Clebsch-Gordan Coefficients for Macdonald Polynomials

Abstract

AbstractIn this paper we use the double affine Hecke algebra to compute the Macdonald polynomial products $$E_\ell P_m$$ E ℓ P m and $$P_\ell P_m$$ P ℓ P m for type $$SL_2$$ S L 2 and type $$GL_2$$ G L 2 Macdonald polynomials. Our method follows the ideas of Martha Yip but executes a compression to reduce the sum from $$2\cdot 3^{\ell -1}$$ 2 · 3 ℓ - 1 signed terms to $$2\ell $$ 2 ℓ positive terms. We show that our rule for $$P_\ell P_m$$ P ℓ P m is equivalent to a special case of the Pieri rule of Macdonald. Our method shows that computing $$E_\ell {\textbf {1}}_0$$ E ℓ 1 0 and $${\textbf {1}}_0 E_\ell {\textbf {1}}_0$$ 1 0 E ℓ 1 0 in terms of a special basis of the double affine Hecke algebra provides universal compressed formulas for multiplication by $$E_\ell $$ E ℓ and $$P_\ell $$ P ℓ . The formulas for a specific products $$E_\ell P_m$$ E ℓ P m and $$P_\ell P_m$$ P ℓ P m are obtained by evaluating the universal formulas at $$t^{-\frac{1}{2}}q^{-\frac{m}{2}}$$ t - 1 2 q - m 2 .