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  • Journal article
    Liebeck MW, Schul G, Shalev A, 2017,

    Rapid growth in finite simple groups

    , 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.
  • Journal article
    Franchi C, Ivanov AA, Mainardis M, 2017,

    Permutation modules for the symmetric group

    , Proceedings of the American Mathematical Society, Vol: 145, Pages: 3249-3262, ISSN: 0002-9939

    In this paper we present a general method for computing the irreducible components of the permutation modules of the symmetric groups over a field $ F$ of characteristic 0. We apply this machinery to determine the decomposition into irreducible submodules of the $ F[S_n]$-permutation module on the right cosets of the normaliser in $ S_n$ of the subgroup generated by a permutation of type $ (3,3)$.
  • Journal article
    Liebeck MW,

    Character ratios for finite groups of Lie type, and applications

    , 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.
  • Journal article
    Gonshaw S, Liebeck MW, O'Brien E, 2016,

    Unipotent class representatives for finite classical groups

    , 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.
  • Journal article
    Ivanov AA, Franchi C, Mainardis M,

    Standard majorana representations of the symmetric groups

    , Journal of Algebraic Combinatorics, ISSN: 1572-9192

    LetGbe a nite group and letWbe a nitely generatedRG-module with a positive de nite bilinear form (;)W. Assume thatGpermutestransitively a generating setXofWand that (;)Wis constant on eachorbital ofGonX. We show a new method for computing the dimensions ofthe irreducible constituents ofW. Further, we apply that method to Majoranarepresentations of the symmetric groups proving that the symmetric groupSnhas a Majorana representation, in which every permutation of type (2;2) ofSncorresponds to a Majorana axis, if and only ifn≤12
  • Journal article
    Giudici M, Ivanov AA, Morgan L, Praeger CEet al., 2016,

    A characterisation of weakly locally projective amalgams related to A16 and the sporadic simple groups M24 and He

    , Journal of Algebra, Vol: 460, Pages: 340-365, ISSN: 0021-8693

    A simple undirected graph is weakly G-locally projective, for a group of automorphisms G, if for each vertex x , the stabiliser G(x) induces on the set of vertices adjacent to x a doubly transitive action with socle the projective group Lnx(qx) for an integer nx and a prime power qx. It is G-locally projective if in addition G is vertex transitive. A theorem of Trofimov reduces the classification of the G -locally projective graphs to the case where the distance factors are as in one of the known examples. Although an analogue of Trofimov's result is not yet available for weakly locally projective graphs, we would like to begin a program of characterising some of the remarkable examples. We show that if a graph is weakly locally projective with each qx=2 and nx=2 or 3, and if the distance factors are as in the examples arising from the rank 3 tilde geometries of the groups M24 and He , then up to isomorphism there are exactly two possible amalgams. Moreover, we consider an infinite family of amalgams of type Un (where each qx=2 and n=nx+1≥4) and prove that if n≥5 there is a unique amalgam of type Un and it is unfaithful, whereas if n=4 then there are exactly four amalgams of type U4, precisely two of which are faithful, namely the ones related to M24 and He , and one other which has faithful completion A16.
  • Journal article
    Liebeck MW, Praeger CE, Saxl J,

    The classification of 3/2-transitive permutation groups and 1/2-transitive linear groups

    , Proceedings of the American Mathematical Society, ISSN: 1088-6826

    A linear group G ≤ GL(V ), where V is a finite vector space, is called 12-transitive if allthe G-orbits on the set of nonzero vectors have the same size. We complete the classificationof all the 12-transitive linear groups. As a consequence we complete the determination of thefinite 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 groupsfor integers k ≥ 2.
  • Journal article
    Schedler TJ, Proudfoot NJ, 2016,

    Poisson–de Rham homology of hypertoric varieties and nilpotent cones

    , 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.
  • Journal article
    Ivanov AA, Franchi C, Mainardis M, 2016,

    Standard Majorana representations of the symmetric groups.

    , Journal of Algebric Combinations, ISSN: 1572-9192

    Let G be a finite group, W be a R[G]-module equipped with a G-invariant positive definite bilinear form (,)W, and X a finite generating set of W such that X is transitively permuted by G. We show a new method for computing the dimensions of the irreducible constituents of W. Further, we apply this method to Majorana representations of the symmetric groups and prove that the symmetric group Sn has a Majorana representation in which every permutation of type (2, 2) of Sn corresponds to a Majorana axis if and only if n≤12.
  • Journal article
    Evans DM, Tsankov T, 2016,

    Free actions of free groups on countable structures and property (T)

    , 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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