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

Parabolic Deligne-Lusztig varieties

Motivated by the Brou\'e conjecture on blocks with abelian defect groups for finite reductive groups, we study "parabolic" Deligne-Lusztig varieties and construct on those which occur in the Brou\'e conjecture an action of a braid monoid, whose action on their $\ell$-adic cohomology will conjecturally factor trough a cyclotomic Hecke algebra. In order to construct this action, we need to enlarge the set of varieties we consider to varieties attached to a "ribbon category"; this category has a {\em Garside family}, which plays an important role in our constructions, so we devote the first part of our paper to the necessary background on categories with Garside families

The center of pure complex braid groups

Brou\'e, Malle and Rouquier conjectured in that the center of the pure braid group of an irreducible finite complex reflection group is cyclic. We prove this conjecture, for the remaining exceptional types, using the analogous result for the full braid group due to Bessis, and we actually prove the stronger statement that any finite index subgroup of such braid group has cyclic center

EL-Shellability and Noncrossing Partitions Associated with Well-Generated Complex Reflection Groups

In this article we prove that the lattice of noncrossing partitions is EL-shellable when associated with the well-generated complex reflection group of type $G(d,d,n)$, for $d,n\geq 3$, or with the exceptional well-generated complex reflection groups which are no real reflection groups. This result was previously established for the real reflection groups and it can be extended to the well-generated complex reflection group of type $G(d,1,n)$, for $d,n\geq 3$, as well as to three exceptional groups, namely $G_{25},G_{26}$ and $G_{32}$, using a braid group argument. We thus conclude that the lattice of noncrossing partitions of any well-generated complex reflection group is EL-shellable. Using this result and a construction by Armstrong and Thomas, we conclude further that the poset of $m$-divisible noncrossing partitions is EL-shellable for every well-generated complex reflection group. Finally, we derive results on the M\"obius function of these posets previously conjectured by Armstrong, Krattenthaler and Tomie.Comment: 37 pages, 4 figures. Moved the technical details of the proof of the EL-shellability of $NC_{G(d,d,n)}$ to the appendix. More references adde