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 \(SL_2\) and type \(GL_2\) 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 \(E_\ell {\textbf {1}}_0\) and \({\textbf {1}}_0 E_\ell {\textbf {1}}_0\) 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 \(t^{-\frac{1}{2}}q^{-\frac{m}{2}}\).