442 research outputs found

    Quasi-hereditary structure of twisted split category algebras revisited

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    Let kk be a field of characteristic 00, let C\mathsf{C} be a finite split category, let α\alpha be a 2-cocycle of C\mathsf{C} with values in the multiplicative group of kk, and consider the resulting twisted category algebra A:=kαCA:=k_\alpha\mathsf{C}. Several interesting algebras arise that way, for instance, the Brauer algebra. Moreover, the category of biset functors over kk is equivalent to a module category over a condensed algebra εAε\varepsilon A\varepsilon, for an idempotent ε\varepsilon of AA. In [2] the authors proved that AA is quasi-hereditary (with respect to an explicit partial order \le on the set of irreducible modules), and standard modules were given explicitly. Here, we improve the partial order \le by introducing a coarser order \unlhd leading to the same results on AA, but which allows to pass the quasi-heredity result to the condensed algebra εAε\varepsilon A\varepsilon describing biset functors, thereby giving a different proof of a quasi-heredity result of Webb, see [26]. The new partial order \unlhd has not been considered before, even in the special cases, and we evaluate it explicitly for the case of biset functors and the Brauer algebra. It also puts further restrictions on the possible composition factors of standard modules.Comment: 39 page

    A ghost algebra of the double Burnside algebra in characteristic zero

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    For a finite group GG, we introduce a multiplication on the \QQ-vector space with basis \scrS_{G\times G}, the set of subgroups of G×GG\times G. The resulting \QQ-algebra \Atilde can be considered as a ghost algebra for the double Burnside ring B(G,G)B(G,G) in the sense that the mark homomorphism from B(G,G)B(G,G) to \Atilde is a ring homomorphism. Our approach interprets \QQ B(G,G) as an algebra eAeeAe, where AA is a twisted monoid algebra and ee is an idempotent in AA. The monoid underlying the algebra AA is again equal to \scrS_{G\times G} with multiplication given by composition of relations (when a subgroup of G×GG\times G is interpreted as a relation between GG and GG). The algebras AA and \Atilde are isomorphic via M\"obius inversion in the poset \scrS_{G\times G}. As an application we improve results by Bouc on the parametrization of simple modules of \QQ B(G,G) and also of simple biset functors, by using results by Linckelmann and Stolorz on the parametrization of simple modules of finite category algebras. Finally, in the case where GG is a cyclic group of order nn, we give an explicit isomorphism between \QQ B(G,G) and a direct product of matrix rings over group algebras of the automorphism groups of cyclic groups of order kk, where kk divides nn.Comment: 41 pages. Changed title from "Ghost algebras of double Burnside algebras via Schur functors" and other minor changes. Final versio

    Permutation resolutions for Specht modules of Hecke algebras

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    In [Boltje,Hartmann: Permutation resolutions for Specht modules, J. Algebraic Combin. 34 (2011), 141-162], a chain complex was constructed in a combinatorial way which conjecturally is a resolution of the (dual of the) integral Specht module for the symmetric group in terms of permutation modules. In this paper we extend the definition of the chain complex to the integral Iwahori Hecke algebra and prove the same partial exactness results that were proved in the symmetric group case. A complete proof of the exactness conjecture in the symmetric group case was recently given by Santana and Yudin, Adv. in Math. 229 (2012), 2578-2601.Comment: 23 pages, fixed mistake in proof of Prop. 2.2, other minor changes, to appear in Journal of Algebr

    The Brou\'e invariant of a pp-permutation equivalence

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    A perfect isometry II (introduced by Brou\'e) between two blocks BB and CC is a frequent phenomenon in the block theory of finite groups. It maps an irreducible character ψ\psi of CC to ±\pm an irreducible character of BB. Brou\'e proved that the ratio of the codegrees of ψ\psi and I(ψ)I(\psi) is a rational number with pp-value zero and that its class in Fp\mathbb{F}_p is independent of ψ\psi. We call this element the Brou\'e invariant of II. The goal of this paper is to show that if II comes from a pp-permutation equivalence or a splendid Rickard equivalence between BB and CC then, up to a sign, the Brou\'e invariant of II is determined by local data of BB and CC and is independent of the pp-permutation equivalence or splendid Rickard equivalence. Apart from results on pp-permutation equivalences, our proof requires new results on extended tensor products and bisets that are also proved in this paper.Comment: 14 page

    Monomial structures, I

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    The goal of a series of papers is to define GG-actions on various AA-fibered structures, where GG is a finite group and AA is an abelian group. One prominent such example is the AA-fibered Burnside ring. If A=C×A=\mathbb{C}^\times, it is also called the ring of monomial representations (introduced by Dress in \cite{Dress1971}) and is the natural home for the canonical induction formula (see \cite{Boltje1990}). In this first part of the series, motivated by constructions in \cite{BoucMutlu}, we introduce AA-fibered structures on posets, on abstract simplicial complexes, and on AA-bundles over topological spaces, together with natural notions of homotopy, and functors between these structures respecting homotopy. In a sequel we will continue with GG-representations in these AA-fibered structures and associate to them elements in the AA-fibered Burnside ring