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In previous work, the authors confirmed the speculation of J. G. Thompson that certain multiquadratic fields are generated by specified character values of sufficiently large alternating groups $A_n$. Here we address the natural generalization of this speculation to the finite general linear groups $mathrm{GL}_mleft(mathbb{F}_qright)$ and $mathrm{SL}_2left(mathbb{F}_qright)$.
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For a finite group $G$, let $K(G)$ denote the field generated over $mathbb{Q}$ by its character values. For $n>24$, G. R. Robinson and J. G. Thompson proved that $$K(A_n)=mathbb{Q}left ({ sqrt{p^*} : pleq n {text{ an odd prime with } p eq n-2}}right),$$ where $p^*:=(-1)^{frac{p-1}{2}}p$. Confirming a speculation of Thompson, we show that arbitrary suitable multiquadratic fields are similarly generated by the values of $A_n$-characters restricted to elements whose orders are only divisible by ramified primes. To be more precise, we say that a $pi$-number is a positive integer whose prime factors belong to a set of odd primes $pi:= {p_1, p_2,dots, p_t}$. Let $K_{pi}(A_n)$ be the field generated by the values of $A_n$-characters for even permutations whose orders are $pi$-numbers. If $tgeq 2$, then we determine a constant $N_{pi}$ with the property that for all $n> N_{pi}$, we have $$K_{pi}(A_n)=mathbb{Q}left(sqrt{p_1^*}, sqrt{p_2^*},dots, sqrt{p_t^*}right).$$
We characterize finite groups G generated by orthogonal transformations in a finite-dimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient piecewise linear structure.
In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l at least 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains Sp(n, l). This result is for instance useful for proving big image results for symplectic Galois representations.
We study multiplicities of unipotent characters in tensor products of unipotent characters of GL(n,q). We prove that these multiplicities are polynomials in q with non-negative integer coefficients. We study the degree of these polynomials and give a necessary and sufficient condition in terms of the representation theory of symmetric groups for these polynomials to be non-zero.
Let $q$ be a power of a prime $p$, let $G$ be a finite Chevalley group over $mathbb{F}_q$ and let $U$ be a Sylow $p$-subgroup of $G$; we assume that $p$ is not a very bad prime for $G$. We explain a procedure of reduction of irreducible complex characters of $U$, which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of $U$ along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when $G$ is of type $mathrm{F}_4$, where we observe that the parametrization is uniform over good primes $p > 3$, but differs for the bad prime $p = 3$. We also explain how it has been applied for all groups of rank $4$ or less.
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