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 :

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

• Branching rules for Macdonald polynomials
• Interpolation Macdonald polynomials
• Lifted BCn interpolation polynomials
• Koornwinder polynomials
• BCn interpolation polynomial in Macdonald polynomial basis

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.