442 research outputs found

### 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

### Permutation resolutions for Specht modules of Hecke algebras

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 $p$-permutation equivalence

A perfect isometry $I$ (introduced by Brou\'e) between two blocks $B$ and $C$
is a frequent phenomenon in the block theory of finite groups. It maps an
irreducible character $\psi$ of $C$ to $\pm$ an irreducible character of $B$.
Brou\'e proved that the ratio of the codegrees of $\psi$ and $I(\psi)$ is a
rational number with $p$-value zero and that its class in $\mathbb{F}_p$ is
independent of $\psi$. We call this element the Brou\'e invariant of $I$. The
goal of this paper is to show that if $I$ comes from a $p$-permutation
equivalence or a splendid Rickard equivalence between $B$ and $C$ then, up to a
sign, the Brou\'e invariant of $I$ is determined by local data of $B$ and $C$
and is independent of the $p$-permutation equivalence or splendid Rickard
equivalence. Apart from results on $p$-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

The goal of a series of papers is to define $G$-actions on various
$A$-fibered structures, where $G$ is a finite group and $A$ is an abelian
group. One prominent such example is the $A$-fibered Burnside ring. If
$A=\mathbb{C}^\times$, 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
$A$-fibered structures on posets, on abstract simplicial complexes, and on
$A$-bundles over topological spaces, together with natural notions of homotopy,
and functors between these structures respecting homotopy. In a sequel we will
continue with $G$-representations in these $A$-fibered structures and associate
to them elements in the $A$-fibered Burnside ring

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