Invariants of (-1)-Skew Polynomial Rings under Permutation Representations

TL;DR

This paper studies the invariants of skew polynomial rings under permutation group actions, revealing their Gorenstein property and conditions for being a complete intersection, with bounds on generators and comparisons to classical cases.

Contribution

It introduces the properties of invariant rings of (-1)-skew polynomial rings under permutation actions, including Gorensteinness and criteria for complete intersections, extending classical invariant theory.

Findings

A^G is always Artin-Schelter Gorenstein.

Bounds on degrees of algebra generators are established.

Comparison with classical invariant rings highlights differences and similarities.

Abstract

The symmetric group S_n acts on the skew polynomial ring A=k_{-1}[x_1,..., x_n] as permutations. For subgroups G of S_n we consider properties of the ring of invariants A^G. We show that A^G is always Artin-Schelter Gorenstein, and we investigate when A^G is a "complete intersection", in a sense we define. We obtain bounds on the degrees of the algebra generators. The properties of A^G are compared with those in the classical case, k[x_1, ..., x_n]^G.