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Group Determinants and Invariant Rings

Yuka Yamaguchi, Naoya Yamaguchi

TL;DR

The paper investigates invariant rings arising from Frobenius' partial differential operators acting on the group determinant $\Theta_G(x_g)$ of a finite group $G$. By leveraging representation-theoretic tools—notably the characters $\chi_{\varphi}$, second orthogonality relations, and the Frobenius determinant theorem—it proves that the intersections of the invariants $\mathbb{C}[\mathbf{y}]^{D_g}$ (and $\mathbb{C}[\mathbf{y}]^{\Delta_g}$) for non-identity conjugacy classes are precisely generated by $\Theta_G(x_g)$, with explicit abelian- and quotient-group refinements. The work also provides concrete invariant generators for the symmetric group $S_3$ and discusses a direct, historically grounded proof of Theorem 1, enriching the linkage between group determinants, invariant theory, and representation theory. Collectively, these results yield explicit descriptions of invariant rings associated with Frobenius operators and illuminate how group structure governs invariant generators and relations.

Abstract

In the study of group determinants, Frobenius introduced certain partial differential operators. This paper presents several results concerning the invariant rings derived from these partial differential operators.

Group Determinants and Invariant Rings

TL;DR

The paper investigates invariant rings arising from Frobenius' partial differential operators acting on the group determinant of a finite group . By leveraging representation-theoretic tools—notably the characters , second orthogonality relations, and the Frobenius determinant theorem—it proves that the intersections of the invariants (and ) for non-identity conjugacy classes are precisely generated by , with explicit abelian- and quotient-group refinements. The work also provides concrete invariant generators for the symmetric group and discusses a direct, historically grounded proof of Theorem 1, enriching the linkage between group determinants, invariant theory, and representation theory. Collectively, these results yield explicit descriptions of invariant rings associated with Frobenius operators and illuminate how group structure governs invariant generators and relations.

Abstract

In the study of group determinants, Frobenius introduced certain partial differential operators. This paper presents several results concerning the invariant rings derived from these partial differential operators.
Paper Structure (5 sections, 5 theorems, 35 equations, 1 table)

This paper contains 5 sections, 5 theorems, 35 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorem $\mathbf{\ref{['thm:2']}}$
  3. Proof of Theorem $\mathbf{\ref{['thm:5']}}$
  4. Proof of Theorem $\mathbf{\ref{['thm:6']}}$
  5. Note on Theorem $\mathbf{\ref{['thm:1']}}$

Key Result

Theorem 1

Let $G$ be a finite group. For any representation $\varphi$ of $G$ over $\mathbb{C}$ and any $g \in G$, the following holds:

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Remark 4
  • Theorem 5
  • Theorem 6