A conjecture of Procesi and a new basis for the decomposition of the graded left regular representation of \(S_ n\)
Let \(\mathbb{R}^*\) denote the quotient of the ring of polynomials in \(n\) variables over the ideal of symmetric polynomials in these variables. The graded character \(\text{char}_ q \mathbb{R}^*\) decomposes into irreducible constituents by \[ \text{char}_ q \mathbb{R}^*= \sum_ \lambda \chi^ \lambda \sum_ S q^{| C(S)|}, \] where \(\chi^ \lambda\) denotes the irreducible character indexed by the partition \(\lambda\), \(S\) ranges over all standard tableaux of shape \(\lambda\), and \(| C(S)|\) is the sum of the entries in the cocharge tableau obtained from \(S\). The author finds a basis for \(\mathbb{R}^*\) that has natural triangularity and explicitly exhibits this decomposition. Furthermore, when this basis is restricted to \(\mathbb{R}_ \mu\), one of a family of \(S_ n\)-modules recently studied by Garsia and Procesi, it yields a basis for this module. This then suggests a new approach to the charge interpretation for the \(q\)-Kostka-Foulkes polynomials \(K_{\lambda\mu} (q)\).
scientific article; zbMATH DE number 490809