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q,t-Kostka Polynomials

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    Let \(\Lambda\) denote the algebra of symmetric functions in a finite or infinite alphabet \(X = \{x_1,x_2,\ldots\}\) with coefficients in the field of rational functions \(\mathbb{Q}(q,t)\). Also denote by \(\Lambda_{\mathbb{Z}[q,t]}\) the algebra of symmetric functions in \(X\) with coefficients in \(\mathbb{Z}[q,t]\). Inside the plethystic brackets \([]\), we adopt the convention that \(X\) stands for \(x_1+x_2+\ldots\). For \(P \in \Lambda\), \(P[X]\) means \(P(x_1,x_2,\ldots)\). We will also let \(X_n\) denote the finite alphabet \(x_1+x_2+\ldots+x_n\).

    We denote partitions by their French Ferrers diagrams, that is with rows decreasing from bottom to top. For a partition \(\mu\) of length no greater than \(k\), denote by \(\mu+1^k\) the partition obtained by prepending a column of length \(k\) to the diagram of \(\mu\).

    Definition of q,t-Kostka Polynomials

    For a partition \(\lambda\), let \(P_{\lambda}(x;q,t)\) denote the Macdonald polynomial of shape \(\lambda\) and let \(Q_{\lambda}(x;q,t)\) denote its dual. Define the integral form \(J_{\lambda}(x;q,t)\) of these Macdonald polynomials as \[J_{\lambda}(x;q,t) = h_{\lambda}(q,t)P_{\lambda}(x;q,t) = h'_{\lambda}(q,t)Q_{\lambda}(x;q,t)\] with \[h_{\lambda}(q,t) = \prod_{s \in \lambda}{(1-q^{a_{\lambda}(s)}t^{l_{\mu}(s)+1})}, \;\;\;\; h'_{\lambda}(q,t) = \prod_{s \in \lambda}{1-q^{a_{\lambda}(s)+1}t^{l_{\mu}(s)}}\] where, for a cell \(s \in \lambda\), \(a_{\lambda}(s)\) and \(l_{\lambda}(s)\) represent respectively the arm and leg of s in \(\lambda\), that is the number of cells of \(\lambda\) that are respectively strictly east and north of \(s\).

    Macdonald showed that \[J_{\mu}[X;q,t] = \sum_{\lambda}{S_{\lambda}[X(1-t)]K_{\lambda\mu}(q,t)}\] for coefficients \(K_{\lambda\mu}(q,t)\) in \(\mathbb{Q}(q,t)\). It was conjectured that these coefficients, called the \(q,t\)-Kostka polynomials, are in fact polynomials in \(\mathbb{Z}[q,t]\).

    Proof of Polynomiality

    Let \[H_{\mu}[X;q,t] = J_{\mu}[\frac{X}{1-t};q,t].\] We now have direct access to the \(q,t\)-Kostka coefficients because \[H_{\mu}[X;q,t] = \sum_{\lambda}{S_{\lambda}[X]K_{\lambda \mu}(q,t)}.\] Also let \[H_{\mu}[X;t] = Q_{\mu}[\frac{X}{1-t};t] = \sum_{\lambda}{S_{\lambda}[X]K_{\lambda\mu}(t)}.\]

    The following theorem is central to Garsia and Zabrocki's proof of the polynomiality of the \(q,t\)-Kostka coefficients.

    Theorem: For any linear operator \(V\) acting on \(\Lambda\) and \(P \in \Lambda\) set \[\tilde{V}^qP[X] = V^YP[qX+(1-q)Y]|_{Y=X}\] where \(V^Y\) is simply \(V\) acting on polynomials in the \(Y\) variables. This given, if \(G_k = G_k(X,t)\) is any linear operator on \(\Lambda\) with the property that \[G_kH_{\mu}[X;t] = H_{\mu+1^k}[X;t]\] for all \(\mu\) of length no greater than \(k\), then \(\tilde{G_k}^q\) has the property \[\tilde{G_k}^qH_{\mu}[X;q,t] = H_{\mu+1^k}[X;q,t]\] for all \(\mu\) of length no greater than \(k\). In particular, the modified Macdonald polynomials \(H_{\mu}[X;q,t]\) may be obtained from the "Rodriguez" formula: \[H_{\mu}[X;q,t] = \tilde{G_{\mu'_1}}^q\tilde{G_{\mu'_2}}^q \cdots \tilde{G_{\mu'_h}}^q \textbf{1}\] where \(\mu'= (\mu'_1,\mu'_2,\ldots,\mu'_h)\) denotes the conjugate of \(\mu\).

    Given this theorem, the polynomiality of the \(q,t\)-Kostka coefficients follows:

    Define the "trivial" operator \(TG_k = TG_k(X;t)\) by setting for the \(\{H_{\mu}[X;t]\}_{\mu}\) basis \[TG_kH_{\mu}[X;t] = \begin{cases}
    H_{\mu+1^k}[X;t] & \mbox{if } l(\mu) \leq k \\
    0 & \mbox{otherwise. }
    \end{cases} \] The Kostka-Foulkes matrix \(K(t) = \|K_{\lambda\mu}(t)\|\) is the transition matrix between the \(\{H_{\mu}[X;t]\}_{\mu}\) basis and the Schur functions. Since \(K(t)\) is unitriangular with entries in \(\mathbb{Z}[t]\), it follows that its inverse \(H(t) = K(t)^{-1}\) has entries in \(\mathbb{Z}[t]\). This implies that \(TG_kS_{\lambda}[X]\) is an integral linear combination of Schur functions. Since the operator \(TG_k\) acts integrally on the Schur basis, the desired result \(K_{\lambda\mu}(q,t) \in \mathbb{Z}[q,t]\) is an immediate consequence of the Rodriguez formula in the above theorem with \(G = TG\).


    Adriano M. Garsia and Mike Zabrocki, Polynomiality of the q,t-Kostka Revisited. Algebraic Combinatorics and Computer Science, 2001, pp 473-491.


    • Roger Tian (UC Davis)

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