Narrow Results

We prove that the finite exceptional groups $F_4(q)$, $E_7{(q)}_{\mathrm{\text{ad}}}$, and $E_8(q)$ have no irreducible complex characters with Frobenius–Schur indicator $-1$, and we list exactly which irreducible characters of these groups are not real-valued. We also give a complete list of complex irreducible characters of the Ree groups ${}^2{F}_4(q^2)$ which are not real-valued, and we show the only character of this group which has Frobenius–Schur indicator $-1$ is the cuspidal unipotent character ${\chi }_{21}$ found by Geck.

In 1993, Sim proved that all the faithful irreducible representations of a finite metacyclic group over any field of positive characteristic have the same degree. In this paper, we restrict our attention to non-modular representations and generalize this result for — (1) finite metabelian groups, over fields of positive characteristic coprime to the order of groups, and (2) finite groups having a cyclic quotient by an abelian normal subgroup, over number fields.

We study realization fields and integrality of characters of finite subgroups of ${\mathrm{\text{GL}}}_n(C)$ and related lattices with a focus on the integrality of characters of finite groups $G$. We are interested in the arithmetic aspects of the integral realizability of representations of finite groups, order generated by the character values, the number of minimal realization splitting fields, and in particular, consider the conditions of realizability in the terms of Hilbert symbols and quaternion algebras and some orders generated by character values over the rings of rational and algebraic integers.