Combinatorial Representation Theory – TRC

I propose to combine the tools developed by B. Külshammer, J. B. Olsson and G. R. Robinson on generalized blocks, in particular for
symmetric groups, and my own research, especially recent results on defect groups and defects for characters and on basic sets.

Existence of perfect isometries between blocks of the same weight of covering groups of the symmetric and alternating groups, and between blocks of Coxeter groups of types B and D.

These perfect isometries will be used to prove Broué's Conjecture in some cases, as well as to exhibit basic sets and new decomposition numbers.

J.-B. Gramain and J. B. Olsson, On bar lengths in partitions, Proceedings of the Edinburgh Mathematical Society (2013) 56, 535-550.

O. Brunat and J.-B. Gramain, Perfect Isometries and Murnaghan-Nakayama Rules, to appear in Transactions of the American Mathematical Society, doi.org/10.1090/tran/6860

The representation theory of finite groups has been a very active area of mathematics for the past century. One reason for this is its numerous applications, not only within mathematics, but also in chemistry and physics.

In several infinite families of groups, it turns out that there is a very strong link between the representation theory of these groups
and the combinatorics involving certain objects (like integer partitions, crystal graphs or cell decompositions). In order to get important information about the irreducible representations, general representation theoretic methods are intertwined with the study of combinatorial properties of the objects labelling them.


The aim of this project is to make progress on important conjectures in modular representation theory pertaining to the symmetric and other groups. To do this I propose to combine the tools developed by B. Külshammer, J. B. Olsson and G. R. Robinson on generalized blocks, in particular for
symmetric groups, and my own research, especially recent results on defect groups and defects for characters and on basic sets.