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

Ayumu Hoshino ; Jun'ichi Shiraishi

Symmetry, integrability and geometry: methods and applications, Tome 16 (2020)

Résumé

We present an explicit formula for the transition matrix $\mathcal{C}$ from the type $BC_n$ Koornwinder polynomials $P_{(1^r)}(x|a,b,c,d|q,t)$ with one column diagrams, to the type $BC_n$ monomial symmetric polynomials $m_{(1^{r})}(x)$. The entries of the matrix $\mathcal{C}$ enjoy a set of four terms recursion relations. These recursions provide us with the branching rules for the Koornwinder polynomials with one column diagrams, namely the restriction rules from $BC_n$ to $BC_{n-1}$. To have a good description of the transition matrices involved, we introduce the following degeneration scheme of the Koornwinder polynomials: $P_{(1^r)}(x|a,b,c,d|q,t) \longleftrightarrow P_{(1^r)}(x|a,-a,c,d|q,t)\longleftrightarrow P_{(1^r)}(x|a,-a,c,-c|q,t) \longleftrightarrow P_{(1^r)}\big(x|t^{1/2}c,-t^{1/2}c,c,-c|q,t\big) \longleftrightarrow P_{(1^r)}\big(x|t^{1/2},-t^{1/2},1,-1|q,t\big)$. We prove that the transition matrices associated with each of these degeneration steps are given in terms of the matrix inversion formula of Bressoud. As an application, we give an explicit formula for the Kostka polynomials of type $B_n$, namely the transition matrix from the Schur polynomials $P^{(B_n,B_n)}_{(1^r)}(x|q;q,q)$ to the Hall–Littlewood polynomials $P^{(B_n,B_n)}_{(1^r)}(x|t;0,t)$. We also present a conjecture for the asymptotically free eigenfunctions of the $B_n$ $q$-Toda operator, which can be regarded as a branching formula from the $B_n$ $q$-Toda eigenfunction restricted to the $A_{n-1}$ $q$-Toda eigenfunctions.

Détail

Keywords: Koornwinder polynomial, degeneration scheme, Kostka polynomial of type $B_n$, $q$-Toda eigenfunction.

Bibliographie

[1] Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., 54, 1985, iv+55 pp. | DOI | MR

[2] Braverman A., Finkelberg M., Shiraishi J., “Macdonald polynomials, Laumon spaces and perverse coherent sheaves”, Perspectives in Representation Theory, Contemp. Math., 610, Amer. Math. Soc., Providence, RI, 2014, 23–41, arXiv: 1206.3131 | DOI | MR | Zbl

[3] Bressoud D. M., “A matrix inverse”, Proc. Amer. Math. Soc., 88 (1983), 446–448 | DOI | MR | Zbl

[4] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 35, Cambridge University Press, Cambridge, 1990 | MR | Zbl

[5] Hoshino A., Noumi M., Shiraishi J., “Some transformation formulas associated with Askey–Wilson polynomials and Lassalle's formulas for Macdonald–Koornwinder polynomials”, Mosc. Math. J., 15 (2015), 293–318, arXiv: 1406.1628 | DOI | MR | Zbl

[6] Hoshino A., Shiraishi J., “Macdonald polynomials of type $C_n$ with one-column diagrams and deformed Catalan numbers”, SIGMA, 14 (2018), 101, 33 pp., arXiv: 1801.09939 | DOI | MR | Zbl

[7] Koornwinder T. H., “Askey–Wilson polynomials for root systems of type $BC$”, Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., 138, Amer. Math. Soc., Providence, RI, 1992, 189–204 | DOI | MR | Zbl

[8] Lassalle M., “Some conjectures for Macdonald polynomials of type $B$, $C$, $D$”, Sém. Lothar. Combin., 52 (2004), B52h, 24 pp., arXiv: math.CO/0503149 | MR | Zbl

[9] Macdonald I. G., “Orthogonal polynomials associated with root systems”, Sém. Lothar. Combin., 45 (2000), B45a, 40 pp., arXiv: math.QA/0011046 | MR | Zbl

[10] Noumi M., Shiraishi J., “A direct approach to the bispectral problem for the Ruijsenaars–Macdonald $q$-difference operators”, arXiv: 1206.5364 | MR

[11] Rains E. M., Warnaar S. O., “Bounded Littlewood identities”, Mem. Amer. Math. Soc. (to appear) | MR

[12] Stokman J. V., “Macdonald–Koornwinder polynomials”, Multivariable Special Functions, eds. T. H. Koornwinder, J.V. Stokman, Cambridge University Press, Cambridge (to appear) | Zbl