Problems on character theory when we restrict the field of values

Nicola Grittini (University of Florence)

03-Dec-2020, 14:30-15:30 (3 years ago)

Abstract: Irreducible characters of rational and real values have always attracted the attention of researchers in Character theory of finite groups. One of the questions which naturally arise when these characters are studied, among the others, is whether some of the most famous results in character theory have variants involving rational or real valued characters. In fact, some of these results have such variants and, maybe not surprisingly, the variants often involve the prime number 2. An example of this fact is a theorem proved in 2007 by Dolfi, Navarro and Tiep. The theorem is a real-valued version of Ito-Michler Theorem and says that, if no real-valued irreducible character of a finite group $G$ has even order, then the group has a normal Sylow 2-subgroup.

On the other hand, it is quite difficult to work with rational and real valued characters if we consider a prime number different from 2. This suggests that, if we want to find variants of some classical results involving character fields of values and an odd prime number $p$, we may not consider as fields $\mathbb{Q}$ and $\mathbb{R}$ but some other fields, whose definition involves the prime $p$ and which are equal to $\mathbb{Q}$ or $\mathbb{R}$ when $p= 2$.

In this talk we will see two cases in which this generalization is possible,one involving rational valued and one involving real valued characters. The part involving real-valued characters has been published in a preprint and has to be considered as a work in progress, nevertheless the approach followed in the study of the problem could be interesting for many researchers in character theory.

group theoryrings and algebras

Audience: researchers in the topic


Al@Bicocca take-away

Organizer: Claudio Quadrelli*
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