Lifted Koornwinder polynomials
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}
- Via the binomial formula, the lifted BCn interpolation polynomials lead immediately to a lifting for Koornwinder polynomials
- 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
- this is analogous to the following formula for Koornwinder polynomials
\[ K_{\lambda}(x;q,t;t_0,t_1,t_2,t_3) =\sum_{\mu\subseteq\lambda} {\lambda \brack \mu}_{q,t,s} \, \frac{K_{\lambda}\big(t_0(1,t,\dots,t^{n-1});q,t;t_0,t_1,t_2,t_3\big)} {K_{\mu}\big(t_0(1,t,\dots,t^{n-1});q,t;t_0,t_1,t_2,t_3\big)}\, \bar{P}_{\mu}^{\ast(n)}(x;q,t,t_0), \]
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
\[ \tilde{K}_1(q,t,T;t_0,t_1,t_2,t_3)= m_1+\frac{1-T}{(1-t)(1-t_0t_1t_2t_3T^2/t^2)} \sum_{r=0}^3 \Big(\frac{t_0t_1t_2t_3T}{t_rt}-t_r\Big). \]
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.