"Koornwinder polynomials"의 두 판 사이의 차이
imported>Pythagoras0 |
imported>Pythagoras0 |
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| 8번째 줄: | 8번째 줄: | ||
Then the Koornwinder density is given by | Then the Koornwinder density is given by | ||
\begin{equation}\label{Eq_Kdensity} | \begin{equation}\label{Eq_Kdensity} | ||
| − | \Delta(x;q,t; | + | \Delta(x;q,t;t_0,t_1,t_2,t_3):= |
\prod_{i=1}^n \frac{(x_i^{\pm 2};q)_{\infty}} | \prod_{i=1}^n \frac{(x_i^{\pm 2};q)_{\infty}} | ||
{\prod_{r=0}^3 (t_r x_i^{\pm};q)_{\infty}} | {\prod_{r=0}^3 (t_r x_i^{\pm};q)_{\infty}} | ||
| 22번째 줄: | 22번째 줄: | ||
\end{align*} | \end{align*} | ||
For complex $q,t,t_0,\dots,t_3$ such that | 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 $\ | + | defines a scalar product on $\mathbb{C}[x^{\pm 1}]$ via |
\[ | \[ | ||
| − | \ | + | \langle{f}{g}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)}:= |
| − | \int_{\mathbb{T}^n} f(x) g(x^{-1})\Delta(x;q,t; | + | \int_{\mathbb{T}^n} f(x) g(x^{-1})\Delta(x;q,t;t_0,t_1,t_2,t_3) \,d T(x), |
\] | \] | ||
where | where | ||
\[ | \[ | ||
| − | + | d T(x):=\frac{1}{2^n n! (2\pi i)^n}\, | |
| − | \frac{ | + | \frac{d x_1}{x_1}\cdots \frac{d x_n}{x_n}. |
\] | \] | ||
| − | Let $W=\ | + | Let $W=\mathfrak{S}_n\ltimes (\Z/2\Z)^n$ be the hyperoctahedral group |
| − | with natural action on $\ | + | with natural action on $\mathbb{C}[x^{\pm}]$. For $\lambda$ a partition |
| − | of length at most $n$, let $m_{\ | + | of length at most $n$, let $m_{\lambda}^W$ be the $W$-invariant monomial |
symmetric function | symmetric function | ||
\[ | \[ | ||
| − | m_{\ | + | m_{\lambda}^W(x):=\sum_{\alpha} x^{\alpha} |
\] | \] | ||
| − | summed over all $\alpha$ in the $W$-orbit of $\ | + | summed over all $\alpha$ in the $W$-orbit of $\lambda$. |
In analogy with the Macdonald polynomials, the Koornwinder | In analogy with the Macdonald polynomials, the Koornwinder | ||
| − | polynomials $K_{\ | + | 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 | are defined as the unique family of polynomials in | ||
| − | $\ | + | $\Lambda^{\mathrm{BC}_n}:=\mathbb{C}[x^{\pm}]^W$ such that \cite{Koornwinder92} |
\[ | \[ | ||
| − | K_{\ | + | K_{\lambda}=m^W_{\lambda}+\sum_{\mu<\lambda} c_{\lambda\mu} m^W_{\mu} |
\] | \] | ||
and | and | ||
\begin{equation}\label{Eq_KKnul} | \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 }\ | + | =0 \qquad\text{if }\lambda\neq\mu. |
\end{equation} | \end{equation} | ||
| − | From the definition it follows that the $K_{\ | + | From the definition it follows that the $K_{\lambda}$ are symmetric under |
permutation of the $t_r$. | permutation of the $t_r$. | ||
The quadratic norm was first evaluated in \cite{vDiejen96} (selfdual case) | The quadratic norm was first evaluated in \cite{vDiejen96} (selfdual case) | ||
| 59번째 줄: | 59번째 줄: | ||
For our purposes we only need | For our purposes we only need | ||
\begin{equation}\label{Eq_Gus} | \begin{equation}\label{Eq_Gus} | ||
| − | \ | + | \langle{1}{1}\rangle_{q,t;t_0,t_1,t_2,t_3}^{(n)} |
=\prod_{i=1}^n | =\prod_{i=1}^n | ||
\frac{(t,t_0t_1t_2t_3t^{n+i-2};q)_{\infty}} | \frac{(t,t_0t_1t_2t_3t^{n+i-2};q)_{\infty}} | ||
| 66번째 줄: | 66번째 줄: | ||
known as Gustafson's integral \cite{Gustafson90}. | known as Gustafson's integral \cite{Gustafson90}. | ||
| − | The $\ | + | The $\mathrm{BC}_n$ analogue of the Cauchy identity |
is given by \cite[Theorem 2.1]{Mimachi01} | is given by \cite[Theorem 2.1]{Mimachi01} | ||
| − | \begin{ | + | \begin{align}\label{Eq_Mim} |
| − | \sum_{\ | + | \sum_{\lambda\subseteq m^n} (-1)^{\lvert{\lambda}\rvert} |
| − | K_{m^n-\ | + | 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 \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), | + | &=\prod_{i=1}^n\prod_{j=1}^m x_i^{-1} \big(1-x_iy_j^{\pm}\big), |
| − | \end{ | + | \end{align} |
where $y=(y_1,\dots,y_m)$ and $(a-b^{\pm}):=(a-b)(a-b^{-1})$. | where $y=(y_1,\dots,y_m)$ and $(a-b^{\pm}):=(a-b)(a-b^{-1})$. | ||
2015년 8월 9일 (일) 23:37 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;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 \cite{Koornwinder92} \[ K_{\lambda}=m^W_{\lambda}+\sum_{\mu<\lambda} c_{\lambda\mu} m^W_{\mu} \] 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 \cite{Gustafson90}.
The $\mathrm{BC}_n$ analogue of the Cauchy identity is given by \cite[Theorem 2.1]{Mimachi01} \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.
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.