The decomposition of a bigraded left regular representation of the diagonal action of \(S_ n\)
Let \(\mu= (\mu_1\geq \mu_2\geq\cdots\geq \mu_{k+ 1})= (k, 1^{n- k})\) be a partition of \(n\). The diagonal action of the symmetric group \(S_n\) on the space of harmonic polynomials \({\mathcal H}_\mu\) affords the left regular representation \(\rho\) of \(S_n\). In their forthcoming paper \textit{A. M. Garsia} and \textit{M. Haiman} define a bigraded character of the diagonal action of \(S_n\) on \({\mathcal H}_\mu\) and show that the character multiplicities are polynomials \(\widetilde K_{\lambda, \mu}(q, t)\) that are closely related to the Macdonald-Kostka polynomials \(K_{\lambda, \mu}(q, t)\). In the paper under review the author constructs a collection of polynomials \(B(\mu)\) that form a basis \(\beta(\mu)\) for \({\mathcal H}_\mu\) which exhibits the decomposition of \({\mathcal H}_\mu\) into its irreducible parts. Let \(B_\lambda(\mu)\) be a collection of bihomogeneous polynomials that span the irreducible representations of shape \(\lambda\). The author constructs \(B_\lambda(\mu)\) as a collection of polynomials indexed by a pair \((S, C)\) of tableaux of shape \(\lambda\) where \(S\) is standard and \(C\) is a biletter cocharge tableau. He also defines the diagonal action of \(S_n\) on \(\beta(\mu)\) and gives a combinatorial interpretation of \(\widetilde K_{\lambda, \mu}(q, t)\). As the author notes \textit{John R. Stembridge} in [Some particular entries of the two-parameter Kostka matrix, Proc. Am. Math. Soc. 121, No. 2, 367-373 (1994; Zbl 0805.05087)] has obtained also a combinatorial interpretation for the latter coefficients of \(\widetilde K_{\lambda, \mu}(q, t)\). His methods are not related to those of the reviewed paper and not concerned with the Garsia-Haiman modules.
scientific article; zbMATH DE number 774647