Representation theory of symmetric groups and the strong Lefschetz property

TL;DR

This paper explores the representation theory of symmetric groups acting on specific Artinian monomial complete intersections, providing explicit bases and multiplicity formulas for their module decompositions.

Contribution

It constructs explicit bases compatible with symmetric group actions and derives multiplicity formulas for irreducible components in the module decomposition.

Findings

Explicit homogeneous bases for A(n,d) with n=3

Recursive and closed-form multiplicity formulas

Decomposition of modules into irreducible components

Abstract

We investigate the structure and properties of an Artinian monomial complete intersection quotient $A(n,d)=\mathbf{k} [x_{1}, \ldots, x_{n}] \big / (x_{1}^{d}, \ldots, x_{n}^d)$. We construct explicit homogeneous bases of $A(n,d)$ that are compatible with the $S_{n}$-module structure for $n=3$, all exponents $d \ge 3$ and all homogeneous degrees $j \ge 0$. Moreover, we derive the multiplicity formulas, both in recursive form and in closed form, for each irreducible component appearing in the $S_{3}$-module decomposition of homogeneous subspaces. 4, 5$.