The composition series of modules induced from Whittaker modules

We study a category of representations over a semisimple Lie algebra, which contains category as well as the so-called Whittaker modules, and prove a generalization of the Kazhdan-Lusztig conjectures in this context. Received: January 15, 1996     

Coprime action and arithmetical conditions on invariant conjugacy classes

Let A and G be finite groups and suppose that A acts coprimely on G via automorphisms. We show that if 4 divides no A-invariant conjugacy class size of G, then G is solvable. We also characterize the A-invariant structure of G under certain arithmetical conditions on the set of A-invariant class sizes of G by means of the fixed point subgroup, some of which imply the solvability of G. Thus, we extend, for coprime action, several results appeared in the literature on class sizes.     

Cyclic conjugacy separability and conjugacy separability of certain HNN extensions

Abstract

In this note, we give a criterion for certain HNN extensions of cyclic conjugacy separable (respectively conjugacy separable) groups with infinite cyclic associated subgroups to be again cyclic conjugacy separable (respectively conjugacy separable).

Communicated by Alexander Olshanskii     

Categorification of (induced) cell modules and the rough structure of generalised Verma modules

This paper presents categorifications of (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. In type A we show that these categorifications depend only on the isomorphism class of the cell module, not on the cell itself. Our main application is multiplicity formulas for parabolically induced modules over a reductive Lie algebra of type A, which finally determines the so-called rough structure of generalised Verma modules. On the way we present several categorification results and give a positive answer to Kostant's problem from [A. Joseph, Kostant's problem, Goldie rank and the Gelfand-Kirillov conjecture, Invent. Math. 56 (3) (1980) 191-213] in many cases. We also present a general setup of decategorification, precategorification and categorification.     

The conjugacy problem for centre-by-metabelian groups

Archiv der Mathematik -     

The conjugacy and isomorphism problems for combable groups

There exist combable groups in which the conjugacy problem is unsolvable. The isomorphism problem is unsolvable for certain recursive sequences of finite presentations of combable groups. Mathematics Subject Classification(2000): 20F67, 20F10, 20F65This research was supported by an EPSRC Advanced Fellowship     

Determination of conjugacy class sizes from products of characters

In Robinson [5] Robinson showed that the character degrees are determined by knowing, for all n, the number of ways that the identity can be expressed as a product of n commutators. Earlier, in Strunkov [6], Strunkov showed that the existence of characters of p-defect 0 can be determined by counting solutions to certain equations involving commutators and conjugates. In this paper, we prove analogs to Robinson’s and Strunkov’s theorems by switching conjugacy classes and characters. We show that counting the multiplicity of the trivial character in certain products of characters determines the conjugacy class sizes and existence of conjugacy classes with p-defect 0.     

The equations of conjugacy classes of nilpotent matrices

Partially supported by NSF Grant     

The conjugacy vector of ap-group of maximal class

We find the conjugacy vector, i.e., we determine the number of conjugacy classes which compose the sets of the elements with centralizers of equal order, for several general families ofp-groups of maximal class which include those of order up top ⁹. As a consequence, we obtain the number of conjugacy classes,r(G), for the groups in these families. Also, we provide upper and lower bounds forr(G) and characterize when they are attained. Examples are given showing that the bounds are actually attained. This work has been supported by DGICYT grant PB91-0446 and by the University of the Basque Country.     

On the unitary globalization of cohomologically induced modules

We describe the unitary globalization of cohomologically induced modules _Aq(λ)$A_{\mathfrak{q}}(\lambda )$. The purpose of the paper is to give a geometric realization of the unitarizable modules. Our results do not constitute a proof of unitarity.     

The permutation action of finite symplectic groups of odd characteristic on their standard modules

We study the space of functions on a finite-dimensional vector space over a field of odd order as a module for a symplectic group. We construct a basis of this module with the following special properties. Each submodule generated by a single basis element under the symplectic group action is spanned as a vector space by a subset of the basis and has a unique maximal submodule. From these properties, the dimension and composition factors of the submodule generated by any subset of the basis can be determined. These results apply to incidence geometry of the symplectic polar space, yielding the symplectic analogue of Hamada's additive formula for the p-ranks of the incidence matrices between points and flats. A special case leads to a closed formula for the p-rank of the incidence matrix between the points and lines of the symplectic generalized quadrangle over a field of odd order. Together with earlier results on the 2-ranks, this result completes the determination of the p-ranks for these quadrangles.     

Partial cotilting modules and the lattices induced by them

We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary ring. Dualizing a result of Bongartz, we show that a module P is partial cotilting iff P is a direct summand of a cotilting module C such that the left Ext-orthogonal class ⊥P coincides with ⊥C. As an application, we characterize all cotilting torsion-free classes. Each partial cotilting module P defines a lattice L = [Cogen P₁P] of torsion-free classes. Similarly, each partial tilting module P′ defines a lattice L′ = [[Gen P′,P′⊥]] of torsion classes. Generalizing a result of Assem and Kerner, we show that the elements of L are determined by their Rej_p-torsion parts, and the elements of L′ by their Tr_p-torsion-free parts.