A theorem of O’Nan and Scott [6]; [2, Chapter 4] restricts the possibilities for maximal subgroups of finite symmetric groups: they are of six types of which the first four are explicitly known, the fifth involves a finite simple group, and the sixth an action of a simple group. This result, in conjunction with the Classification of Finite Simple Groups, has a number of consequences. In particu...

We construct explicit generating sets Sn and S̃n of the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), Sn) and C(Sym(n), S̃n) into a family of bounded degree expanders for all n. This answers affirmatively an old question which has been asked many times in the literature. These expanders have many applications in the theory of random walks on groups, card shuffling a...

We study the moduli spaces and compute the fundamental groups of plane sextics of torus type with at least two type E6 singular points. As a simple application, we compute the fundamental groups of 125 other sextics, most of which

Over the last few weeks, we have seen lots of things in representation theory—we began by looking at an onslaught of examples to see explicitly what representations look like, we discussed character theory and looked at character tables, we talked about representations from a module-theoretic perspective, and then we saw how we can model the representation theory of the symmetric group via the ...

This paper is based on a talk given at the conference ”Representation theory of real reductive groups”, Salt Lake City, July 2009. We fix an algebraically closed field k of characteristic exponent p. (We assume, except in §18, that either p = 1 or p ≫ 0.) We also fix a symmetric space that is a triple (G, θ,K) where G is a connected reductive algebraic group over k, θ : G −→ G is an involution ...