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On the Character Degrees of a Sylow p-Subgroup of a Finite Chevalley Group G(pf) Over a Bad Prime

Published online by Cambridge University Press:  15 April 2019

By
Tung Le ,
Kay Magaard  and
Alessandro Paolini
Edited by
C. M. Campbell ,
C. W. Parker ,
M. R. Quick ,
E. F. Robertson  and
C. M. Roney-Dougal
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
C. W. Parker
Affiliation:
University of Birmingham
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
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Summary

Let q be a power of a prime p and let U(q) be a Sylow p-subgroup of a finite Chevalley group G(q) defined over the field with q elements. We first give a parametrization of the set Irr(U(q)) of irreducible characters of U(q) when G(q) is of type G2. This is uniform for primes p ≥ 5, while the bad primes p = 2 and p = 3 have to be considered separately. We then use this result and the contribution of several authors to show a general result, namely that if G(q) is any finite Chevalley group with p a bad prime, then there exists a character χ \in Irr(U(q)) such that χ(1) = qn/p for some n \in Z≥0 . In particular, for each G(q) and every bad prime p, we construct a family of characters of such degree as inflation followed by an induction of linear characters of an abelian subquotient V(q) of U(q).

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Groups St Andrews 2017 in Birmingham , pp. 427 - 442
Publisher: Cambridge University Press
Print publication year: 2019

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