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Journal articleLiebeck MW, Praeger CE, Saxl J, 2019, , Proceedings of the American Mathematical Society, Vol: 147, Pages: 5023-5037, ISSN: 1088-6826
A linear group G ≤ GL(V ), where V is a finite vector space, is called 12-transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the 12-transitive linear groups. As a consequence we complete the determination of the finite 32-transitive permutation groups – the transitive groups for which a point-stabilizerhas all its nontrivial orbits of the same size. We also determine the (k +12)-transitive groups for integers k ≥ 2.
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Journal articleLiebeck MW, Schul G, Shalev A, 2017, , Transactions of the American Mathematical Society, Vol: 369, Pages: 2765-8779, ISSN: 1088-6850
We show that small normal subsets A of finite simple groups growvery rapidly – namely, |A2| ≥ |A|2− , where > 0 is arbitrarily small.Extensions, consequences, and a rapid growth result for simple algebraicgroups are also given.
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Journal articleLiebeck MW, 2017, , Contemporary Mathematics, Vol: 694, ISSN: 0271-4132
For a nite groupG, acharacter ratiois a complex number of the form (x) (1),wherex2Gand is an irreducible character ofG. Upper bounds for absolutevalues of character ratios, particularly for simple groups, have long been of interest,for various reasons; these include applications to covering numbers, mixing timesof random walks, and the study of word maps. In this article we shall survey someresults on character ratios for nite groups of Lie type, and their applications.Character ratios for alternating and symmetric groups have been studied in greatdepth also { see for example [32, 33] { culminating in the de nitive results andapplications to be found in [20]; but we shall not discuss these here.It is not hard to see the connections between character ratios and group struc-ture. Here are three well known, elementary results illustrating these connections.The rst two go back to Frobenius. Denote by Irr(G) the set of irreducible charac-ters ofG.
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Journal articleGonshaw S, Liebeck MW, O'Brien E, 2016, , Journal of Group Theory, Vol: 20, Pages: 505-525, ISSN: 1435-4446
We describe explicitly representatives of the conjugacy classes ofunipotent elements of the finite classical groups.
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Journal articleSchedler TJ, Proudfoot NJ, 2016, , Selecta Mathematica, Vol: 23, Pages: 179-202, ISSN: 1022-1824
We prove a conjecture of Etingof and the second author for hypertoric varieties that the Poisson–de Rham homology of a unimodular hypertoric cone is isomorphic to the de Rham cohomology of its hypertoric resolution. More generally, we prove that this conjecture holds for an arbitrary conical variety admitting a symplectic resolution if and only if it holds in degree zero for all normal slices to symplectic leaves. The Poisson–de Rham homology of a Poisson cone inherits a second grading. In the hypertoric case, we compute the resulting 2-variable Poisson–de Rham–Poincaré polynomial and prove that it is equal to a specialization of an enrichment of the Tutte polynomial of a matroid that was introduced by Denham (J Algebra 242(1):160–175, 2001). We also compute this polynomial for S3-varieties of type A in terms of Kostka polynomials, modulo a previous conjecture of the first author, and we give a conjectural answer for nilpotent cones in arbitrary type, which we prove in rank less than or equal to 2.
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Journal articleGinzburg V, Schedler TJ, 2016, , Proceedings of the London Mathematical Society, Vol: 112, Pages: 549-587, ISSN: 0024-6115
Based on the ideas of Cuntz and Quillen, we give a simple construction of cyclic homology of unital algebras in terms of the noncommutative de Rham complex and a certain differential similar to the equivariant de Rham differential. We describe the Connes exact sequence in this setting. We define equivariant Deligne cohomology and construct, for each 饾憶猢1 , a natural map from cyclic homology of an algebra to the GL饾憶 鈥恊quivariant Deligne cohomology of the variety of 饾憶 鈥恉imensional representations of that algebra. The bridge between cyclic homology and equivariant Deligne cohomology is provided by extended cyclic homology, which we define and compute here, based on the extended noncommutative de Rham complex introduced previously by the authors.
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Journal articleEvans DM, Ghadernezhad Z, Tent K, 2016, , Annals of Pure and Applied Logic, Vol: 167, Pages: 22-48, ISSN: 1873-2461
We show that the automorphism groups of certain countable structures obtained using the Hrushovski amalgamation method are simple groups. The structures we consider are the ‘uncollapsed’ structures of infinite Morley rank obtained by the ab initio construction and the (unstable) ℵ0-categorical pseudoplanes. The simplicity of the automorphism groups of these follows from results which generalize work of Lascar and of Tent and Ziegler.
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Journal articleEvans DM, Tsankov T, 2016, , Fundamenta Mathematicae, Vol: 232, Pages: 49-63, ISSN: 1730-6329
We show that if G is a non-archimedean, Roelcke precompact Polish group, then G has Kazhdan's property (T). Moreover, if G has a smallest open subgroup of finite index, then G has a finite Kazhdan set. Examples of such G include automorphism groups of countable ω-categorical structures, that is, the closed, oligomorphic permutation groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation groups. The latter allows the construction of a non-abelian free subgroup of G acting freely in all infinite transitive permutation representations of G.
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Journal articleLiebeck MW, Seitz GM, Testerman DM, 2015, , Pacific Journal of Mathematics, Vol: 279, Pages: 357-382, ISSN: 0030-8730
For G a simple algebraic group over an algebraically closed field of characteristic 0, we determine the irreducible representations ρ:G→I(V), where I(V) denotes one of the classical groups SL(V), Sp(V), SO(V), such that ρ sends some distinguished unipotent element of G to a distinguished element of I(V). We also settle a base case of the general problem of determining when the restriction of ρ to a simple subgroup of G is multiplicity-free.
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Journal articleLiebeck MW, O'Brien EA, 2015, , Transactions of the American Mathematical Society, Vol: 368, Pages: 6189-6226, ISSN: 1088-6850
Let q be a prime power and let G be an absolutely irreducible subgroup ofGLd(F), where F is a finite field of the same characteristic as Fq, the field of q elements. Assume that G ∼= G(q), a quasisimple group of exceptional Lie type over Fq which is neither a Suzuki nor a Ree group. We present a Las Vegas algorithm that constructs an isomorphism from G to the standard copy of G(q). If G 6∼=3D4(q) with q even, then the algorithm runs in polynomial time, subject to the existence of a discrete log oracle.
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