Zeros and roots of unity in character tables
@inproceedings{Miller2020ZerosAR,
title={Zeros and roots of unity in character tables},
author={Alexander R. Miller},
year={2020}
}. 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 average class g G , χ ( g ) is a root of unity or zero for more than a third of the irreducible characters χ ∈ Irr( G ). We show that in many cases “more than a third” can be replaced by “more than half”.
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$p$-groups and zeros of characters
- Mathematics
- 2023
. Fix a prime p and an integer n ≥ 0. Among the non-linear irreducible characters of the p -groups of order p n , what is the minimum number of elements that take the value 0?
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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 $.
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