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Representations of finite groups of Lie type and associated algebras

Final Activity Report Summary - OXFORDALGEBRA (Representations of finite groups of Lie type and associated algebras)

This research is in the representation theory of finite groups and algebras. Three major aspects have been studied with success.

The main part of the project concerned Alvis-Curtis duality, for finite groups of Lie type. This is a duality on characters of finite reductive groups, which turns out to be a consequence of a derived equivalence. This then gives the possibility of using high-level mathematical tools.

The first result in this project studies Alvis-Curtis duality and its connection to to q-Schur algebras and Hecke algebras. q-Schur algebras can be constructed for example as quotients of quantised enveloping algebras, and Hecke algebras are deformations of group algebras of finite Coxeter groups.

The second major result, in joint work with B. Ackermann, investigates the link between modular Alvis-Curtis duality and decomposition numbers for general linear groups. Decomposition numbers are absolutely essential for modular representation theory. Exploiting these results, the fellow together with K.M. Tan computes decomposition numbers for general linear groups, for arbitrary blocks of weight 2.

Two further main results extend the range. First, the study of Alvis-Curtis duality required construction of complexes, through permtuation modules. In joint work with K. Erdmann, permutation modules of finite general linear groups acting on partial flags in the natural module were studies explicitly. The results included a complete parametrisation of their indecomposable direct summands, and determine their vertices and Green correspondents. It also gives rise to a new invariant of tilting modules for q-Schur algebras.
Second, a joint project with A. Clark and K. Erdmann, extends very recent discoveries in the representation theory of symmetric groups and results on dynamical systems arising from tilings of the plane. This studies rhombal algebras, which on one hand model parts of blocks of symmetric groups and general linear groups, and open up a new way to approach the homological properties of the representations, and on the other hand describe an aperiodic tiling of the plane.