Jonathan R. Kujawa

A New Proof of the Mullineux Conjecture

Let Sd denote the symmetric group on d letters. In 1979 Mullineux conjectured a combinatorial algorithm for calculating the effect of tensoring with an irreducible Sd-module with the one dimensional sign module when the ground field has positive characteristic. Kleshchev proved the Mullineux conjecture in 1996. In the present article we provide a new proof of the Mullineux conjecture which is entirely independent of Kleshchevs approach. Applying the representation theory of the supergroup GL(m | n) and the supergroup analogue of Schur-Weyl Duality it becomes straightforward to calculate the combinatorial effect of tensoring with the sign representation and, hence, to verify Mullineuxs conjecture. Similar techniques also allow us to classify the irreducible polynomial representations of GL(m | n) of degree d for arbitrary m, n, and d.

Cohomology and support varieties for Lie superalgebras II

In [2] (Preprint, 2006, arXiv:math.RT/0609363) the authors initiated a study of the representation theory of classical Lie superalgebras via a cohomological approach. Detecting subalgebras were constructed and a theory of support varieties was developed. The dimension of a detecting subalgebra coincides with the defect of the Lie superalgebra, and the dimension of the support variety for a simple supermodule was conjectured to equal the atypicality of the supermodule. In this paper the authors compute the support varieties of Kac supermodules for Type-I Lie superalgebras and of the simple supermodules for (m|n). The latter result verifies our earlier conjecture for (m|n). In our investigation we also delineate several of the major differences between Type-I versus Type-II classical Lie superalgebras. Finally, the connection between atypicality, defect and superdimension is made more precise by using the theory of support varieties and representations of Clifford superalgebras.

Cohomology and support varieties for Lie superalgebras

Unlike Lie algebras, the finite dimensional complex representations of a simple Lie superalgebra are usually not semisimple. As a consequence, despite over thirty years of study, these remain mysterious objects. In this paper we introduce a new tool: the notion of cohomological support varieties for the finite dimensional supermodules for a classical Lie superalgebra g = g 0 ⊕ g 1 which are completely reducible over g 0 . They allow us to provide a new, functorial description of the previously combinatorial notions of defect and atypicality. We also introduce the detecting subalgebra of g. Its role is analogous to the defect subgroup in the theory of finite groups in positive characteristic. Using invariant theory we prove that there are close connections between the cohomology and support varieties of g and the detecting subalgebra.

Geometric Schur Duality of Classical Type

This is a generalization of the classic work of Beilinson, Lusztig and MacPherson. In this paper (and an Appendix) we show that the quantum algebras obtained via a BLM-type stabilization procedure in the setting of partial Flag varieties of type B/C are two (modified) coideal subalgebras of the quantum general linear Lie algebra, U.

Crystal structures arising from representations of (

\overset{.}{\mathbf{U}}

Presenting Schur superalgebras

ℐ and U.

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

\overset{.}{\mathbf{U}}