Meinolf Geck’s research while affiliated with University of Stuttgart and other places

These are unpublished notes from about 1992-1993 which, retrospectively, may be regarded as a complement to Lusztig's recent paper on the trace of Coxeter elements. Our notes include explicit tables for those traces. The proofs rely on a connection with Lusztig's work on Coxeter orbits and eigenspaces of Frobenius, which may be of independent interest.

Let $\mathfrak {g}$ g be a finite-dimensional simple Lie algebra over $\mathbb {C}$ C . In the 1950s Chevalley showed that $\mathfrak {g}$ g admits particular bases, now called “Chevalley bases”, for which the corresponding structure constants are integers. Such bases are not unique but, using Lusztig’s theory of canonical bases, one can single out a “canonical” Chevalley basis which is unique up to a global sign. In this paper, we give explicit formulae for the structure constants with respect to such a basis.

Roger Carter (1934--2022) was a very well known mathematician working in algebra, representation theory and Lie theory. He spent most of his mathematical career in Warwick. Roger was a great communicator of mathematics: the clarity, precision and enthusiasm of his lectures delivered in his beautiful handwriting were hallmark features recalled by numerous students and colleagues. His books have been described as marvelous pieces of scholarship and service to the general mathematical community. We both met Roger early in our careers, and were encouraged and influenced by him~ -- ~and his lovely sense of humour. This text is our tribute, both to his mathematical achievements, and to his kindness and generosity towards his students, his colleagues, his collaborators, and his family.

In the literature on finite groups of Lie type, there exist two different conventions about the labelling of the irreducible characters of Weyl groups of type $F_4$ F 4 . We point out some issues concerning these two conventions and their effect on tables about unipotent characters or the Springer correspondence. Using experiments related to these issues with the computer algebra system , we spotted an error in Spaltenstein’s tables for the generalised Springer correspondence in type $E_7$ E 7 .

The workshop Representations of Finite Groups was organised by Olivier Dudas (Marseille), Meinolf Geck (Stuttgart), Radha Kessar (Manchester), and Gabriel Navarro (Valencia). It covered a wide variety of aspects of the representation theory of finite groups and related topics, and showcased several recent breakthrough results.

Let q be a prime power and $F_4(q)$ F 4 ( q ) be the Chevalley group of type $F_4$ F 4 over a finite field with q elements. Marcelo and Shinoda (Tokyo J Math 18:303–340, 1995) determined the values of the unipotent characters of $F_4(q)$ F 4 ( q ) on all unipotent elements, extending earlier work by Kawanaka and Lusztig to small characteristics. Assuming that q is a power of 2, we explain how to construct the complete character table of $F_4(q)$ F 4 ( q ) .

Let q be a prime power and $F_4(q)$ be the Chevalley group of type $F_4$ over a finite field with q elements. Marcelo--Shinoda (1995) determined the values of the unipotent characters of $F_4(q)$ on all unipotent elements, extending earlier work by Kawanaka and Lusztig to small characteristics. Assuming that q is a power of~2, we explain how to construct the complete character table of~$F_4(q)$.

In the literature on finite groups of Lie type, there exist two different conventions about the labelling of the irreducible characters of Weyl groups of type~$F_4$. We point out some issues concerning these two conventions and their effect on tables about unipotent characters or the Springer correspondence. Using experiments related to these issues with the computer algebra system {\sf CHEVIE}, we spotted an error in Spaltenstein's tables for the generalised Springer correspondence in type~$E_7$.

The purpose of this note is to advertise an elegant algorithmic proof for the Jordan--Chevalley decomposition of a matrix, following and (slightly) revising the discussion of Couty, Esterle und Zarouf (2011). The basic idea of that method goes back to Chevalley (1951).

The article “Generalised Gelfand-Graev Represenations in Bad Characteristic?,” written by Meinolf Geck, was originally published Online First without Open Access.