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Journal articleLiebeck MW, Praeger CE, Saxl J, 2019,
The classification of 3/2transitive permutation groups and 1/2transitive linear groups
, Proceedings of the American Mathematical Society, Vol: 147, Pages: 50235037, ISSN: 10886826A linear group G ≤ GL(V ), where V is a finite vector space, is called 12transitive if all the Gorbits on the set of nonzero vectors have the same size. We complete the classification of all the 12transitive linear groups. As a consequence we complete the determination of the finite 32transitive permutation groups – the transitive groups for which a pointstabilizerhas all its nontrivial orbits of the same size. We also determine the (k +12)transitive groups for integers k ≥ 2.

Journal articleLiebeck MW, Schul G, Shalev A, 2017,
Rapid growth in finite simple groups
, Transactions of the American Mathematical Society, Vol: 369, Pages: 27658779, ISSN: 10886850We show that small normal subsets A of finite simple groups growvery rapidly – namely, A2 ≥ A2− , where > 0 is arbitrarily small.Extensions, consequences, and a rapid growth result for simple algebraicgroups are also given.

Journal articleFranchi C, Ivanov AA, Mainardis M, 2017,
Permutation modules for the symmetric group
, Proceedings of the American Mathematical Society, Vol: 145, Pages: 32493262, ISSN: 00029939In 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 articleLiebeck MW, 2017,
Character ratios for finite groups of Lie type, and applications
, Contemporary Mathematics, Vol: 694, ISSN: 02714132For 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 structure. 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 characters ofG.

Journal articleGonshaw S, Liebeck MW, O'Brien E, 2016,
Unipotent class representatives for finite classical groups
, Journal of Group Theory, Vol: 20, Pages: 505525, ISSN: 14354446We describe explicitly representatives of the conjugacy classes ofunipotent elements of the finite classical groups.

Journal articleIvanov AA, Franchi C, Mainardis M,
Standard majorana representations of the symmetric groups
, Journal of Algebraic Combinatorics, ISSN: 15729192LetGbe a nite group and letWbe a nitely generatedRGmodule 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 articleIvanov AA, Franchi C, Mainardis M, 2016,
Standard Majorana representations of the symmetric groups
, Journal of Algebric Combinations, Vol: 44, Pages: 265292, ISSN: 15729192Let G be a finite group, W be a R[G]module equipped with a Ginvariant 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 articleGiudici M, Ivanov AA, Morgan L, et 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: 340365, ISSN: 00218693A simple undirected graph is weakly Glocally 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 Glocally 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 articleSchedler TJ, Proudfoot NJ, 2016,
Poisson–de Rham homology of hypertoric varieties and nilpotent cones
, Selecta Mathematica, Vol: 23, Pages: 179202, ISSN: 10221824We 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 2variable 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 S3varieties 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 articleGinzburg V, Schedler TJ, 2016,
A new construction of cyclic homology
, Proceedings of the London Mathematical Society, Vol: 112, Pages: 549587, ISSN: 00246115Based 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 𝑛⩾1 , a natural map from cyclic homology of an algebra to the GL𝑛 ‐equivariant Deligne cohomology of the variety of 𝑛 ‐dimensional 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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