"Lifted Koornwinder polynomials"의 두 판 사이의 차이

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imported>Pythagoras0
 
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==introduction==
 
==introduction==
$
+
<math>
 
\newcommand{\la}{\lambda}
 
\newcommand{\la}{\lambda}
 
\newcommand{\La}{\Lambda}
 
\newcommand{\La}{\Lambda}
$
+
</math>
  
  
* The lifted Koornwinder polynomials $\tilde{K}_{\lambda}$ are a $7$-parameter family of inhomogeneous symmetric functions
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* The lifted Koornwinder polynomials <math>\tilde{K}_{\lambda}</math> are a <math>7</math>-parameter family of inhomogeneous symmetric functions
* They are invariant under permutations of the $t_r$ and form a $\mathbb{Q}(q,t,T,t_0,t_1,t_2,t_3)$ basis of $\Lambda$.
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* They are invariant under permutations of the <math>t_r</math> and form a <math>\mathbb{Q}(q,t,T,t_0,t_1,t_2,t_3)</math> basis of <math>\Lambda</math>.
* As a function of the $t_r$ the lifted Koornwinder polynomial $\tilde{K}_{\la}$ has poles at
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* As a function of the <math>t_r</math> the lifted Koornwinder polynomial <math>\tilde{K}_{\la}</math> has poles at
 
\begin{equation}\label{Eq_poles}
 
\begin{equation}\label{Eq_poles}
 
t_0t_1t_2t_3=q^{2-\la_i-j} t^{i+\la'_j}T^{-2}, \qquad (i,j)\in\la.
 
t_0t_1t_2t_3=q^{2-\la_i-j} t^{i+\la'_j}T^{-2}, \qquad (i,j)\in\la.
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\tilde{P}^*_\mu(;q,t,T;t_0).
 
\tilde{P}^*_\mu(;q,t,T;t_0).
 
\]
 
\]
Here $\tilde{P}^*_\mu(;q,t,T;t_0)$ denotes [[Lifted BCn interpolation polynomials]]
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Here <math>\tilde{P}^*_\mu(;q,t,T;t_0)</math> denotes [[Lifted BCn interpolation polynomials]]
 
* this is analogous to the following formula for [[Koornwinder polynomials]]
 
* this is analogous to the following formula for [[Koornwinder polynomials]]
$$
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:<math>
 
K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3) =\sum_{\mu\subseteq\lambda}
 
K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3) =\sum_{\mu\subseteq\lambda}
 
{\lambda \brack \mu}_{q,t,s} \,
 
{\lambda \brack \mu}_{q,t,s} \,
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{K_{\mu}\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),
 
\bar{P}_{\mu}^{\ast(n)}(x;q,t,t_0),
$$
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</math>
  
 
==Koornwinder polynomial==
 
==Koornwinder polynomial==
* Let $K^{(n)}_\lambda$ be Koornwinder polynomials
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* Let <math>K^{(n)}_\lambda</math> be Koornwinder polynomials
  
 
;thm
 
;thm
For any integer $n>0$ and partition $\lambda$, and for generic values
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For any integer <math>n>0</math> and partition <math>\lambda</math>, and for generic values
 
of the parameters,
 
of the parameters,
 
\[
 
\[
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==examples==
 
==examples==
* For example, $\tilde{K}_0=1$ and  
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* For example, <math>\tilde{K}_0=1</math> and  
 
\[
 
\[
 
\tilde{K}_1(q,t,T;t_0,t_1,t_2,t_3)=
 
\tilde{K}_1(q,t,T;t_0,t_1,t_2,t_3)=

2020년 11월 16일 (월) 05:32 기준 최신판

introduction

\( \newcommand{\la}{\lambda} \newcommand{\La}{\Lambda} \)


  • The lifted Koornwinder polynomials \(\tilde{K}_{\lambda}\) are a \(7\)-parameter family of inhomogeneous symmetric functions
  • They are invariant under permutations of the \(t_r\) and form a \(\mathbb{Q}(q,t,T,t_0,t_1,t_2,t_3)\) basis of \(\Lambda\).
  • As a function of the \(t_r\) the lifted Koornwinder polynomial \(\tilde{K}_{\la}\) has poles at

\begin{equation}\label{Eq_poles} t_0t_1t_2t_3=q^{2-\la_i-j} t^{i+\la'_j}T^{-2}, \qquad (i,j)\in\la. \end{equation}

definition

The lifted Koornwinder polynomials are defined by the expansion \[ \tilde{K}_\lambda(;q,t,T;t_0,t_1,t_2,t_3) = \sum_{\mu\subset\lambda} {\lambda \brack \mu}_{q,t,(T/t)\sqrt{t_0t_1t_2t_3/q}} \frac{k^0_\lambda(q,t,T;t_0{:}t_1,t_2,t_3)} {k^0_\mu(q,t,T;t_0{:}t_1,t_2,t_3)} \tilde{P}^*_\mu(;q,t,T;t_0). \] Here \(\tilde{P}^*_\mu(;q,t,T;t_0)\) denotes Lifted BCn interpolation polynomials

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

Koornwinder polynomial

  • Let \(K^{(n)}_\lambda\) be Koornwinder polynomials
thm

For any integer \(n>0\) and partition \(\lambda\), and for generic values of the parameters, \[ \tilde{K}_\lambda(x_1^{\pm 1},\dots x_n^{\pm 1};q,t,t^n;t_0,t_1,t_2,t_3) = \begin{cases} K^{(n)}_\lambda(x_1,\dots x_n;q,t;t_0,t_1,t_2,t_3) & \ell(\lambda)\le n\\ 0 & \text{otherwise.} \end{cases} \]

examples

  • For example, \(\tilde{K}_0=1\) and

\[ \tilde{K}_1(q,t,T;t_0,t_1,t_2,t_3)= m_1+\frac{1-T}{(1-t)(1-t_0t_1t_2t_3T^2/t^2)} \sum_{r=0}^3 \Big(\frac{t_0t_1t_2t_3T}{t_rt}-t_r\Big). \]


related items


articles

  • 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.