p-Groups and the Polynomial Ring of Invariants Question

TL;DR

This paper characterizes when the ring of invariants of a finite p-group acting on a polynomial ring is itself a polynomial ring, based on the group's structure and geometric properties of the invariant ring.

Contribution

It provides a complete characterization of polynomial invariant rings for p-groups acting on symmetric algebras, extending known results to higher dimensions.

Findings

For dim(V)=3, polynomial invariants occur iff G is generated by transvections.

For dim(V)>3, polynomial invariants occur iff certain subring conditions and Cohen-Macaulay property hold.

The results connect algebraic and geometric conditions for polynomial invariants.

Abstract

Let G be a finite p-subgroup of GL(V), where p = char(F), and V is finite-dimensional over the field F. Let S(V) be the symmetric algebra of V, S(V)^G the subring of G-invariants, and V* the F-dual space of V. The following presents our solution to the above question. Theorem A. Suppose dim(V)=3. Then S(V)^G is a polynomial ring if and only if G is generated by transvections. Theorem B. Suppose dim(V) > 3. Then S(V)^G is a polynomial ring if and only if: (1) S(V)^{G_U} is a polynomial ring for each subspace U of V* with dim(U)=2, where G_U = {g in G | g(u) = u, for all u in U}, and (2) S(V)^G is Cohen-Macaulay. Alternatively, (1) can be replaced by the equivalent condition: (3) dim (non-smooth locus of S(V)^G) < 2.