Polynomiality of the \(q,t\)-Kostka revisited.

*(English)*Zbl 0969.05063
Crapo, H. (ed.) et al., Algebraic combinatorics and computer science. A tribute to Gian-Carlo Rota. Milano: Springer. 473-491 (2001).

From the introduction: The polynomiality problem for the \(q, t\)-Kostka coefficients was posed by Macdonald in the fall 1988 meeting of the Lotharingian seminar. It remained open for quite a few years, when suddenly, in 1996, several proofs of varied difficulty appeared in a period of only a few months. At the present there are three basically different approaches to proving the polynomiality of the \(q, t\)-Kostka coefficients: (1) via plethystic formulas (Garsia-Tesler, Garsia-Remmel); (2) via vanishing properties (Sahi, Knop); (3) via Rodrigues formulas (Lapointe-Vinet, Kirillov-Noumi).

Each of these approaches has its own special advantages. The plethystic approach led to very efficient algorithms for computing these coefficients and ultimately produced remarkably simple explicit formulas. The vanishing properties approach led to the discovery of basic non-symmetric variants of the Macdonald polynomials with remarkable combinatorial implications that still remain to be fully explored. The approach via Rodrigues formulas stems from a pioneering paper of Lapointe-Vinet on Jack polynomials. Although originally it was based on deep affine Hecke algebra identities, eventually the idea led to some of the most elementary proofs of the polynomiality result. In particular it produced a family of symmetric function operators \(\{B^{q,t}_k\}_{k= 1,2,\dots}\) which permitted the construction of the Macdonald “integral forms” \(J_\mu(x; q,t)\), one part at a time, starting from 1, according to an identity of the form \[ J_{\mu'}(x; q,t)= B^{q,t}_{\mu_1} B^{q,t}_{\mu_2}\cdots B^{q,t}_{\mu_k}\text{\textbf{1}}. \] Our main contribution here is a remarkably simple argument which shows that families of operators \(B^{q,t}_k\), yielding such a formula, may be obtained by \(q\)-twisting in a minor way any sequence of operators \(\{B^t_k\}_{k= 1,2,\dots}\) which yields the analogous formula \[ Q_{\mu'}(x; t)= B^t_{\mu_1} B^t_{\mu_2}\cdots B^t_{\mu_k}\text{\textbf{1}} \] for the Hall-Littlewood polynomial \(Q_\mu(x; q,t)\). As a byproduct we obtain that the polynomiality of the \(q, t\)-Kostka is an immediate consequence of the polynomiality of the Kostka-Foulkes coefficients. What is surprising is that this fact was missed for so many years by researchers in this area. What might be even more surprising is that we obtain a remarkably general result by further simplifying some of the arguments used in L. Lapointe and L. Vinet [Int. Math. Res. Not. 1995, No. 9, 419-424 (1995; Zbl 0868.33009)] and A. Kirillov and M. Noumi [CRM Proc. Lect. Notes 22, 227-243 (1999; Zbl 0947.33015)].

For the entire collection see [Zbl 0956.00018].

Each of these approaches has its own special advantages. The plethystic approach led to very efficient algorithms for computing these coefficients and ultimately produced remarkably simple explicit formulas. The vanishing properties approach led to the discovery of basic non-symmetric variants of the Macdonald polynomials with remarkable combinatorial implications that still remain to be fully explored. The approach via Rodrigues formulas stems from a pioneering paper of Lapointe-Vinet on Jack polynomials. Although originally it was based on deep affine Hecke algebra identities, eventually the idea led to some of the most elementary proofs of the polynomiality result. In particular it produced a family of symmetric function operators \(\{B^{q,t}_k\}_{k= 1,2,\dots}\) which permitted the construction of the Macdonald “integral forms” \(J_\mu(x; q,t)\), one part at a time, starting from 1, according to an identity of the form \[ J_{\mu'}(x; q,t)= B^{q,t}_{\mu_1} B^{q,t}_{\mu_2}\cdots B^{q,t}_{\mu_k}\text{\textbf{1}}. \] Our main contribution here is a remarkably simple argument which shows that families of operators \(B^{q,t}_k\), yielding such a formula, may be obtained by \(q\)-twisting in a minor way any sequence of operators \(\{B^t_k\}_{k= 1,2,\dots}\) which yields the analogous formula \[ Q_{\mu'}(x; t)= B^t_{\mu_1} B^t_{\mu_2}\cdots B^t_{\mu_k}\text{\textbf{1}} \] for the Hall-Littlewood polynomial \(Q_\mu(x; q,t)\). As a byproduct we obtain that the polynomiality of the \(q, t\)-Kostka is an immediate consequence of the polynomiality of the Kostka-Foulkes coefficients. What is surprising is that this fact was missed for so many years by researchers in this area. What might be even more surprising is that we obtain a remarkably general result by further simplifying some of the arguments used in L. Lapointe and L. Vinet [Int. Math. Res. Not. 1995, No. 9, 419-424 (1995; Zbl 0868.33009)] and A. Kirillov and M. Noumi [CRM Proc. Lect. Notes 22, 227-243 (1999; Zbl 0947.33015)].

For the entire collection see [Zbl 0956.00018].

##### MSC:

05E05 | Symmetric functions and generalizations |

33D52 | Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) |