Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions

Abstract

We present an explicit formula for the transition matrix 𝒞 from the type 𝐵𝐶ₙ Koornwinder polynomials 𝘗₍₁ᵣ₎(𝑥|𝑎, 𝘣, c, 𝑑|𝑞, 𝘵) with one column diagrams, to the type 𝐵𝐶ₙ monomial symmetric polynomials m₍₁ᵣ₎(𝑥). The entries of the matrix C enjoy a set of four-term recursion relations. These recursions provide us with the branching rules for the Koornwinder polynomials with one column diagrams, namely the restriction rules from 𝐵𝐶ₙ to 𝐵𝐶ₙ₋₁. To have a good description of the transition matrices involved, we introduce the following degeneration scheme of the Koornwinder polynomials: 𝘗₍₁ᵣ₎(𝑥|𝑎, 𝘣, c, 𝑑|𝑞, 𝘵) ⟷ 𝘗₍₁ᵣ₎(𝑥|𝑎, −𝑎, c, 𝑑|𝑞, 𝘵) ⟷ 𝘗₍₁ᵣ₎(𝑥|𝑎, −𝑎, c, −c|𝑞, 𝘵) ⟷ 𝘗₍₁ᵣ₎(𝑥|𝘵¹/²c, −𝘵¹/²c, c, −c|𝑞, 𝘵) ⟷ 𝘗₍₁ᵣ₎(𝑥|𝘵¹/², −𝘵¹/², 1, −1|𝑞, 𝘵). We prove that the transition matrices associated with each of these degeneration steps are given in terms of the matrix inversion formula of Bressoud. As an application, we give an explicit formula for the Kostka polynomials of type Bₙ, namely the transition matrix from the Schur polynomials 𝘗⁽ᴮⁿ 'ᴮⁿ ⁾₍₁ᵣ₎(𝑥|𝑞; 𝑞, 𝑞) to the Hall-Littlewood polynomials 𝘗⁽ᴮⁿ 'ᴮⁿ ⁾₍₁ᵣ₎(𝑥|𝘵; 0, 𝘵). We also present a conjecture for the asymptotically free eigenfunctions of the 𝐵ₙ 𝑞-Toda operator, which can be regarded as a branching formula from the 𝐵ₙ 𝑞-Toda eigenfunction restricted to the 𝘈ₙ₋₁ 𝑞-Toda eigenfunctions.