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  • 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
    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
    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.

  • Journal article
    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: 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.

  • Journal article
    Ginzburg V, Schedler TJ,

    A new construction of cyclic homology

    , Proceedings of the London Mathematical Society
  • Journal article
    Franchi C, Ivanov AA, Mainardis M, 2015,

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

    , ARS Mathematica Contemporanea, Vol: 11, 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.

  • Journal article
    Amato D, Evans DM, 2015,

    Infinite primitive and distance transitive directed graphs of finite out-valency

    , Journal of Combinatorial Theory Series B, Vol: 114, Pages: 33-50, ISSN: 1096-0902

    We give certain properties which are satisfied by the descendant set of a vertex in an infinite, primitive, distance transitive digraph of finite out-valency and provide a strong structure theory for digraphs satisfying these properties. In particular, we show that there are only countably many possibilities for the isomorphism type of such a descendant set, thereby confirming a conjecture of the first Author. As a partial converse, we show that certain related conditions on a countable digraph are sufficient for it to occur as the descendant set of a primitive, distance transitive digraph.

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