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9 - Character bounds for finite simple groups and applications
- By Pham Huu Tiep
Edited by C. M. Campbell, University of St Andrews, Scotland, M. R. Quick, University of St Andrews, Scotland, E. F. Robertson, University of St Andrews, Scotland, C. M. Roney-Dougal, University of St Andrews, Scotland, D. I. Stewart, University of Manchester
Book:

Groups St Andrews 2022 in Newcastle

Published online:

21 November 2024

Print publication:

12 December 2024, pp 360-406
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Summary

Given the current knowledge of complex representations of finite quasi-simple groups, obtaining good upper bounds for their characters values still remains a difficult problem, a satisfactory solution of which would have significant implications in a number of applications. We will report on recent results that produce such character bounds, and discuss some such applications, in and outside of group theory.

Trivial source character tables of $\operatorname{SL}_2(q)$, part II
Part of
- Grothendieck groups and $K_0$
- Representation theory of groups
Niamh Farrell, Caroline Lassueur
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 66 / Issue 3 / August 2023

Published online by Cambridge University Press:

30 June 2023, pp. 689-709
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We compute the trivial source character tables (also called species tables of the trivial source ring) of the infinite family of finite groups $\operatorname{SL}_{2}(q)$ for q even over a large enough field of odd characteristics. This article is a continuation of our article Trivial Source Character Tables of $\operatorname{SL}_{2}(q)$, where we considered, in particular, the case in which q is odd in non-defining characteristic.

DETERMINING ASCHBACHER CLASSES USING CHARACTERS
Part of
- Linear algebraic groups and related topics
- Representation theory of groups
SEBASTIAN JAMBOR
Journal:

Journal of the Australian Mathematical Society / Volume 98 / Issue 3 / June 2015

Published online by Cambridge University Press:

11 November 2014, pp. 355-363

Print publication:

June 2015
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Let ${\rm\Delta}:G\rightarrow \text{GL}(n,K)$ be an absolutely irreducible representation of an arbitrary group $G$ over an arbitrary field $K$; let ${\it\chi}:G\rightarrow K:g\mapsto \text{tr}({\rm\Delta}(g))$ be its character. In this paper, we assume knowledge of ${\it\chi}$ only, and study which properties of ${\rm\Delta}$ can be inferred. We prove criteria to decide whether ${\rm\Delta}$ preserves a form, is realizable over a subfield, or acts imprimitively on $K^{n\times 1}$. If $K$ is finite, we can decide whether the image of ${\rm\Delta}$ belongs to certain Aschbacher classes.