142 research outputs found
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
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
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 is coprime to a prime
if and only if the Sylow -subgroups of 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 . In this follow-up paper, we
obtain a full improvement for all primes.Comment: 19 page
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
Exponential sums and total Weil representations of finite symplectic and unitary groups
We construct explicit local systems on the affine line in characteristic
, whose geometric monodromy groups are the finite symplectic groups
for all , and others whose geometric monodromy groups are
the special unitary groups for all odd , and any power
of , 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 , whose geometric monodromy groups are
the finite symplectic groups for any , and others whose
geometric monodromy groups are the finite general unitary groups for
any odd .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
which act irreducibly on some symmetric power with . 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
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 of a finite classical group in any nontrivial
irreducible cross characteristic representation. With a few explicit
exceptions, this degree is at least .Comment: 57 pages. Proc. London Math. Soc., to appea
Cross characteristic representations of are Reducible over proper subgroups
We prove that the restriction of any absolutely irreducible representation of
Steinberg's triality groups in characteristic coprime to q to any
proper subgroup is reducibleComment: 12 pages; with an appendix by Frank Himsted
Irreducible characters of even degree and normal Sylow -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 is coprime to a given prime , then has a normal Sylow
-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 is less
than then has a normal Sylow -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
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