Representation theory of symmetric groups and the strong Lefschetz property

Abstract

We investigate the structure and properties of an Artinian monomial complete intersection quotient $A(n,d)=\mathbf{k}[x_1,\dots ,x_n]/(x_1^d,\dots ,x_n^d)$. 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$.