Research of explicit formulas and associated algebraic structures for Macdonald polynomials

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K05186
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Hiroshima Institute of Technology (2017-2018); Kagawa National College of Technology (2016)
• Principal Investigator: Hoshino Ayumu 広島工業大学, 工学部, 准教授 (30598280)
• Co-Investigator(Kenkyū-buntansha): 白石 潤一 東京大学, 大学院数理科学研究科, 准教授 (20272536)
• Research Collaborator: Noumi Masatoshi
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: Macdonald 多項式 / Koornwinder 多項式 / matrix inversion

Outline of Final Research Achievements

1.We constructed explicit formulas for the Koornwinder and Macdonald polynomials of type B, C and D with one column diagrams. We constructed combinatorial expressions for the Macdonald polynomials of type C and D. 2.We constructed explicit forms of transition matrices from the type C Macdonald polynomials to the type C monomial symmetric polynomials with one column diagrams by using b,q,t-deformations of Catalan numbers. 3.We constructed t-Kostka polnomials with one column diagrams of type B, C and D. 4.We constructed transition matrices in terms of certain degenerated polynomials for the Koornwinder polynomials. 5.We conjectured certain Pieri rules for the Macdonald polynomials of type C. 6.We constructed polyhedral realizations of crystal bases for the integrable highest weight modules of nonexceptional affine quantum algebras.

Free Research Field

無限可積分系，量子群の表現論

Academic Significance and Societal Importance of the Research Achievements

本研究では，一列型のKoornwinder多項式やB,C,D型Macdonald多項式の明示公式を構成し，また，一列型Koornwinder多項式のパラメータを特殊化した多項式間の遷移行列達をBressoudやKrattenthalerのmatrix inversionを用いて記述し，これらの遷移行列達が多項式の階数に依存しないことを発見した．この事実はA型以外のMacdonald多項式においては知られていない．また，応用として，Catalan数のb,q,t変形や一列型B,C,D型Kostka多項式のt-変形の具体的な表示を構成した．これらから，本研究成果は学術的に価値があると考えている．