"Koornwinder polynomials"의 두 판 사이의 차이
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* They depend on six parameters, except for n = 1 when they correspond to the 5-parameter Askey–Wilson polynomials [3]. | * They depend on six parameters, except for n = 1 when they correspond to the 5-parameter Askey–Wilson polynomials [3]. | ||
| + | |||
| + | ==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;\tees):= | ||
| + | \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 | ||
| + | $\abs{q},\abs{t},\abs{t_0},\dots,\abs{t_3}<1$ this | ||
| + | defines a scalar product on $\Complex[x^{\pm 1}]$ via | ||
| + | \[ | ||
| + | \ip{f}{g}_{q,t;\tees}^{(n)}:= | ||
| + | \int_{\mathbb{T}^n} f(x) g(x^{-1})\Delta(x;q,t;\tees) \dup T(x), | ||
| + | \] | ||
| + | where | ||
| + | \[ | ||
| + | \dup T(x):=\frac{1}{2^n n! (2\pi\iup)^n}\, | ||
| + | \frac{\dup x_1}{x_1}\cdots \frac{\dup x_n}{x_n}. | ||
| + | \] | ||
| + | Let $W=\Symm_n\ltimes (\Z/2\Z)^n$ be the hyperoctahedral group | ||
| + | with natural action on $\Complex[x^{\pm}]$. For $\la$ a partition | ||
| + | of length at most $n$, let $m_{\la}^W$ be the $W$-invariant monomial | ||
| + | symmetric function | ||
| + | \[ | ||
| + | m_{\la}^W(x):=\sum_{\alpha} x^{\alpha} | ||
| + | \] | ||
| + | summed over all $\alpha$ in the $W$-orbit of $\la$. | ||
| + | In analogy with the Macdonald polynomials, the Koornwinder | ||
| + | polynomials $K_{\la}=K_{\la}(x;q,t;\tees)$ | ||
| + | are defined as the unique family of polynomials in | ||
| + | $\La^{\BC_n}:=\Complex[x^{\pm}]^W$ such that \cite{Koornwinder92} | ||
| + | \[ | ||
| + | K_{\la}=m^W_{\la}+\sum_{\mu<\la} c_{\la\mu} m^W_{\mu} | ||
| + | \] | ||
| + | and | ||
| + | \begin{equation}\label{Eq_KKnul} | ||
| + | \ip{K_{\la}}{K_{\mu}}_{q,t;\tees}^{(n)} | ||
| + | =0 \qquad\text{if }\la\neq\mu. | ||
| + | \end{equation} | ||
| + | From the definition it follows that the $K_{\la}$ 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} | ||
| + | \ip{1}{1}_{q,t;\tees}^{(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 \cite{Gustafson90}. | ||
| + | |||
| + | The $\BC_n$ analogue of the Cauchy identity \eqref{Eq_Mac-Cauchy-a}, | ||
| + | is given by \cite[Theorem 2.1]{Mimachi01} | ||
| + | \begin{multline}\label{Eq_Mim} | ||
| + | \sum_{\la\subseteq m^n} (-1)^{\abs{\la}} | ||
| + | K_{m^n-\la}(x;q,t;\tees) K_{\la'}(y;t,q;\tees) \\[-1mm] | ||
| + | =\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{multline} | ||
| + | where $y=(y_1,\dots,y_m)$ and $(a-b^{\pm}):=(a-b)(a-b^{-1})$. | ||
==history== | ==history== | ||
2015년 8월 9일 (일) 23:21 판
introduction
- The Koornwinder polynomials [43] are a generalisation of the Macdonald polynomials to the root system BCn.
- They depend on six parameters, except for n = 1 when they correspond to the 5-parameter Askey–Wilson polynomials [3].
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;\tees):= \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 $\abs{q},\abs{t},\abs{t_0},\dots,\abs{t_3}<1$ this defines a scalar product on $\Complex[x^{\pm 1}]$ via \[ \ip{f}{g}_{q,t;\tees}^{(n)}:= \int_{\mathbb{T}^n} f(x) g(x^{-1})\Delta(x;q,t;\tees) \dup T(x), \] where \[ \dup T(x):=\frac{1}{2^n n! (2\pi\iup)^n}\, \frac{\dup x_1}{x_1}\cdots \frac{\dup x_n}{x_n}. \] Let $W=\Symm_n\ltimes (\Z/2\Z)^n$ be the hyperoctahedral group with natural action on $\Complex[x^{\pm}]$. For $\la$ a partition of length at most $n$, let $m_{\la}^W$ be the $W$-invariant monomial symmetric function \[ m_{\la}^W(x):=\sum_{\alpha} x^{\alpha} \] summed over all $\alpha$ in the $W$-orbit of $\la$. In analogy with the Macdonald polynomials, the Koornwinder polynomials $K_{\la}=K_{\la}(x;q,t;\tees)$ are defined as the unique family of polynomials in $\La^{\BC_n}:=\Complex[x^{\pm}]^W$ such that \cite{Koornwinder92} \[ K_{\la}=m^W_{\la}+\sum_{\mu<\la} c_{\la\mu} m^W_{\mu} \] and \begin{equation}\label{Eq_KKnul} \ip{K_{\la}}{K_{\mu}}_{q,t;\tees}^{(n)} =0 \qquad\text{if }\la\neq\mu. \end{equation} From the definition it follows that the $K_{\la}$ 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} \ip{1}{1}_{q,t;\tees}^{(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 \cite{Gustafson90}.
The $\BC_n$ analogue of the Cauchy identity \eqref{Eq_Mac-Cauchy-a}, is given by \cite[Theorem 2.1]{Mimachi01} \begin{multline}\label{Eq_Mim} \sum_{\la\subseteq m^n} (-1)^{\abs{\la}} K_{m^n-\la}(x;q,t;\tees) K_{\la'}(y;t,q;\tees) \\[-1mm] =\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{multline} 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.
encyclopedia
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.
- Stokman, J. V. “Koornwinder Polynomials and Affine Hecke Algebras.” arXiv:math/0002090, February 11, 2000. http://arxiv.org/abs/math/0002090.
- [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
- [3] R. Askey and J. A. Wilson, Some basic hypergeometric polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. Vol. 319, AMS, Providence, RI, 1985.