Invariant Ring Structure and Finite Linear Actions

Definitions and mathematical statement

The supplied record poses a broad classification question, rather than a specific conjecture. The following gives conventional meanings to its notation; the scope qualifications below remain unresolved.

Let $K$ be a field, let $V$ be a finite-dimensional $K$-vector space, and let $\rho\colon G\to\operatorname{GL}_K(V)$ be a linear action of a finite group $G$. Write $V^*=\operatorname{Hom}_K(V,K)$ and $K[V]=\operatorname{Sym}_K(V^*)$, the polynomial coordinate algebra, including when $K$ is finite. The induced action is determined by $g\cdot\lambda=\lambda\circ\rho(g)^{-1}$ for $\lambda\in V^*$. Define \[ R=K[V]^G=\{f\in K[V]:g\cdot f=f\text{ for every }g\in G\}. \] Here a polynomial invariant ring means an isomorphism $R\cong K[t_1,\ldots,t_m]$ of $K$-algebras for some integer $m\geq 0$. A global complete-intersection presentation means \[ R\cong K[x_1,\ldots,x_n]/(f_1,\ldots,f_c), \] where $n,c\geq 0$ and $f_1,\ldots,f_c$ is a regular sequence: its ideal is proper and each $f_i$ is a non-zero-divisor modulo its predecessors. For a commutative Noetherian ring $R$, the Cohen--Macaulay property means that $\operatorname{depth}R_{\mathfrak p}=\dim R_{\mathfrak p}$ for every prime ideal $\mathfrak p$. The Gorenstein property means that each $R_{\mathfrak p}$ has finite injective dimension as a module over itself.

Requested investigation. Determine how properties of the finite linear action $\rho$ characterize or constrain the polynomial, complete-intersection, Gorenstein, and Cohen--Macaulay properties of $K[V]^G$, including the hypotheses under which particular implications or equivalences hold.

Unresolved scope. The source does not specify the characteristic or other restrictions on $K$, whether $\operatorname{char}K$ divides $|G|$, whether the action must be faithful, or explicit dimension assumptions on $V$. Finite dimensionality above is a conventional interpretation requiring confirmation. It also does not specify graded versus ungraded polynomial isomorphisms, global versus local complete-intersection conventions, which properties of the action are intended, or a particular implication to establish. Thus this investigation cannot be extracted as a uniquely quantified conjecture from the supplied record; the definitions above do not resolve these source omissions.

Short English statement

The question asks how a finite group's linear action determines whether its polynomial invariant ring is polynomial, a complete intersection, Gorenstein, or Cohen--Macaulay. The supplied formulation leaves the field hypotheses, action assumptions, and intended classification target unspecified, so this is a statement-only research-question draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001858$, title Invariant Ring Structure and Finite Linear Actions, exact source code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied. The recorded proposed year is 2011.
• Supplied source citation: Computational Group Theory (2012), Oberwolfach Reports, pp. 2113--2161, DOI 10.4171/owr/2011/37, https://doi.org/10.4171/owr/2011/37. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The supplied literature-source label is "OWR report 37," with the same link: https://doi.org/10.4171/owr/2011/37.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that no general classification resolving this broad invariant-theory question was found. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and asserts that the official report confirms the classification question. These are supplied verification claims; the report has not been independently consulted here. The citation's title year 2012, DOI year component 2011, and proposed year 2011 are retained without resolving their relationship.