Plethystic Formulas and Positivity for q,t-Kostka Coefficients

Abstract

Our results concern the Macdonald q, t-Kostka coefficients K λµ(q, t). More precisely we work here with the expressions \(\tilde K_{\lambda \mu } (q,t) = t^{n(\mu )} K_{\lambda \mu } (q,1/t)\) where n(µ) denotes the sum of the legs of the cells of µ. The K λµ (q, t) have been conjectured by Macdonald to be polynomials in q, t with positive integer coefficients. We prove here the Macdonald conjecture for arbitrary µ when λ is an augmented hook. Our proof is based on explicit formulas yielding \(\tilde K_{\lambda \mu } (q,t)\) as a symmetric polynomial plethystically evaluated at \(\tilde K_{(n - 1,1),\mu } (q,t)\).More precisely, it can be shown that for each γ ⊢ k there is a unique symmetric polynomial kγ (x; q, t), of degree ≤ k in x, such that for any λ of the form λ = (n − k, γ), we have \(\tilde K_{(n - 1,1),\mu } (q,t) = k_\lambda \left[ {\tilde K_{(n - 1,1),\mu } (q,t);q,t} \right]\). The proof of existence of the polynomials kγ (x; q, t) is algorithmic. There are now two separate algorithms yielding kγ (x; q, t). We present here the chronologically first one. The second one is presented in a paper by Garsia-Tesler [12] where it yields that kγ (x;q, t) is a polynomial with integer coefficients in x, q, t, 1/q, 1/t. Although the first algorithm is not as good with denominators, we present it here since, combined with some recent work of Lapointe-Vinet, it yields what may be the simplest proof that the K λµ (q, t) are polynomials with integer coefficients.