New Perspectives in Representation Theory of Finite Groups (17w5003)

Arriving in Banff, Alberta Sunday, October 15 and departing Friday October 20, 2017

Organizers

Gabriel Navarro (Universidad de Valencia)

(Bergische Universität Wuppertal)

(Technische Universität Kaiserslautern)

(University of Arizona)

Objectives

In this meeting we intend to discuss recent breakthrough results on deep global-local conjectures as well as new research directions in representation theory of finite groups that opened up in the past few years. Four major long-standing conjectures have shaped the research in this area:

- the Brauer height zero conjecture from 1955,
- the Alperin-McKay conjecture, going back to 1972,
- the Alperin weight conjecture from 1986,
- the Broue ́ abelian defect group conjecture from 1988.


All four of these conjectures have seen remarkable progress in the last few years. At the same time, new perspectives have opened up following some new conjectures on representations of finite groups of a somewhat different nature:

-- the Eaton–Moreto ́ extension of Brauer’s height zero conjecture,
-- the Kessar–Linckelmann conjecture on blocks with one simple module,
-- the Malle–Robinson conjecture on the number of simple modules in a block,
-- the basic set conjecture.

The investigations around the proof of McKay’s conjecture and the partial proofs of other conjectures have shown the extent to which our knowledge of the ordinary representation theory of finite reductive groups is still incomplete, even after the groundbreaking work of Lusztig. Questions on canonical parametrisation of characters, and thus on their behaviour under automorphisms, and even the very simple question of determining character degrees of almost simple groups still remain unsolved.

The meeting will bring together the leading experts on the representation theory of abstract finite groups as well as of finite reductive groups to exploit the substantial advances on these fundamental old and new conjectures. The recent progress in our field has attracted a number of young researchers to this area. The conference will provide them with a unique opportunity to learn more about the current exciting developments and become directly involved in this fast moving area of research. It is our hope that the meeting will again strengthen existing and foster new collaborations, as well as facilitating significant progress along the lines of all these fundamental conjectures.