Hilbert Functions of $\mathfrak S_n$-Stable Artinian Gorenstein Algebras

TL;DR

This paper investigates the structure of symmetric group-stable Artinian Gorenstein algebras, focusing on their Hilbert functions and graded characters, revealing new insights into their algebraic and representation-theoretic properties.

Contribution

It characterizes the Hilbert functions and graded characters of specific symmetric group-stable Artinian Gorenstein algebras with trivial socles and orbit-generated apolar submodules.

Findings

Explicit descriptions of Hilbert functions for these algebras

Character formulas for their graded representations

Connections between algebraic structure and symmetric group actions

Abstract

We describe the graded characters and Hilbert functions of certain graded artinian Gorenstein quotients of the polynomial ring which are also representations of the symmetric group. Specifically, we look at those algebras whose socles are trivial representations and whose principal apolar submodules are generated by the sum of the orbit of a power of a linear form.