[Paper Review] The interplay of Invariant Theory with Multiplicative Ideal Theory and with Arithmetic Combinatorics

K. Cziszter, M. Domokos+1 more|arXiv (Cornell University)|May 22, 2015

Commutative Algebra and Its Applications4 citations

TL;DR

This paper establishes deep connections between invariant theory, multiplicative ideal theory, and arithmetic combinatorics by analyzing the factorization structure of invariant rings $\mathbb{F}[V]^G$ for finite group actions. It provides an explicit divisor theory for these rings, links the Noether number $\beta(G)$ to the Davenport constant $\mathsf{D}(G)$, and proves that for abelian groups, $\sigma(G) = \exp(G)$ and $\eta(G)$ is finite and related to zero-sum sequences over $\widehat{G}$, unifying key invariants across these fields.

ABSTRACT

This paper surveys and develops links between polynomial invariants of finite groups, factorization theory of Krull domains, and product-one sequences over finite groups. The goal is to gain a better understanding of the multiplicative ideal theory of invariant rings, and connections between the Noether number and the Davenport constants of finite groups.

Motivation & Objective

• To unify and deepen the interplay between invariant theory, multiplicative ideal theory, and arithmetic combinatorics.
• To determine the divisor theory and class group structure of invariant rings $\mathbb{F}[V]^G$ for finite group actions.
• To clarify the relationship between the Noether number $\beta(G)$ and the Davenport constant $\mathsf{D}(G)$, especially in the abelian case.
• To investigate the invariants $\sigma(G,V)$ and $\eta(G,V)$, particularly their finiteness and structural properties in the context of module generation over subrings.
• To pose and initiate the study of $\eta(G)$ for non-abelian groups, relating it to the monoid of zero-sum sequences over $\widehat{G}$.

Proposed method

• Construct an explicit divisor theory for the multiplicative monoid $R^\bullet = (\mathbb{F}[V]^G)^\bullet$ via a divisibility-preserving homomorphism into a free abelian monoid.
• Use the canonical transfer homomorphism $\theta: R^\bullet \to \mathcal{B}(\mathcal{C}(R)^*)$ to link factorization in $\mathbb{F}[V]^G$ to zero-sum sequences over the class group.
• Apply the isomorphism $\psi: M \to \mathcal{F}(\widehat{G})$ between monomials in $\mathbb{F}[V]^G$ and sequences over the character group $\widehat{G}$ to translate group-invariant monomial structure into zero-sum sequence problems.
• Leverage results from zero-sum theory (e.g., Davenport constant, $\mathsf{D}(G)$) to analyze the Noether number $\beta(G)$, showing $\beta(G) = \mathsf{D}(G)$ for abelian $G$.
• Define $\sigma(G,V)$ as the minimal degree $d$ such that $\mathbb{F}[V]^G$ is finitely generated over a subring of invariants of degree $\leq d$, and $\eta(G,V)$ as the maximal degree of generators of $\mathbb{F}[V]^G_+$ over the subring generated up to degree $\sigma(G,V)$.
• Use induction and transfer homomorphism techniques to bound $\beta_k(G,V)$ via $\beta_k(G,V) \leq (k-1)\sigma(G,V) + \eta(G,V)$.

Experimental results

Research questions

• RQ1 How can the divisor theory of invariant rings $\mathbb{F}[V]^G$ be explicitly described in terms of invariant theory?
• RQ2 What is the precise relationship between the Noether number $\beta(G)$ and the Davenport constant $\mathsf{D}(G)$ for finite groups?
• RQ3 Is $\eta(G) = \sup\{\eta(G,W)\}$ finite for finite non-abelian groups $G$, and how is it related to $\eta(\mathcal{B}(G))$?
• RQ4 Can the invariants $\sigma(G,V)$ and $\eta(G,V)$ be systematically bounded or computed, especially in the modular case?
• RQ5 To what extent do results on minimal zero-sum sequences in $\mathcal{F}(\widehat{G})$ translate into structural results on $G$-invariant monomials in $\mathbb{F}[V]^G$?

Key findings

• The paper provides a self-contained, explicit divisor theory for $\mathbb{F}[V]^G$, recovering the class group result of Benson and Nakajima and extending it to the full arithmetic structure.
• For finite abelian groups, $\sigma(G) = \exp(G) = \mathsf{e}(G)$, showing that the minimal degree generating subring is determined by the exponent of the group.
• It is shown that $\eta(G) = \eta(\widehat{G})$, and since $\widehat{G} \cong G$ for abelian groups, $\eta(G)$ is finite and computable via zero-sum sequences.
• The Noether number $\beta(G)$ equals the Davenport constant $\mathsf{D}(G)$ for finite abelian groups, a result established via the isomorphism between monomials and zero-sum sequences.
• For the symmetric group $S_4$, $\beta(S_4) = 9$, correcting a previous claim of 10, based on the degree of the invariant $\sigma_3\Delta_4$.
• For the non-abelian group generated by Pauli matrices, $\beta(G) = 7$, computed via $b(G,V) = 6$ and the bound $\beta(G) \leq \mathsf{D}_3(C_2 \oplus C_2) = 7$, with equality confirmed.