# Degrees, class sizes and divisors of character values

@article{Gallagher2012DegreesCS,
title={Degrees, class sizes and divisors of character values},
author={Patrick X. Gallagher},
journal={Journal of Group Theory},
year={2012},
volume={15},
pages={455 - 467}
}
• P. Gallagher
• Published 1 July 2012
• Mathematics
• Journal of Group Theory
Abstract. In the character table of a finite group there is a tendency either for the character degree to divide the conjugacy class size or the character value to vanish. There is also a partial divisibility where the determinant of the character is not 1. There are versions of these depending on a subgroup, based on an arithmetic property of spherical functions which generalizes the integrality of the values of the characters and the central characters.
6 Citations
. For any ﬁnite group G , Thompson proved that, for each χ ∈ Irr( G ), χ ( g ) is a root of unity or zero for more than a third of the elements g ∈ G , and Gallagher proved that, for each larger than
• Mathematics
Representation Theory of the American Mathematical Society
• 2020
In the high rank limit, the fraction of non-zero character table entries of finite simple groups of Lie type goes to zero.
If $\lambda$ and $\mu$ are two non-empty Young diagrams with the same number of squares, and $\boldsymbol\lambda$ and $\boldsymbol\mu$ are obtained by dividing each square into $d^2$ congruent
For any finite group $G$, Thompson proved that, for each $\chi\in {\rm Irr}(G)$, $\chi(g)$ is a root of unity or zero for more than a third of the elements $g\in G$, and Gallagher proved that, for
• Mathematics
International Mathematics Research Notices
• 2020
For $G=\textrm{GL}(n,q)$, the proportion $P_{n,q}$ of pairs $(\chi ,g)$ in $\textrm{Irr}(G)\times G$ with $\chi (g)\neq 0$ satisfies $P_{n,q}\to 0$ as $n\to \infty$.