Clebsch-Gordan coefficients for Macdonald polynomials

2023/10/16 by Aritra Bhattacharya, Arun Ram, Bhattacharya, Aritra +1
Mathematics · #05E05 #33D52 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2310.10846

openalex publication_date 2023/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we use the double affine Hecke algebra to compute the Macdonald polynomial products E_ℓ P_m and P_ℓ P_m for type SL₂ and type GL₂ Macdonald polynomials. Our method follows the ideas of Martha Yip but executes a compression to reduce the sum from 2⋅ 3^ℓ-1 signed terms to 2ℓ positive terms. We show that our rule for P_ℓ P_m is equivalent to a special case of the Pieri rule of Macdonald. Our method shows that computing E_ℓ1₀ and 1₀ E_ℓ 1₀ in terms of a special basis of the double affine Hecke algebra provides universal compressed formulas for multiplication by E_ℓ and P_ℓ. The formulas for a specific products E_ℓ P_m and P_ℓ P_m are obtained by evaluating the universal formulas at t^-\frac12q^-(m)/(2).

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