Daniel K. Nakano

Representation type of Hecke algebras of type

In this paper we provide a complete classification of the representation type for the blocks for the Hecke algebra of type A, stated in terms of combinatorical data. The computation of the complexity of Young modules is a key component in the proof of this classification result.

SPECHT FILTRATIONS FOR HECKE ALGEBRAS OF TYPE A

Let Hq (d) be the Iwahori–Hecke algebra of the symmetric group, where q is a primitive lth root of unity. Using results from the cohomology of quantum groups and recent results about the Schur functor and adjoint Schur functor, it is proved that, contrary to expectations, for l 4 the multiplicities in a Specht or dual Specht module filtration of an Hq (d)-module are well defined. A cohomological criterion is given for when an Hq (d)-module has such a filtration. Finally, these results are used to give a new construction of Young modules that is analogous to the Donkin–Ringel construction of tilting modules. As a corollary, certain decomposition numbers can be equated with extensions between Specht modules. Setting q = 1, results are obtained for the symmetric group in characteristic p 5. These results are false in general for p = 2 or 3.

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.

Endotrivial modules for finite groups of Lie type.

Abstract 1. Introduction Let G be a finite group and k be a field of characteristic p > 0. An endotrivial kG-module is a finitely generated kG-module M whose k-endomorphism ring is isomorphic to a trivial module in the stable module category. That is, M is an endotrivial module provided where P is a projective kG-module. Now recall that as kG-modules, where M * = Hom k (M, k) is the k-dual of M. Hence, the functor “ ” induces an equivalence on the stable module category and the collection of all endotrivial modules makes up a part of the Picard group of all stable equivalences of kG-modules. In particular, equivalence classes of endotrivial modules modulo projective summands form a group that is an essential part of the group of stable self-equivalences.

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.

ENDOTRIVIAL MODULES FOR THE SYMMETRIC AND ALTERNATING GROUPS

In this paper we determine the group of endotrivial modules for certain symmetric and alternating groups in characteristic

Cohomology for quantum groups via the geometry of the nullcone

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Support varieties for modules over Chevalley groups and classical Lie algebras

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