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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2020
• Main Authors: Hoshino, Ayumu, Shiraishi, Jun'ichi
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2020
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/210764
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions. Ayumu Hoshino and Jun'ichi Shiraishi. SIGMA 16 (2020), 084, 28 pages

We present an explicit formula for the transition matrix from the type ₙ Koornwinder polynomials ₍₁ᵣ₎(|, , c, |, ) with one column diagrams, to the type ₙ monomial symmetric polynomials m₍₁ᵣ₎(). 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 ₙ to ₙ₋₁. To have a good description of the transition matrices involved, we introduce the following degeneration scheme of the Koornwinder polynomials: ₍₁ᵣ₎(|, , c, |, ) ⟷ ₍₁ᵣ₎(|, −, c, |, ) ⟷ ₍₁ᵣ₎(|, −, c, −c|, ) ⟷ ₍₁ᵣ₎(|¹/²c, −¹/²c, c, −c|, ) ⟷ ₍₁ᵣ₎(|¹/², −¹/², 1, −1|, ). 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 ⁽ᴮⁿ 'ᴮⁿ ⁾₍₁ᵣ₎(|; , ) to the Hall-Littlewood polynomials ⁽ᴮⁿ 'ᴮⁿ ⁾₍₁ᵣ₎(|; 0, ). We also present a conjecture for the asymptotically free eigenfunctions of the ₙ -Toda operator, which can be regarded as a branching formula from the ₙ -Toda eigenfunction restricted to the ₙ₋₁ -Toda eigenfunctions.