BCn interpolation polynomials

introduction

introduced by Okounkov as an analogous objects to Interpolation Macdonald polynomials

notation

We define relations \(\prec\) and \(\succ\) such that \(\kappa\prec\lambda\) (equivalently \(\lambda\succ\kappa\)) for two partitions iff \(\lambda/\kappa\) is a vertical strip; that is, \(\kappa_i\le \lambda_i\le \kappa_i+1\) for all \(i\).
we frequently use the product of the form

\[ \prod_{(i,j)\in \lambda} f(i,j), \] where \((i,j)\in \lambda\) means that \(1\le i\) and \(1\le j\le \lambda'_i\)

let us define Generalized q-shifted factorials

\begin{align} C^+_\lambda(x;q,t)&:=\prod_{(i,j)\in \lambda} (1-q^{\lambda_i+j-1} t^{2-\lambda'_j-i} x)\\ &\phantom{:}= \prod_{1\le i\le l} \frac{(q^{\lambda_i} t^{2-l-i} x;q)} {(q^{2\lambda_i} t^{2-2i} x;q)} \prod_{1\le i<j\le l} \frac{(q^{\lambda_i+\lambda_j} t^{3-i-j} x;q)} {(q^{\lambda_i+\lambda_j} t^{2-i-j} x;q)},\\ C^-_\lambda(x;q,t)&:=\prod_{(i,j)\in \lambda} (1-q^{\lambda_i-j} t^{\lambda'_j-i} x)\\ &\phantom{:}= \prod_{1\le i\le l} \frac{(x;q)} {(q^{\lambda_i} t^{l-i} x;q)} \prod_{1\le i<j\le l} \frac{(q^{\lambda_i-\lambda_j} t^{j-i} x;q)} {(q^{\lambda_i-\lambda_j} t^{j-i-1} x;q)},\\ C^0_\lambda(x;q,t)&:=\prod_{(i,j)\in \lambda} (1-q^{j-1} t^{1-i} x)\\ &\phantom{:}= \prod_{1\le i\le l} (t^{1-i} x;q)_{\lambda_i}. \end{align}

\(\bar{P}^{*(n)}_\lambda(\mu;q,t,s):=\bar{P}^{*(n)}_\lambda(q^{\mu_i} t^{n-i} s;q,t,s)=\bar{P}^{*(n)}_\lambda(q^{\mu_1}t^{n-1},q^{\mu_2}t^{n-2},\cdots, q^{\mu_n}t^{0};q,t,s)\)

definition

Let \(\lambda\) be a partition with at most \(n\) parts
The BCn interpolation \(\bar{P}^{*(n)}_\lambda(x_1,\dots,x_n;q,t,s)\) is the unique polynomial in \(\Lambda_{t,s}\) satisfying the following conditions :

\(\deg \bar{P}^{*(n)}_\lambda(x;q,t,s)\leq |\lambda|\)
\(\bar{P}^{*(n)}_\lambda(\mu;q,t,s)=0\) if \(\quad\lambda\not\subset\mu\)
\(\bar{P}^{*(n)}_\lambda(\lambda;q,t,s)=\dots\) (normalization)

branching rule

thm (3.9)

We have \[ \bar{P}^{*(n+m)}_\lambda(x_1,x_2,\dots x_n,t^{m-1} v,t^{m-2} v,\dots v;q,t,s) = \sum_{\substack{\mu\subset\lambda\\\ell(\mu)\le n}} \psi^{(B)}_{\lambda/\mu}(v,vt^m;q,t,s t^n) \bar{P}^{*(n)}_\mu(x_1,x_2,\dots x_n;q,t,s), \] where \[ \psi^{(B)}_{\lambda/\mu}(v,v';q,t,s) = \frac{ C^0_\lambda(s/v;q,t) C^0_\lambda(t/sv';1/q,1/t)} { C^0_\mu(s/v;q,t) C^0_\mu(t/sv';1/q,1/t)} P_{\lambda/\mu}(\left[\frac{v^k-v^{\prime k}}{1-t^k}\right];q,t) \]

cor (3.10)

We have \[ \bar{P}^{*(n+1)}_\lambda(x_1,x_2,\dots x_n,v;q,t,s) = \sum_{\substack{\mu'\prec\lambda'\\\mu_{n+1}=0}} \psi^{(b)}_{\lambda/\mu}(v;q,t,s t^n) \bar{P}^{*(n)}_\mu(x_1,x_2,\dots x_n;q,t,s), \] where \[ \psi^{(b)}_{\lambda/\mu}(v;q,t,s) = \psi_{\lambda/\mu}(q,t) \prod_{(i,j)\in \lambda/\mu} (v+1/v-q^{j-1} t^{1-i}s-q^{1-j}t^{i-1}/s) \]

BCn q-binomial coefficient

see BCn q-binomial coefficient

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

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.
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.