A Refined Waring Problem for Finite Simple Groups

Let v and w be nontrivial words in two free groups. We prove that, for all sufficiently large finite non-abelian simple groups G, there exist subsets C of v(G) and D of w(G) of size such that every element of G can be realized in at least one way as the product of an element of C and an element of D and the average number of such representations is O(log |G|). In particular, if w is a fixed nontrivial word and G is a sufficiently large finite non-abelian simple group, then w(G) contains a thin base of order 2. This is a non-abelian analogue of a result of Van Vu for the classical Waring problem. Further results concerning thin bases of G of order 2 are established for any finite group and for any compact Lie group G.Comment: 20 page

A problem of Kollar and Larsen on finite linear groups and crepant resolutions

The notion of age of elements of complex linear groups was introduced by M. Reid and is of importance in algebraic geometry, in particular in the study of crepant resolutions and of quotients of Calabi-Yau varieties. In this paper, we solve a problem raised by J. Kollar and M. Larsen on the structure of finite irreducible linear groups generated by elements of age at most 1. More generally, we bound the dimension of finite irreducible linear groups generated by elements of bounded deviation. As a consequence of our main results, we derive some properties of symmetric spaces for the unitayr group having shortest closed geodesics of bounded length, and of quotients of affine space by a finite group having a crepant resolution.Comment: 55 page

Exponential sums and total Weil representations of finite symplectic and unitary groups

We construct explicit local systems on the affine line in characteristic $p>2$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for all $n \ge 2$, and others whose geometric monodromy groups are the special unitary groups $SU_n(q)$ for all odd $n \ge 3$, and $q$ any power of $p$, in their total Weil representations. One principal merit of these local systems is that their associated trace functions are one-parameter families of exponential sums of a very simple, i.e., easy to remember, form. We also exhibit hypergeometric sheaves on $G_m$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for any $n \ge 2$, and others whose geometric monodromy groups are the finite general unitary groups $GU_n(q)$ for any odd $n \geq 3$.Comment: 56 page

Symmetric powers and a problem of Kollar and Larsen

We prove a conjecture of Kollar and Larsen on Zariski closed subgroups of $GL(V)$ which act irreducibly on some symmetric power $Sym^{k}(V)$ with $k \geq 4$. This conjecture has interesting implications, in particular on the holonomy group of a stable vector bundle on a smooth projective variety, as shown by the recent work of Balaji and Kollar.Comment: 49 pages. Inventiones Mathematicae, to appea

Cross characteristic representations of 3D4(q)^3D_4(q) are Reducible over proper subgroups

We prove that the restriction of any absolutely irreducible representation of Steinberg's triality groups $^3D_4(q)$ in characteristic coprime to q to any proper subgroup is reducibleComment: 12 pages; with an appendix by Frank Himsted

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

Hall-Higman type theorems for semisimple elements of finite classical groups

We prove an analogue of the celebrated Hall-Higman theorem, which gives a lower bound for the degree of the minimal polynomial of any semisimple element of prime power order $p^{a}$ of a finite classical group in any nontrivial irreducible cross characteristic representation. With a few explicit exceptions, this degree is at least $p^{a-1}(p-1)$.Comment: 57 pages. Proc. London Math. Soc., to appea

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

The average character degree and an improvement of the Ito-Michler theorem

The classical It\^{o}-Michler theorem states that the degree of every ordinary irreducible character of a finite group $G$ is coprime to a prime $p$ if and only if the Sylow $p$-subgroups of $G$ are abelian and normal. In an earlier paper, we used the notion of average character degree to prove an improvement of this theorem for the prime $p=2$. In this follow-up paper, we obtain a full improvement for all primes.Comment: 19 page

Irreducible characters of even degree and normal Sylow 22-subgroups

The classical It\^o-Michler theorem on character degrees of finite groups asserts that if the degree of every complex irreducible character of a finite group $G$ is coprime to a given prime $p$, then $G$ has a normal Sylow $p$-subgroup. We propose a new direction to generalize this theorem by introducing an invariant concerning character degrees. We show that if the average degree of linear and even-degree irreducible characters of $G$ is less than $4/3$ then $G$ has a normal Sylow $2$-subgroup, as well as corresponding analogues for real-valued characters and strongly real characters. These results improve on several earlier results concerning the It\^o-Michler theorem.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1506.0645