Appendix A: Adequacy of representations of finite groups of Lie type

Thorne introduced the notion of adequate representations as a weakening of the big representations used by Wiles and Taylor and others. In this appendix to Dieulefait's paper, Automorphy of Symm5(GL(2)) and base change, we show that certain representations of SL(2,q) are adequate. This is used by Dieulefait to prove results about Hecke eigenforms of level 1 and newforms. We also prove some general results about adequacy for representations of finite groups of Lie type in the natural characteristic.Comment: This is appendix A to Dieulefait's paper, arXiv:1208.3946, mentioned in the abstrac

On the singular value decomposition over finite fields and orbits of GU x GU

The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU(m,q) x GU(n,q) on n by n matrices (which is the analog of the singular value decomposition). The proof involves Kronecker's theory of pencils and the Lang-Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick, Larsen and Tiep where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form AB where B is either the transpose of A or the conjugate transpose.Comment: 12 pages, second version has minor change

Average dimension of fixed point spaces with applications

Let $G$ be a finite group, $F$ a field, and $V$ a finite dimensional $FG$-module such that $G$ has no trivial composition factor on $V$. Then the arithmetic average dimension of the fixed point spaces of elements of $G$ on $V$ is at most $(1/p) \dim V$ where $p$ is the smallest prime divisor of the order of $G$. This answers and generalizes a 1966 conjecture of Neumann which also appeared in a paper of Neumann and Vaughan-Lee and also as a problem in The Kourovka Notebook posted by Vaughan-Lee. Our result also generalizes a recent theorem of Isaacs, Keller, Meierfrankenfeld, and Moret\'o. Various applications are given. For example, another conjecture of Neumann and Vaughan-Lee is proven and some results of Segal and Shalev are improved and/or generalized concerning BFC groups

On isometry groups of self-adjoint traceless and skew-symmetric matrices

This note is concerned with isometries on the spaces of self-adjoint traceless matrices. We compute the group of isometries with respect to any unitary similarity invariant norm. This completes and extends the result of Nagy on Schatten $p$-norm isometries. Furthermore, we point out that our proof techniques could be applied to obtain an old result concerning isometries on skew-symmetric matrices

On the non-coprime k(GV) problem

Let V be a finite faithful completely reducible FG-module for a finite field F and a finite group G. In various cases explicit linear bounds in |V| are given for the numbers of conjugacy classes k(GV) and k(G) of the semidirect product GV and of the group G respectively. These results concern the so-called non-coprime k(GV)-problem.Comment: 26 page

Spinors and essential dimension

We prove that spin groups act generically freely on various spinor modules, in the sense of group schemes and in a way that does not depend on the characteristic of the base field. As a consequence, we extend the surprising calculation of the essential dimension of spin groups and half-spin groups in characteristic zero by Brosnan--Reichstein--Vistoli (Annals of Math., 2010) and Chernousov--Merkurjev (Algebra & Number Theory, 2014) to fields of characteristic different from 2.Comment: v2 adds an appendix by Alexander Premet on generic stabilizer in HSpin_16 in characteristic

Polynomials with PSL(2) monodromy

Let K be a field of characteristic p>0, and let q be a power of p. We determine all polynomials f in K[t]\K[t^p] of degree q(q-1)/2 such that the Galois group of f(t)-u over K(u) has a transitive normal subgroup isomorphic to PSL_2(q), subject to a certain ramification hypothesis. As a consequence, we describe all polynomials f in K[t], of degree not a power of p, such that f is functionally indecomposable over K but f decomposes over an extension of K. Moreover, except for one ramification setup (which is treated in the companion paper arxiv:0707.1837), we describe all indecomposable polynomials f in K[t] of non-p-power degree which are exceptional, in the sense that x-y is the only absolutely irreducible factor of f(x)-f(y) which lies in K[x,y]. It is known that, when K is finite, a polynomial f is exceptional if and only if it induces a bijection on infinitely many finite extensions of K.Comment: 44 pages; changed notation throughout and made various minor change

Decompositions of Small Tensor Powers and Larsen's Conjecture

We classify all pairs (G,V) with G a closed subgroup in a classical group with natural module V over the complex numbers such that G has the same composition factors on the kth tensor power of V, for a fixed (small) k. In particular, we prove Larsen's conjecture stating that for dim(V) > 6 and k = 4, there are no such G aside from those containing the derived subgroup of the classical group. We also find all the examples where this fails for dim(V) < 7. As a consequence of our results, we obtain a short proof of a related conjecture of Katz. These conjectures are used in Katz's recent works on monodromy groups attached to Lefschetz pencils and to character sums over finite fields. Modular versions of these conjectures are also studied, with a particular application to random generation in finite groups of Lie type.Comment: 83 pages, to appear in Representation Theor

Sectional rank and Cohomology

We prove that there is a bound on the dimension of the first cohomology group of a finite group with coefficients in an absolutely irreducible in characteristic p in terms of the sectional p-rank of the group

Permutation representations of nonsplit extensions involving alternating groups

L. Babai has shown that a faithful permutation representation of a nonsplit extension of a group by an alternating group $A_k$ must have degree at least $k^2(\frac{1}{2}-o(1))$, and has asked how sharp this lower bound is. We prove that Babai's bound is sharp (up to a constant factor), by showing that there are such nonsplit extensions that have faithful permutation representations of degree $\frac{3}{2}k(k-1)$. We also reprove Babai's quadratic lower bound with the constant $\frac{1}{2}$ improved to 1 (by completely different methods)