The decomposition of a bigraded left regular representation of the diagonal action of Sn

AbstractLet μ = (μ1 ⩾ μ2 ⩾ ⋯ ⩾ μk + 1) = (k, 1n − k) be a partition of n. In [GH] Garsia and Haiman show that the diagonal action of Sn on the space of harmonic polynomials Hμ affords the left regular representation p of Sn. Furthermore, Garsia and Haiman define a bigraded character of the diagonal action of Sn on Hμ and show that the character multiplicities are polynomials K̃λ, μ(q, t) that are closely related to the Macdonald-Kostka polynomials Kλ, μ(q, t). In this paper we construct a collection of polynomials B(μ) that form a basis for Hμ which exhibits the decomposition of Hμ into its irreducible parts. Through this connection we give a combinatorial interpretation of the polynomials K̃λ, μ(q, t)