| 研究実績の概要 |
The main purpose of the Project " Combinatorics around ti VI" was to study combinatorial properties of algebraic solutions of the Painleve VI equation and associated polynomials which have been introduced and intensively studied by K. Okamoto and H. Umemura in the middle of 70's of the last century. Nowadays these polynomials are commonly known as the Umemura polynomials. It was observed by K. Okamoto and H.Umemura that these polynomials depend on two discrete parameters and satisfy very complicated recurrence relations, but nevertheless have only integer coefficients. In the case when one discrete parameter is equal to 0, explicit formula for Umemura polynomials has been conjectured by S.Okada and has been proved by K.Okamoto, H. Umemura, S.Okada and M.Noumi. Surprisingly, each coefficient has an interpretation as dimension of certain irreducible representation of the Lie group of type A.
The main results of this Project are:1) equivalence of Kirillov--Taneda and Noumi's conjectural formulas.2) Combinatorial interpretation of some coefficients of 2d-Umemura.We also study Lorentzian properties of 2d-Umemura polynomials For that goal we organize at RIMS and carry out the International Workshop "P-positivity in Matroid Theory and related topics", October 4-8,2021. During this Workshop several leading specialists in Combinatorics and Matroid Theory delivered lectures concerning Lorentzian polynomials.log-concavity,tropical geometry, which we expect cab be applied to the study of q-Painleve VI and related equation.
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