Twisted Frobenius-Schur Indicators and Character Degree Sums in Dihedral Groups

TL;DR

This paper explores the relationship between the sum of irreducible representation degrees and twisted involutions in finite groups, especially dihedral groups, revealing inequalities and classifications based on automorphisms.

Contribution

It establishes that for dihedral groups, the sum of degrees exceeds or equals the number of twisted involutions for all automorphisms, with a complete classification of these involutions.

Findings

Proves $T(D_n) \,\geq\, m_\sigma$ for all automorphisms of dihedral groups.

Classifies twisted involutions in dihedral groups using number theory.

Extends known relations from real characters to automorphisms in dihedral groups.

Abstract

Let $G$ be a finite group and $T(G)$ be the sum of the degrees of its irreducible complex representations. We investigate the relationship between $T(G)$ and the number of twisted involutions $m_\sigma = |\{g \in G \mid \sigma(g) = g^{-1}\}|$ for an automorphism $\sigma$. While it is known that $T(G) = m_e$ for the identity automorphism $e$ in certain cases (e.g., real characters), we analyze this relation for non-identity automorphisms of groups of order $p, 2p, p^2$. We prove that for the family of Dihedral groups $D_n$, the inequality $T(D_n) \geq m_\sigma$ holds for all $\sigma \in \mathrm{Aut}(D_n)$. We provide a complete classification of $m_\sigma$ using number-theoretic properties of the automorphism parameters.