The representation theory of the supergroup GL(m-n).

摘要

In this dissertation we consider the modular representation theory of the supergroup GL(m|n). Let T be the diagonal subgroup of GL( m|n) and let X(T) denote the character group of T. A main theme of our work is that the combinatorics of a certain crystal structure on X( T) describes aspects of the representation theory of GL( m|n). For example, we prove that the central characters of GL(m|n) are determined by the crystal structure of X(T) and from this obtain a linkage principle for GL(m| n). Also, one can define translation functors Fr and Er for r ∈ Z/pZ by tensoring with the natural GL( m|n)-supermodule or its dual and projecting onto certain central characters. We prove that the crystal structure of X(T) describes the effect of these functors on an irreducible GL(m|n)-supermodule. Consequently, the crystal structure describes the effect of tensoring an irreducible supermodule with the natural GL(m| n)-supermodule or its dual. We also prove, independent of the work described above, a version of the Steinberg Tensor Product Theorem for GL(m|n).