Clebsch-Gordan coefficients for Macdonald polynomials

Abstract

In this paper we use the double affine Hecke algebra to compute the Macdonald polynomial products $E_{\ell }{P}_m$ and $P_{\ell }{P}_m$ for type $S{L}_2$ and type $G{L}_2$ Macdonald polynomials. Our method follows the ideas of Martha Yip but executes a compression to reduce the sum from $2\cdot 3^{\ell -1}$ signed terms to $2\ell$ positive terms. We show that our rule for $P_{\ell }{P}_m$ is equivalent to a special case of the Pieri rule of Macdonald. Our method shows that computing $E_{\ell }{1}_0$ and $1_0{E}_{\ell }{1}_0$ in terms of a special basis of the double affine Hecke algebra provides universal compressed formulas for multiplication by $E_{\ell }$ and $P_{\ell }$. The formulas for a specific products $E_{\ell }{P}_m$ and $P_{\ell }{P}_m$ are obtained by evaluating the universal formulas at $t^{-\frac{1}{2}}{q}^{-\frac{m}{2}}$.

44 pages, including a section with Examples