Crystal structures arising from representations of $GL(m\mathrm{\mid }n)$

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