We consider non‐trivial irreducible tensor products of modular representations of a symmetric group Σn in characteristic 2 for even n completing the proof of a classification conjecture of Gow and Kleshchev about such products.

We study irreducible restrictions of modules over symmetric groups to subgroups. We get reduction results which substantially restrict the classes of subgroups and modules for which this is possible. Such results are known when the characteristic of the ground field is greater than 3, but the small characteristics cases require a substantially more delicate analysis and new ideas. This work fits into the Aschbacher-Scott program on maximal subgroups of finite classical groups.

In this paper we construct explicit sampling sets and present reconstruction algorithms for Fourier signals on finite vector spaces G, with |G| = p r for a suitable prime p. The two sampling sets have sizes of order O(pt 2 r 2 ) and O(pt 2 r 3 log(p)) respectively, where t is the number of large coefficients in the Fourier transform. The algorithms approximate the function up to a small constant of the best possible approximation with t non-zero Fourier coefficients. The fastest of the algorithms has complexity O(p 2 t 2 r 3 log(p)).

A conjugacy class C of a finite group G is a sign conjugacy class if every irreducible character of G takes value 0, 1 or -1 on C. In this paper we classify the sign conjugacy classes of the symmetric groups and thereby verify a conjecture of Olsson.

We call an irreducible character p-singular if p divides its degree. We prove a number of equivalent conditions for a character of the symmetric group S n to be p-singular, involving a certain family of conjugacy classes. This answers in part a question by Navarro.