Twisted Frobenius-Schur Indicators and Character Degree Sums in Dihedral Groups
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| Format: | Preprint |
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2026
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| author | Yerrapati, Venkata Subbaiah Dixit, Rahul Shukla, Ajay Kumar |
| author_facet | Yerrapati, Venkata Subbaiah Dixit, Rahul Shukla, Ajay Kumar |
| contents | 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 $m_σ= |\{g \in G \mid σ(g) = g^{-1}\}|$ for an automorphism $σ$. 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 $T(D_n) \geq m_σ$ holds for all $σ\in \mathrm{Aut}(D_n)$. We provide a complete classification of $m_σ$ using number-theoretic properties of the automorphism parameters. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_22127 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Twisted Frobenius-Schur Indicators and Character Degree Sums in Dihedral Groups Yerrapati, Venkata Subbaiah Dixit, Rahul Shukla, Ajay Kumar Group Theory Representation Theory Primary 20C15, 20F28, 20F55, Secondary 20B25, 20B35 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 $m_σ= |\{g \in G \mid σ(g) = g^{-1}\}|$ for an automorphism $σ$. 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 $T(D_n) \geq m_σ$ holds for all $σ\in \mathrm{Aut}(D_n)$. We provide a complete classification of $m_σ$ using number-theoretic properties of the automorphism parameters. |
| title | Twisted Frobenius-Schur Indicators and Character Degree Sums in Dihedral Groups |
| topic | Group Theory Representation Theory Primary 20C15, 20F28, 20F55, Secondary 20B25, 20B35 |
| url | https://arxiv.org/abs/2605.22127 |