Koornwinder polynomials

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introduction

  • The Koornwinder polynomials, introduced in [Koornwinder92], are a generalisation of the Macdonald polynomials to the root system BCn.
    • $n = 1$, they correspond to the Askey-Wilson polynomials depending on 5-parameter $(q;t_0,t_1,t_2,t_3)$ or $(q;a,b,c,d)$
    • $n \geq 2$, they depend on six parameters $(q,t;t_0,t_1,t_2,t_3)$
  • It is known that all Macdonald polynomials associated with the classical (nonexceptional) root systems can be considered as their special cases, see Macdonald polynomials associatied with root systems
  • In Macdonald's theory of orthogonal polynomials associated to root systems, the Koornwinder-Macdonald polynomials is a family which corresponds to the non-reduced irreducible affine root system of type $(C^{\vee}_n,C_n)$

definition

def (Koornwinder density)

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 $\lambda$.

def (Koornwinder polynomial)

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}

notation

$$ \newcommand{\la}{\lambda} \newcommand{\La}{\Lambda} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\tees}{t_0,t_1,t_2,t_3} \newcommand{\B}{\mathrm B} \newcommand{\BC}{\mathrm{BC}} \newcommand{\C}{\mathrm C} \newcommand{\qbin}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}} $$

properties

  • From the definition it follows that the $K_{\lambda}$ are symmetric under permutation of the $t_r$.

quadratic norm

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 $(a-b^{\pm}):=(a-b)(a-b^{-1})$.


relation with Macdonald polynomials

  • $P_{\lambda}^{(B_n,B_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;-1,-q^{1/2},t_2^{1/2},q^{1/2})$
  • $P_{\lambda}^{(B_n,C_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;-1,-q^{1/2},t_2, t_2 q^{1/2})$
  • $P_{\lambda}^{(C_n,B_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;q^{1/2},-q^{1/2},t_2^{1/2},-t_2^{1/2})$
  • $P_{\lambda}^{(C_n,C_n)}(x;q,t,t_2)=K_{\lambda}(x;q,t;(t_2q)^{1/2},-(t_2q)^{1/2},t_2^{1/2},-t_2^{1/2})$


expression in terms of interpolation polynomials

  • binomial coefficient

\[ {\lambda \brack \mu}_{q,t,s} := \frac{\bar{P}^{*(n)}_\mu(\lambda;q,t,s t^{1-n})} {\bar{P}^{*(n)}_\mu(\mu;q,t,s t^{1-n})} \]

  • Okounkov's binomial formula (Theorem 7.10, [Okounkov98]) gives an expansion of the Koornwinder polynomials in terms of BCn interpolation polynomials
  • The coefficients in this expansion are given by $BC_n$ $q$-binomial coefficients ${\lambda \brack \mu}_{q,t,s}$ times a ratio of principally specialised Koornwinder polynomials:

$$ K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3) =\sum_{\mu\subseteq\lambda} {\lambda \brack \mu}_{q,t,s} \, \frac{K_{\lambda}\big(t_0(1,t,\dots,t^{n-1});q,t;t_0,t_1,t_2,t_3\big)} {K_{\mu}\big(t_0(1,t,\dots,t^{n-1});q,t;t_0,t_1,t_2,t_3\big)}\, \bar{P}_{\mu}^{\ast(n)}(x;q,t,t_0), $$ where $s=t^{n-1}\sqrt{t_0t_1t_2t_3/q\,}$.


Now let \cite{Rains05} \[ C_{\lambda}^{+}(z;q,t):=\prod_{(i,j)\in \lambda} \big(1-zq^{\lambda_i+j-1}t^{2-\lambda'_j-i}\big). \] By \cite[Proposition 4.1]{Rains05} \[ \qbin{m^n}{\mu}_{q,t,s}=(-q)^{\abs{\mu}} t^{n(\mu)} q^{n(\mu')}\, \frac{(t^n,q^{-m},s^2q^mt^{1-n};q,t)_{\mu}} {C_{\mu}^{-}(q,t;q,t)C_{\mu}^{+}(s^2;q,t)} \] and the specialisation formulas \cite{vDiejen96,Sahi99} \begin{multline*} K_{\lambda}\big(t_0(1,t,\dots,t^{n-1});q,t;\tees\big) \\ =\frac{t^{n(\lambda)}}{(t_0t^{n-1})^{\abs{\la}}} \cdot \frac{(t^n,t_0t_1t^{n-1},t_0t_2t^{n-1},t_0t_3t^{n-1};q,t)_{\la}} {C^{-}_{\la}(t;q,t)C^{+}_{\la}(t_0t_1t_2t_3t^{2n-2}/q;q,t)} \end{multline*} and \cite[Corollary 3.11]{Rains05} \[ \bar{P}_{\mu}^{\ast}\big(z(1,t,\dots,t^{n-1});q,t,s\big) =\frac{t^{2n(\mu)} q^{-n(\mu')}}{(-st^{n-1})^{\abs{\mu}}}\cdot \frac{(t^n,s/z,szt^{n-1};q,t)_{\mu}} {C_{\mu}^{-}(t;q,t)} \]

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


expositions

articles

  • van Diejen, J. F., and E. Emsiz. “Branching Rules for Symmetric Hypergeometric Polynomials.” arXiv:1601.06186 [math-Ph], January 22, 2016. http://arxiv.org/abs/1601.06186.
  • 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. “BCn-Symmetric Polynomials.” Transformation Groups 10, no. 1 (March 2005): 63–132. doi:10.1007/s00031-005-1003-y. http://arxiv.org/abs/math/0112035.
  • [Mimachi01] K. Mimachi, A duality of Macdonald--Koornwinder polynomials and its application to integral representations, Duke Math. J.107 (2001), 265--281.
  • Stokman, J. V. “Koornwinder Polynomials and Affine Hecke Algebras.” arXiv:math/0002090, February 11, 2000. http://arxiv.org/abs/math/0002090.
  • S. Sahi, Nonsymmetric Koornwinder polynomials and duality, Ann. of Math. (2) 150 (1999), 267--282.
  • [Okounkov98] * Okounkov, A. “BC-Type Interpolation Macdonald Polynomials and Binomial Formula for Koornwinder Polynomials.” Transformation Groups 3, no. 2 (June 1998): 181–207. doi:10.1007/BF01236432.
  • J. F. van Diejen, Self-dual Koornwinder--Macdonald polynomials, Invent. Math. 126 (1996), 319--339.
  • van Diejen, J. F. “Commuting Difference Operators with Polynomial Eigenfunctions.” arXiv:funct-an/9306002, June 7, 1993. http://arxiv.org/abs/funct-an/9306002.
  • [Koornwinder92] 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.
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