Modular representations of profinite groups

Our aim is to transfer several foundational results from the modular representation theory of finite groups to the wider context of profinite groups. We are thus interested in profinite modules over the completed group algebra of a profinite group G, where k is a finite field of characteristic p.We define the concept of relative projectivity for a profinite -module. We prove a characterization of finitely generated relatively projective modules analogous to the finite case with additions of interest to the profinite theory. We introduce vertices and sources for indecomposable finitely generated -modules and show that the expected conjugacy properties hold—for sources this requires additional assumptions. Finally we prove a direct analogue of Green’s Indecomposability Theorem for finitely generated modules over a virtually pro-p group.     

Multiplicity-free representations of symmetric groups

Building on work of Saxl, we classify the multiplicity-free permutation characters of all symmetric groups of degree 66 or more. A corollary is a complete list of the irreducible characters of symmetric groups (again of degree 66 or more) which may appear in a multiplicity-free permutation representation. The multiplicity-free characters in a related family of monomial characters are also classified. We end by investigating a consequence of these results for Specht filtrations of permutation modules defined over fields of prime characteristic.     

Regular representations of discrete groups

Kaniuth [3] has given necessary and sufficient conditions for the regular representation of a discrete group to be type I or type II. We give a non-measure-theoretic proof of his results, plus additional information on the structure of the W^∗ algebra generated by the regular representation.     

Factorial representations of path groups

The reduction of the energy representation of the group of mappings from I = [0, 1], S¹1, $\text{R}$₊ or $\text{R}$ into a compact semisimple Lie group G is given. For G = SU(2), the factoriality of the representation, which is of type III in the case I=$\text{R}$, is proved.     

Multiplier representations of abelian groups

The multiplier representations of a locally compact abelian group are classified, in the case there are only type I representations, and a simple criterion is obtained for determining when only type I representations occur. It is also shown that all the irreducible multiplier representations belonging to a fixed multiplier have the same dimension. To obtain these results we have to extend the little group method of Mackey and Blattner to handle multiplier representations of nonseparable groups.     

On almost representations of groups

We say that a group belongs to the class if every nonunit quotient group of has an element of order two.

Let be a Hilbert space and let be its group of unitary operators. Suppose that groups and belong to the class and the order of is more than two. Then the free product has the following property. For any there exists a mapping satisfying the following conditions :

1)

2) for any representation the relation

holds.

     

Local representations of braid groups

We study the local linear representations of the braid group B ₃ and the homogeneous local representations of B _n for n ≥ 2. We investigate the connection of these representations with the Burau representation. The linear representations of B _n are constructed from the Wada representation of B _n in the automorphism group of a free group.     

Quasidiagonal representations of nilpotent groups

We show that every unitary representation of a discrete solvable virtually nilpotent group G is quasidiagonal. Roughly speaking, this says that every unitary representation of G   approximately decomposes as a direct sum of finite dimensional approximate representations. In operator algebraic terms we show that C^?(G)$C^{?}(G)$ is strongly quasidiagonal.     

Schottky groups and maximal representations

We describe a construction of Schottky type subgroups of automorphism groups of partially cyclically ordered sets. We apply this construction to the Shilov boundary of a Hermitian symmetric space and show that in this setting Schottky subgroups correspond to maximal representations of fundamental groups of surfaces with boundary. As an application, we construct explicit fundamental domains for the action of maximal representations into \(\mathrm {Sp}(2n,\mathbb {R})\) on \(\mathbb {RP}^{2n-1}\).