Koornwinder polynomials

introduction

The Koornwinder polynomials [43] are a generalisation of the Macdonald polynomials to the root system BCn.
- $n = 1$, they correspond to the 5-parameter Askey-Wilson polynomials
- $n \geq 2$, they depend on six parameters,

definition

Throughout this section $x=(x_1,\dots,x_n)$. Then the Koornwinder density is given by \begin{equation}\label{Eq_Kdensity} \Delta(x;q,t;t_0,t_1,t_2,t_3):= \prod_{i=1}^n \frac{(x_i^{\pm 2};q)_{\infty}} {\prod_{r=0}^3 (t_r x_i^{\pm};q)_{\infty}} \prod_{1\leq i<j\leq n} \frac{(x_i^{\pm}x_j^{\pm};q)_{\infty}} {(tx_i^{\pm}x_j^{\pm};q)_{\infty}}, \end{equation} where \begin{align*} (x_i^{\pm};q)_{\infty}&:=(x_i,x_i^{-1};q)_{\infty} \\ (x_i^{\pm}x_j^{\pm};q)_{\infty}&:= (x_ix_j,x_ix_j^{-1},x_i^{-1}x_j,x_i^{-1}x_j^{-1};q)_{\infty}. \end{align*} For complex $q,t,t_0,\dots,t_3$ such that $\lvert{q}\rvert,\lvert{t}\rvert,\lvert{t_0}\rvert,\dots,\lvert{t_3}\rvert<1$ this defines a scalar product on $\mathbb{C}[x^{\pm 1}]$ via \[ \langle{f}{g}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)}:= \int_{\mathbb{T}^n} f(x) g(x^{-1})\Delta(x;q,t;t_0,t_1,t_2,t_3) \,d T(x), \] where \[ d T(x):=\frac{1}{2^n n! (2\pi i)^n}\, \frac{d x_1}{x_1}\cdots \frac{d x_n}{x_n}. \] Let $W=\mathfrak{S}_n\ltimes (\Z/2\Z)^n$ be the hyperoctahedral group with natural action on $\mathbb{C}[x^{\pm}]$. For $\lambda$ a partition of length at most $n$, let $m_{\lambda}^W$ be the $W$-invariant monomial symmetric function \[ m_{\lambda}^W(x):=\sum_{\alpha} x^{\alpha} \] summed over all $\alpha$ in the $W$-orbit of $\lambda$. In analogy with the Macdonald polynomials, the Koornwinder polynomials $K_{\lambda}=K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3)$ are defined as the unique family of polynomials in $\Lambda^{\mathrm{BC}_n}:=\mathbb{C}[x^{\pm}]^W$ such that [43] \[ K_{\lambda}=m^W_{\lambda}+\sum_{\mu<\lambda} c_{\lambda\mu} m^W_{\mu} \] and \begin{equation}\label{Eq_KKnul} \langle{K_{\lambda}}{K_{\mu}}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)} =0 \qquad\text{if }\lambda\neq\mu. \end{equation} From the definition it follows that the $K_{\lambda}$ are symmetric under permutation of the $t_r$. The quadratic norm was first evaluated in \cite{vDiejen96} (selfdual case) and \cite{Sahi99} (general case). For our purposes we only need \begin{equation}\label{Eq_Gus} \langle{1}{1}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)} =\prod_{i=1}^n \frac{(t,t_0t_1t_2t_3t^{n+i-2};q)_{\infty}} {(q,t^i;q)_{\infty}\prod_{0\leq r<s\leq 3}(t_rt_st^{i-1};q)_{\infty}}, \end{equation} known as Gustafson's integral [Gustafson90]

Cauchy identity

thm Mimachi ([Mimachi01] thm 2.1)

The $\mathrm{BC}_n$ analogue of the Cauchy identity is given by \begin{align}\label{Eq_Mim} \sum_{\lambda\subseteq m^n} (-1)^{\lvert{\lambda}\rvert} K_{m^n-\lambda}(x;q,t;t_0,t_1,t_2,t_3) K_{\lambda'}(y;t,q;t_0,t_1,t_2,t_3) \\ &=\prod_{i=1}^n\prod_{j=1}^m \big(x_i+x_i^{-1}-y_j-y_j^{-1}\big)\\ &=\prod_{i=1}^n\prod_{j=1}^m x_i^{-1} \big(1-x_iy_j^{\pm}\big), \end{align} where $y=(y_1,\dots,y_m)$ and $(a-b^{\pm}):=(a-b)(a-b^{-1})$.

history

Several years after the work of Askey and Wilson, Koornwinder extended the Askey–Wilson polynomials to a family of multivariable Laurent polynomials labelled by the non-reduced root system $BC_n$
The various families of Macdonald (orthogonal) polynomials for classical root systems are all contained in the Koornwinder polynomials, and for a long time it was assumed they represented the highest possible level of generalisation.

related items

encyclopedia

https://en.wikipedia.org/wiki/Koornwinder_polynomials

expositions

Stokman, Jasper V. “Lecture Notes on Koornwinder Polynomials.” In Laredo Lectures on Orthogonal Polynomials and Special Functions, 145–207. Adv. Theory Spec. Funct. Orthogonal Polynomials. Nova Sci. Publ., Hauppauge, NY, 2004. http://www.ams.org/mathscinet-getitem?mr=2085855.
Stokman, Jasper V. “Macdonald-Koornwinder Polynomials.” arXiv:1111.6112 [math], November 25, 2011. http://arxiv.org/abs/1111.6112.

articles

Corteel, Sylvie, and Lauren Williams. ‘Macdonald-Koornwinder Moments and the Two-Species Exclusion Process’. arXiv:1505.00843 [cond-Mat, Physics:nlin], 4 May 2015. http://arxiv.org/abs/1505.00843.
Stokman, Jasper, and Bart Vlaar. “Koornwinder Polynomials and the XXZ Spin Chain.” Journal of Approximation Theory 197 (September 2015): 69–100. doi:10.1016/j.jat.2014.03.003.
van Diejen, J. F., and E. Emsiz. “Branching Formula for Macdonald-Koornwinder Polynomials.” arXiv:1408.2280 [math], August 10, 2014. http://arxiv.org/abs/1408.2280.
Rains, Eric M. “BC_n-Symmetric Polynomials.” arXiv:math/0112035, December 4, 2001. http://arxiv.org/abs/math/0112035.
K. Mimachi, A duality of Macdonald--Koornwinder polynomials and its application to integral representations, Duke Math. J.107 (2001), 265--281.
S. Sahi, Nonsymmetric Koornwinder polynomials and duality, Ann. of Math. (2) 150 (1999), 267--282.
Stokman, J. V. “Koornwinder Polynomials and Affine Hecke Algebras.” arXiv:math/0002090, February 11, 2000. http://arxiv.org/abs/math/0002090.
J. F. van Diejen, Self-dual Koornwinder--Macdonald polynomials, Invent. Math. 126 (1996), 319--339.
[43] T. H. Koornwinder, Askey–Wilson polynomials for root systems of type BC in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, pp. 189–204, Contemp. Math. 138, Amer. Math. Soc., Providence, 1992. http://oai.cwi.nl/oai/asset/2292/2292A.pdf
[Gustafson90] R. A. Gustafson, A generalization of Selberg's beta integral, Bull. Amer. Math. Soc. (N.S.) 22 (1990), 97--105.