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    Ivanov AA, Franchi C, Mainardis M,

    The 2A Majorana representations of the Harada-Norton group.

    , ARS Mathematica Contemporanea, ISSN: 1855-3966

    We show that all 2A-Majorana representations of the Harada-Norton groupF5have the same shape. IfRis such a representation, wedetermine, using the theory of association schemes, the dimension and theirreducible constituents of the linear spanUof the Majorana axes. Finally, weprove that, ifRis based on the (unique) embedding ofF5in the Monster,Uis closed under the algebra product.

    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

    Liebeck MW,

    Character ratios for finite groups of Lie type, and applications

    , Contemporary Mathematics - American Mathematical Society, 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.

    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.

    Liebeck MW, Schul G, Shalev A,

    Rapid growth in finite simple groups

    , Transactions of the American Mathematical Society, 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.

    Schedler TJ, Ginzburg V,

    A new construction of cyclic homology

    , Proceedings of the London Mathematical Society
    Franchi C, Ivanov AA, Mainardis M, 2017,

    Permutation modules for the symmetric group

    , Proceedings of the American Mathematical Society, Pages: 1-1, ISSN: 0002-9939
    Gonshaw S, Liebeck MW, O'Brien EA, 2017,

    Unipotent class representatives for finite classical groups

    , JOURNAL OF GROUP THEORY, Vol: 20, Pages: 505-525, ISSN: 1433-5883
    Proudfoot N, Schedler T, 2017,

    Poisson–de Rham homology of hypertoric varieties and nilpotent cones

    , Selecta Mathematica, Vol: 23, Pages: 179-202, ISSN: 1022-1824
    Evans DM, Ghadernezhad Z, Tent K, 2016,

    Simplicity of the automorphism groups of some Hrushovski constructions

    , ANNALS OF PURE AND APPLIED LOGIC, Vol: 167, Pages: 22-48, ISSN: 0168-0072

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