Macdonald Polynomials of Type C_n with One-Column Diagrams and Deformed Catalan Numbers

Abstract

We present an explicit formula for the transition matrix C from the type C n degeneration of the Koornwinder polynomials P (1 r ) (x | a, −a, c, −c | q, t) with one column diagrams, to the type C n monomial symmetric polynomials m (1 r ) (x). The entries of the matrix C enjoy a set of three term recursion relations, which can be regarded as a (a, c, t)deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type (C n , C n ) Macdonald polynomials P (x | q; q, q) to the Hall-Littlewood polynomials P (Cn,Cn) (1 r ) (x | t; 0, t).

Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions

“…also has already been studied in [6], presented as a certain fourfold summation. Theorem 1.7 ([6, Theorem 2.2]).…”

“…As another application of our results obtained in this paper, we calculate the transition matrix from the Schur polynomials to the Hall-Littlewood polynomials, namely the Kostka polynomials of type B n , associated with one column diagrams. As for the Kostka polyonomials of types C n and D n associated with one column diagrams, see [6].…”

“…In Section 2, we derive a slightly improved version of the fourfold summation formula for the Koornwinder polynomial with one column diagram. See [6] as for the original version. In Section 3, we give the transition matrices from the Koornwinder polynomials P (1 r ) (x|a, b, c, d|q, t) with one column diagrams, to certain degenerations of the Koornwinder polynomials…”

“…This article is a continuation of [6]. Recall that in the work of Lassalle [8], Bressoud's matrix inversion formula [3] is extensively used to describe the transition matrices associated with the Macdonald polynomials of types B n , C n and D n with one column diagrams.…”

“…One of our motivations in the present paper is to establish a generalization of Lassalle's principle to the case of the Koornwinder polynomials [7] P (1 r ) (x|a, b, c, d|q, t) with full six parameters a, b, c, d, q and t. (As for the definition of P (1 r ) (x|a, b, c, d|q, t), see Section 2.) Our starting point is the new version (Theorem 1.8) of our previous fourfold summation formula obtained in [6] (Theorem 1.7). We show that the new fourfold formula can be understood as a product of four Bressoud matrices (Theorem 2.5), thereby giving us the corresponding inversion formulas automatically (Theorem 3.6).…”