS OF GL(m|n)

This paper provides results on the modular representation theory of the supergroup 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. 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 GL(m|n) has non-conjugate Borel subgroups and we show how Serganova’s odd reflections give rise to canonical crystal isomorphisms.