68 research outputs found
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 . In particular, we complete the
classification of the vertices of simple -modules labelled by hook
partitions
Quasi-hereditary structure of twisted split category algebras revisited
Let be a field of characteristic , let be a finite split
category, let be a 2-cocycle of with values in the
multiplicative group of , and consider the resulting twisted category
algebra . Several interesting algebras arise that way,
for instance, the Brauer algebra. Moreover, the category of biset functors over
is equivalent to a module category over a condensed algebra , for an idempotent of . In [2] the authors
proved that is quasi-hereditary (with respect to an explicit partial order
on the set of irreducible modules), and standard modules were given
explicitly. Here, we improve the partial order by introducing a coarser
order leading to the same results on , but which allows to pass the
quasi-heredity result to the condensed algebra
describing biset functors, thereby giving a different proof of a quasi-heredity
result of Webb, see [26]. The new partial order 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 , we introduce a multiplication on the \QQ-vector
space with basis \scrS_{G\times G}, the set of subgroups of . The
resulting \QQ-algebra \Atilde can be considered as a ghost algebra for the
double Burnside ring in the sense that the mark homomorphism from
to \Atilde is a ring homomorphism. Our approach interprets \QQ
B(G,G) as an algebra , where is a twisted monoid algebra and is
an idempotent in . The monoid underlying the algebra is again equal to
\scrS_{G\times G} with multiplication given by composition of relations (when
a subgroup of is interpreted as a relation between and ).
The algebras 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 is a
cyclic group of order , 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 , where divides .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
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