7.11.2013, 15:30-16:30 Philipp Perepelitsky: $p$-permutation equivalences between blocks of finite groups

In this talk we describe joint work with Robert Boltje. Let F be an algebraically closed eld of positive characteristic p: Let $G$ and $H$ be nite groups. Let A be a block of $FG$ and let B be a block of $FH$: A $p$-permutation equivalence between $A$ and $B$ is an element  in the group of $(A,B)$-$p$-permutation bimodules with twisted diagonal vertices such that 
$\gamma \cdot_H\gamma^\circ = [A]$ and  $\gamma^\circ \cdot_G \gamma  = [B]$. A $p$-permutation equivalence lies between a splendid Rickard equivalence and an isotypy.
We introduce the notion of a $\gamma$-Brauer pair, which generalizes the notion of a Brauer pair for a $p$-block of a finite group. The $\gamma$-Brauer pairs satisfy an appropriate Sylow theorem. Furthermore, each maximal $\gamma$-Brauer pair identies the defect groups, fusion systems and Külshammer-Puig classes of $A$ and $B$: Additionally, the Brauer construction applied to $\gamma$  induces a $p$-permutation equivalence at the local level, and a splendid Morita equivalence between the Brauer correspondents of $A$ and $B$.

14.11.2013, 15:30-16:30 Iulian Simion: Refining the Bruhat decomposition
Following Shimura we will describe Hecke rings. We then specialize to the case of Iwahori-Hecke algebras and consider their structure constants for standard basis elements. A result of Kawanaka describes these constants by means of certain subexpression for elements of the Weyl group. Curtis reinterpreted this result and generalized results of Deodhar related to a refinement of the Bruhat decomposition. We show how, in some particular cases, this point of view allows for a simple description of the structure constants.

21.11.2013 no talk (Baer-Kolloquium, 23.11.2013)

28.11.2013, 15:30-16:30 Fuat Erdem: On the generating graph of a finite group

The generating graph of a finite group $G$, denoted by $\Gamma(G)$, is the graph on the non-identity elements of $G$ in which two distinct vertices are joined by an edge if and only if they generate $G$. An important notion in the study of graphs is that of a Hamiltonian cycle. A Hamiltonian cycle in an undirected simple graph is a cycle that visits every vertex exactly once. We will first mention several aspects of the generating graph and then raise the subject to a more recent question, namely that, for which finite groups $G$ does there exist a Hamiltonian cycle in $\Gamma(G)$? We will mainly discuss the case of the symmetric group.

 5.12.2013, no talk (Darstellungstheorietage und Nikolauskonferenz 2013)

12.12.2013 16:00-17:00 Jürgen Müller: The Abelian Defect Group Conjecture for sporadic simple groups

Let G be a finite group, let A be a prime block of G having an abelian defect group P, let N be the normaliser in G of P, and let B be the Brauer correspondent of A. Then the Abelian Defect Group Donjecture (in various forms due to M. Broue and J. Rickard) says that the bounded derived categories of the module categories of A and B are equivalent as triangulated categories. Although being in the focus of intensive study since two decades now, and considerable progress has been made, a general proof seems to be out of sight.

A possible strategy is a reduction to the simple (and closely related) groups. Hence it seems worth-while to tackle these groups first. In view of this, in this talk we show how a combination of theoretical strategies and techniques from computational representation theory can be pursued to prove the Abelian Defect Group Conjecture for some of the sporadic simple groups. This is joint work with Shigeo Koshitani and Felix Noeske.

19.12.2013 15:30-16:30 Olivier Dudas: Ordering conjugacy classes in Weyl groups

Conjugacy classes in a Weyl group or more generally a Coxeter group do not behave well with respect to the Bruhat order. This difficulty can be overcome by looking at elements of minimal length in a given conjugacy class. I will explain how the order that is induced on these elements should reflect the closure relations on unipotent classes of the corresponding reductive group.

 9.1.2014 15:30-16:30 Caroline Lassueur: The position of endo-$p$-permutation modules and relatives in the Auslander-Reiten quiver

16.1.2014 17:00-18:00 (Room 48-436)Serge Bouc: The poset of posets

In this joint work with Jacques Thévenaz, we study the structure of the algebra $E(X)$ of essential relations on a finite set $X$: this algebra (which doesn't seem to have been considered so far) is the quotient of the algebra of the monoid of all relations on $X$ by the submodule generated by relations which factor through a (strictly) smaller set. In particular, we determine all the simple $E(X)$-modules. These are parametrized in particular by order relations on $X$. This motivates further study of combinatorial and topological properties of the "poset of posets" on $X$.

23.1.2014 15:30-16:30 Elisabeth Schulte: Alperin's Weight Conjecture - On Inductive Conditions for $G_2(q)$

In my talk I give a brief introduction to the Weight Conjecture of J. L. Alperin (1986) and its reduction to finite simple groups, which leads to the so-called inductive Alperin Weight Condition. We will consider this condition for the Chevalley groups $G_2(q)$.

30.1.2014 15:30-16:30 Burkhard Külshammer: Some questions in block theory
In my talk I will address some results in block theory which were recently obtained with various co-authors. These results are concerned with defect groups, fusion systems and Loewy lengths of blocks. One of these results is also related to one of R. Brauer's problems.

10.2.2014 15:30-16:30 (Room 48-436) Alessandro Paolini (University of Birmingham): A Basis of the Gelfand–Graev Algebra of a Chevalley Group

6.3.2014 10:30 - 11:30 (Room 48-436) Michel Enguehard (Institut de Mathématique de Jussieu): Towards a Jordan decomposition of blocks of finite reductive groups

Let $G$ be a connected algebraic reductive group over an algebraically closed field of non zero characteristic $p$, let $F$ be an endomorphism of $G$ defining $G$ over a finite field ${\Bbb F}_q$, let $G^F$ be the subgroup of rational points. Let $\ell$ be a prime number different from $p$. Let $(G^*,F)$ be `` in duality with $(G,F)$ ". By a theorem of Broué-Michel, for any $\ell$-block $B$ of $G^F$ there exist a unique $G^{*\, F}$-conjugacy class $(s)$ of semi-simple $\ell'$-elements in $G^{*\, F}$ such that at least one element of the Lusztig's series of irreducible representations of $G^F$ defined by $(s)$ is a representation of $B$. If $s=1$, $B$ is said to be unipotent.

For a large class of data $((G,F),\ell)$, we associate to $((s),B)$ as above a datum $((G(s),F),B(s))$ where

• $(G(s),F)$ is an algebraic reductive group defined over ${\Bbb F}_q$, with connected component $G(s)^\circ$, and
• $B(s)$ is an $\ell$-block of $G(s)^F$ that dominates a unipotent $\ell$-block of $G(s)^{\circ \, F}$,

in such a way that

• The defect groups of $B$ and of $B(s)$ are isomorphic.
• There is a one-to-one map preserving height between the sets of irreducible representations of $B$ and of $B(s)$.
• The Frobenius categories associated to $B$ and $B(s)$ are equivalent.