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$.
Categories: math.RT