Constructing representations of Hecke algebras for complex reflection groups

We investigate the representations and the structure of Hecke algebras associated to certain finite complex reflection groups. We first describe computational methods for the construction of irreducible representations of these algebras, including a generalization of the concept of $W$-graph to the situation of complex reflection groups. We then use these techniques to find models for all irreducible representations in the case of complex reflection groups of dimension at most three. Using these models we are able to verify some important conjectures on the structure of Hecke algebras

On saturated fusion systems and Brauer indecomposability of Scott modules

Let $p$ be a prime number, $G$ a finite group, $P$ a $p$-subgroup of $G$ and $k$ an algebraically closed field of characteristic $p$. We study the relationship between the category \Ff_P(G) and the behavior of $p$-permutation $kG$-modules with vertex $P$ under the Brauer construction. We give a sufficient condition for \Ff_P(G) to be a saturated fusion system. We prove that for Scott modules with abelian vertex, our condition is also necessary. In order to obtain our results, we prove a criterion for the categories arising from the data of $(b, G)$-Brauer pairs in the sense of Alperin-Brou\'e and Brou\'e-Puig to be saturated fusion systems on the underlying $p$-group

Digital Mathematics Libraries: The Good, the Bad, the Ugly

The idea of a World digital mathematics library (DML) has been around since the turn of the 21th century. We feel that it is time to make it a reality, starting in a modest way from successful bricks that have already been built, but with an ambitious goal in mind. After a brief historical overview of publishing mathematics, an estimate of the size and a characterisation of the bulk of documents to be included in the DML, we turn to proposing a model for a Reference Digital Mathematics Library--a network of institutions where the digital documents would be physically archived. This pattern based rather on the bottom-up strategy seems to be more practicable and consistent with the digital nature of the DML. After describing the model we summarise what can and should be done in order to accomplish the vision. The current state of some of the local libraries that could contribute to the global views are described with more details

Automorphisms and Generalized Involution Models of Finite Complex Reflection Groups

We prove that a finite complex reflection group has a generalized involution model, as defined by Bump and Ginzburg, if and only if each of its irreducible factors is either $G(r,p,n)$ with $\gcd(p,n)=1$; $G(r,p,2)$ with $r/p$ odd; or $G_{23}$, the Coxeter group of type $H_3$. We additionally provide explicit formulas for all automorphisms of $G(r,p,n)$, and construct new Gelfand models for the groups $G(r,p,n)$ with $\gcd(p,n)=1$.Comment: 29 page