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The q, t - Kostka polynomials $K_{\lambda\mu}(q, t)$ are defined as follows (all notations I do not explain here come from the classical book by Macdonald: Symmetric Functions and Hall polynomials, 2nd edition).

Let $J_{\mu}(x; q, t)$ be the integral form of the Macdonald polynomial $P_{\mu}(x; q, t)$ and $S_{\lambda}(x; t)$ be the Schur functions associated with the product $\prod{(1 - tx_i)/(1 - x_i)}$, explicitly $$S_{\lambda}(x; t) = \sum_{\rho}{z_{\rho}^{-1}\chi_{\rho}^{\lambda}p_{\rho}(x; t)},$$ where $$p_{\rho}(x; t) = p_{\rho}(x) \prod_{i=1}^{l(\rho)}(1 - t^{\rho_i}).$$ Then $K_{\lambda\mu}(q, t)$ are defined by the equality $$J_{\mu}(x; q, t) = \sum_{\lambda}{K_{\lambda\mu}(q, t)S_{\lambda}(x; t)}.$$ It is not hard to prove that $K_{\lambda\mu}(0, 0) = \delta_{\lambda\mu}$ and $K_{\lambda\mu}(0, 1) = K_{\lambda\mu}$ are the classical Kostka numbers. There is another equivalent way to define $K_{\lambda\mu}(q, t)$ by using a different form of Macdonald polynomials $H_{\lambda}$ (and Schur polynomials $s_{\mu}$ replace the variations $S_{\lambda}$) that is better-known to combinatorialists.

It was conjectured by Macdonald that $K_{\lambda\mu}(q, t) \in \mathbb{N}[q, t]$ and it was proved a few years later by Haiman, if I'm not mistaken.

My question: has there been research on "some kind of analogues" of q, t - Kostka polynomials for root systems other than that of type A? If so, is there any result or conjecture on positivity or integrality of such analogue? Is such analogue exists, it'd be a polynomial on $q, t_{\alpha}$, where $\alpha$ varies over $W$-orbit representatives of simple roots. Even if not, I'd appreciate some references on known generalizations of q, t - Kostka polynomials that may be relevant.

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  • $\begingroup$ I'm not sure what reference tools you use, but for example a fairly recent preprint (since published in a combinatorics journal) explores Spin analogues: front.math.ucdavis.edu/1102.4901 $\endgroup$ Dec 8, 2015 at 23:25
  • $\begingroup$ You might find some ideas in Jim's book: math.upenn.edu/~jhaglund/books/qtcat.pdf There is definition for Macdonald polynomials for general root systems, and hence also Schur polynomials. You can therefore do the classical plethystic substitution, and expand and see what you get I guess... $\endgroup$ Dec 9, 2015 at 0:00

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