Let G be any of the groups (P )GL(n, q), (P ) SL(n, q). Define a (simple) graph Γ = Γ (G) on the set of elements of G by connecting two vertices by an edge if and only if they generate G. Suppose that n is at least 12. Then the maximum size of a complete subgraph in Γ is equal to the chromatic number of Γ if n ≡ 2 (mod 4), or if n ≡ 2 (mod 4), q is odd and G = (P ) SL(n, q). This work was motivated by a question of Blackburn.

Let J be a finite-dimensional nilpotent algebra over a finite field F q . We formulate a procedure for analysing characters of the group 1 + J . In particular, we study characters of the group U n (q) of unipotent triangular n × n matrices over F q . Using our procedure, we compute the number of irreducible characters of U n (q) of each degree for n 13. Also, we explain and generalise a phenomenon concerning the group U 13 (2) discovered by Isaacs and Karagueuzian.

We define reduced zeta functions of Lie algebras, which can be derived, via the Euler characteristic, from motivic zeta functions counting subalgebras and ideals. We show that reduced zeta functions of Lie algebras possessing a suitably well-behaved basis are easy to analyse. We prove that reduced zeta functions are multiplicative under certain conditions and investigate which reduced zeta functions have functional equations.

Abstract. Turner's Conjecture describes all blocks of symmetric groups and Hecke algebras up to derived equivalence in terms of certain double algebras. With a view towards a proof of this conjecture, we develop a general theory of Turner doubles. In particular, we describe doubles as explicit maximal symmetric subalgebras of certain generalized Schur algebras and establish a Schur-Weyl duality with wreath product algebras.

Abstract. Given natural numbers m and n, we define a deflation map from the characters of the symmetric group S mn to the characters of S n . This map is obtained by first restricting a character of S mn to the wreath product S m o S n , and then taking the sum of the irreducible constituents of the restricted character on which the base group S m S m acts trivially. We prove a combinatorial formula which gives the values of the images of the irreducible characters of S mn under this map. We also prove an analogous result for more general deflation maps in which the base group is not required to act trivially. These results generalize the Murnaghan-Nakayama rule and special cases of the LittlewoodRichardson rule. As a corollary we obtain a new combinatorial formula for the character multiplicities that are the subject of the long-standing Foulkes' Conjecture. Using this formula we verify Foulkes' Conjecture in some new cases.