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Jonas Hetz

Publications and source records attributed to Jonas Hetz.

5 recordsLinked to original sources
arXiv2023

The values of unipotent characters at unipotent elements for groups of type $E_8$ and ${^2\!E}_6$

In order to tackle the problem of generically determining the character tables of the finite groups of Lie type $\mathbf{G}(q)$ associated to a connected reductive group $\mathbf{G}$ over $\overline{\mathbb F}_p$, Lusztig developed the theory of character sheaves in the 1980s. The subsequent work of Lusztig and Shoji in principle reduces this problem to specifying certain roots of unity. The situation is particularly well understood as far as character values at unipotent elements are concerned. We complete the computation of the values of unipotent characters at unipotent elements for the groups $\mathbf{G}(q)$ where $\mathbf{G}$ is the simple group of type $E_8$, by specifying the aforementioned roots of unity for all prime powers $q$. We also resolve this task for the groups ${^2\!E}_6(q)$ when $q$ is a power of $p=2$. Our results thus conclude the project of computing the values of unipotent characters at unipotent elements for the simple exceptional groups of Lie type.

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arXiv2022

On the labelling of characters of Weyl groups of type $F_4$

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~$E_7$.

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arXiv2020

On the values of unipotent characters of finite Chevalley groups of type $E_7$ in characteristic 2

Let $G$ be a finite group of Lie type. In order to determine the character table of $G$, Lusztig developed the theory of character sheaves. In this framework, one has to find the transformation between two bases for the space of class functions on $G$, one of them being the irreducible characters of $G$, the other one consisting of characteristic functions associated to character sheaves. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. The problem of specifying these scalars can be reduced to considering cuspidal character sheaves. We will deal with the latter for the specific case where $G=E_7(q)$, and $q$ is a power of the bad prime $p=2$ for $E_7$.

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arXiv2019

On the values of unipotent characters of finite Chevalley groups of type $E_6$ in characteristic 3

Let $G$ be a finite Chevalley group. We are concerned with computing the values of the unipotent characters of $G$ by making use of Lusztig's theory of character sheaves. In this framework, one has to find the transformation between several bases for the class functions on $G$. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. We shall determine these scalars in the specific case where $G$ is the (twisted or non-twisted) group of type $E_6$ over the finite field with $q$ elements, for $q$ a power of the bad prime $p=3$, by exploiting known facts about the representation theory of the Hecke algebra associated with $G$.

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