Representation theory of symmetric groups and the strong Lefschetz property

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$.