Vertices of simple modules of symmetric groups labelled by hook partitions

In this article we study the vertices of simple modules for the symmetric groups in prime characteristic $p$. In particular, we complete the classification of the vertices of simple $S_n$-modules labelled by hook partitions

Quasi-hereditary structure of twisted split category algebras revisited

Let $k$ be a field of characteristic $0$, let $\mathsf{C}$ be a finite split category, let $\alpha$ be a 2-cocycle of $\mathsf{C}$ with values in the multiplicative group of $k$, and consider the resulting twisted category algebra $A:=k_\alpha\mathsf{C}$. Several interesting algebras arise that way, for instance, the Brauer algebra. Moreover, the category of biset functors over $k$ is equivalent to a module category over a condensed algebra $\varepsilon A\varepsilon$, for an idempotent $\varepsilon$ of $A$. In [2] the authors proved that $A$ 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 $A$, but which allows to pass the quasi-heredity result to the condensed algebra $\varepsilon A\varepsilon$ 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

For a finite group $G$, we introduce a multiplication on the \QQ-vector space with basis \scrS_{G\times G}, the set of subgroups of $G\times G$. The resulting \QQ-algebra \Atilde can be considered as a ghost algebra for the double Burnside ring $B(G,G)$ in the sense that the mark homomorphism from $B(G,G)$ to \Atilde is a ring homomorphism. Our approach interprets \QQ B(G,G) as an algebra $eAe$, where $A$ is a twisted monoid algebra and $e$ is an idempotent in $A$. The monoid underlying the algebra $A$ is again equal to \scrS_{G\times G} with multiplication given by composition of relations (when a subgroup of $G\times G$ is interpreted as a relation between $G$ and $G$). The algebras $A$ 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 $G$ is a cyclic group of order $n$, 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 $k$, where $k$ divides $n$.Comment: 41 pages. Changed title from "Ghost algebras of double Burnside algebras via Schur functors" and other minor changes. Final versio

Signed Young Modules and Simple Specht Modules

By a result of Hemmer, every simple Specht module of a finite symmetric group over a field of odd characteristic is a signed Young module. While Specht modules are parametrized by partitions, indecomposable signed Young modules are parametrized by certain pairs of partitions. The main result of this article establishes the signed Young module labels of simple Specht modules. Along the way we prove a number of results concerning indecomposable signed Young modules that are of independent interest. In particular, we determine the label of the indecomposable signed Young module obtained by tensoring a given indecomposable signed Young module with the sign representation. As consequences, we obtain the Green vertices, Green correspondents, cohomological varieties, and complexities of all simple Specht modules and a class of simple modules of symmetric groups, and extend the results of Gill on periodic Young modules to periodic indecomposable signed Young modules.Comment: To appear in Adv. Math. 307 (2017) 369--416. Proposition 4.3 (F4), (F5) corrected, Lemma 4.9 adjusted accordingl

Source Algebras of Blocks, Sources of Simple Modules, and a Conjecture of Feit

We verify a finiteness conjecture of Feit on sources of simple modules over group algebras for various classes of finite groups related to the symmetric groups.Comment: 28 pages; minor corrections; added Section

Vertices of Lie Modules

Let Lie(n) be the Lie module of the symmetric group S_n over a field F of characteristic p>0, that is, Lie(n) is the left ideal of FS_n generated by the Dynkin-Specht-Wever element. We study the problem of parametrizing non-projective indecomposable summands of Lie(n), via describing their vertices and sources. Our main result shows that this can be reduced to the case when n is a power of p. When n=9 and p=3, and when n=8 and p=2, we present a precise answer. This suggests a possible parametrization for arbitrary prime powers.Comment: 26 page

Theoretische und algorithmische Methoden zur Berechnung von Vertizes irreduzibler Moduln symmetrischer Gruppen

Diese Arbeit befasst sich mit der Bestimmung von Vertizes und Quellen irreduzibler Moduln endlicher symmetrischer Gruppen. Dabei werden sowohl theoretische als auch algorithmische Methoden zur Vertexberechnung entwickelt und auf verschiedene Klassen von irreduziblen Moduln symmetrischer Gruppen angewandt. Zu diesen zählen beispielsweise gewisse äußere Potenzen des natürlichen irreduziblen Moduls in gerader Charakteristik, die sogenannten vollständig zerfallenden Moduln sowie die verallgemeinerten Young-Moduln. Ferner wurden mit einer Ausnahme auch die Vertizes aller irreduziblen Moduln symmetrischer Gruppen in 2-Blöcken vom Gewicht < 5 beziehungsweise in 3-Blöcken vom Gewicht < 4 bestimmt

The Centralizer of a subgroup in a group algebra

If R is a commutative ring, G is a nite group, and H is a subgroup of G, then the centralizer algebra RGH is the set of all elements of RG that commute with all elements of H. The algebra RGH is a Hecke algebra in the sense that it is isomorphic to EndRHG(RG) = EndRHG(1H HG). The authors have been studying the representation theory of these algebras in several recent and not so recent papers [4], [5], [6], [7], [10], [11], mainly in cases where G is p-solvable and H is normal, or when G = Sn and H = Sm for

The Centralizer of a subgroup in a group algebra

If R is a commutative ring, G is a nite group, and H is a subgroup of G, then the centralizer algebra RGH is the set of all elements of RG that commute with all elements of H. The algebra RGH is a Hecke algebra in the sense that it is isomorphic to EndRHG(RG) = EndRHG(1H HG). The authors have been studying the representation theory of these algebras in several recent and not so recent papers [4], [5], [6], [7], [10], [11], mainly in cases where G is p-solvable and H is normal, or when G = Sn and H = Sm for