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Crystal structures arising from representations of $GL(m\vert n)$

Author: Jonathan Kujawa
Journal: Represent. Theory 10 (2006), 49-85
MSC (2000): Primary 20C20, 05E99; Secondary 17B10
Posted: February 16, 2006
MathSciNet review: 2209849
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Abstract: This paper provides results on the modular representation theory of the supergroup $GL(m\vert n).$ 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 $\mathcal{O}$ . 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 $GL(m\vert n)$ has non-conjugate Borel subgroups and we show how Serganova's odd reflections give rise to canonical crystal isomorphisms.

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