First and Second Fundamental Theorems for Invariant Rings Generated by Circulant Determinants

TL;DR

This work determines explicit generators and relations for invariant rings arising from circulant determinants under the actions generated by the differential operators $D$ and $\Delta$, and it characterizes SL-invariant rings of circulant type. By applying a Fourier-type variable change, the authors diagonalize these operators and identify key determinants $\Theta_p(\bm{y}^{(p)}_i)$ and their products, establishing first and second fundamental theorems when $n$ has at most two prime factors. They prove that for $n$ with one prime factor $p$, the invariants are generated by $\Theta_p(\bm{y}^{(p)}_i)$; for two primes $p,q$, by $\Theta_p(\bm{y}^{(p)}_i)$ and $\Theta_q(\bm{y}^{(q)}_j)$, with precise kernel descriptions for the associated maps $\rho$ and $\rho'$, and provide kernel relations $t_i$. They further show a product decomposition $\Theta_n(\bm{x}) = \prod_{i} \Theta_p(\bm{y}^{(p)}_i)$ and prove that $\mathbb{C}[\bm{x}]^{\mathrm{SL}(\mathbb{C}G)} = \mathbb{C}[\Theta_n(\bm{x})]$, embedding the circulant-determinant framework into the invariant theory landscape and connecting to Dedekind-Laquer-type results. Overall, the paper yields explicit generators and relations for these invariant rings in the two-prime-factor setting and clarifies how higher factorization alters the equality with the prime-factor subalgebras.

Abstract

In this paper, we give the first and second fundamental theorems of invariant theory for certain invariant rings whose generators are expressed by circulant determinants.

Theorems & Definitions (31)

• Theorem 1: First Fundamental Theorem
• Theorem 2: Second Fundamental Theorem
• Theorem 3
• Theorem 4
• Theorem 5
• Corollary 6
• Lemma 7
• proof
• Lemma 8
• proof