Stephen Doty

Polynomial representations, algebraic monoids, and Schur algebras of classical type

First we study Zariski-closed subgroups of general linear groups over an infinite field with the property that their polynomial representation theory is graded in a natural way. There are “Schur algebras” associated with such a group and their representations completely determine the polynomial representations of the original subgroup. Moreover, the polynomial representations of the subgroup are equivalent with the rational representations of a certain algebraic monoid associated with the subgroup, and the aforementioned Schur algebras are the linear duals of the graded components of the coordinate bialgebra on the monoid. Then we study the monoids and Schur algebras associated with the classical groups. We obtain some structural information on the monoids. Then we characterize the associated Schur algebras as centralizer algebras for the Brauer algebras, in characteristic zero.

Presenting generalized -Schur algebras

We obtain a presentation by generators and relations for generalized Schur algebras and their quantizations. This extends earlier results obtained in the type A case. The presentation is compatible with Lusztigs modified form of a quantized enveloping algebra. We show that generalized Schur algebras inherit a canonical basis from the modified form of the quantized enveloping algebra, that this gives them a cellular structure, and thus they are quasihereditary over a field.

Representation type of -Schur algebras

Abstract. We give a complete classification of the classical Schur algebras and the infinitesimal Schur algebras which have tame representation type. In combination with earlier work of some of the authors on semisimplicity and finiteness, this completes the classification of representation type of all classical and infinitesimal Schur algebras in all characteristics.

Extensions of Modules over Schur Algebras, Symmetric Groups and Hecke Algebras

We study the relation between the cohomology of general linear and symmetric groups and their respective quantizations, using Schur algebras and standard homological techniques to build appropriate spectral sequences. As our methods fit inside a much more general context within the theory of finite-dimensional algebras, we develop our results first in that general setting, and then specialize to the above situations. From this we obtain new proofs of several known results in modular representation theory of symmetric groups. Moreover, we reduce certain questions about computing extensions for symmetric groups and Hecke algebras to questions about extensions for general linear groups and their quantizations.

Brauer algebras, symplectic Schur algebras and Schur-Weyl duality

In this paper we prove the Schur-Weyl duality between the symplectic group and the Brauer algebra over an arbitrary infinite field K. We show that the natural homomorphism from the Brauer algebra B n (-2m) to the endomorphism algebra of the tensor space (K 2m )⊗ n as a module over the symplectic similitude group GSp 2m (K) (or equivalently, as a module over the symplectic group Sp 2m (K)) is always surjective. Another surjectivity, that of the natural homomorphism from the group algebra for GSp 2m (K) to the endomorphism algebra of (K 2m )⊗ n as a module over B n (-2m), is derived as an easy consequence of S. Oehmss results [S. Oehms, J. Algebra (1) 244 (2001), 19-44].

The Quantized Walled Brauer Algebra and Mixed Tensor Space

In this paper we investigate a multi-parameter deformation

The composition factors of Fp[x1, x2, x3] as a GL(3, p)-module

\mathfrak{B}_{r,s}^n(a,\lambda,\delta)

Semisimple Schur Algebras

of the walled Brauer algebra which was previously introduced by Leduc (1994). We construct an integral basis of

Truncated symmetric powers and modular representations of GL n

\mathfrak{B}_{r,s}^n(a,\lambda,\delta)

The rational Schur algebra

consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of