Combinatorics around Painleve VI

• Project/Area Number: 16K05057
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyoto University
• Principal Investigator: Kirillov Anatoli 京都大学, 数理解析研究所, 研究員 (20314057)
• Project Period (FY): 2016-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: Painleve / Poschl-Teller models / Umemura polynomials / Combinatorics / Painleve equations / Hessenberg varies / Fomin-Kirillov algebras / FK algebras / Classical Schubert / Quantum Schubert / Grothendieck Calculi / Classical RSK / Tropical RSK / Probabilistic models / Integrable Systems / FK-algebras / Cohomology theories / Flag varieties / Integrable systems / Quadratic algebras / Quantum K-theory / Painleve & Bethe Ansatz / Quantum cohomology / Elliptic Functions / 代数学 / 代数的組み合わせ論

Outline of Final Research Achievements

Proof of old standing Conjecture that the all coefficients of the two parameter Umemura polynomials are integer numbers and gave their explicit formulas (partly joint with M.Taneda and P. van Diejen). Some quantum and elliptic generalizations are suggested.

Academic Significance and Societal Importance of the Research Achievements

Our results explained and proved Conjecture that coefficients of Umemura polynomials are integers , and pointed out on surprising connections of Painleve and Poschl-Teller models and Combinatorics and Representation Theory.

Research Products

• [Int'l Joint Research] University of Oregon/UCB, Berkeley(米国)
• [Int'l Joint Research] Warwick University(英国)
• [Int'l Joint Research] National Research University HES(ロシア連邦)
• [Int'l Joint Research] JINR(ロシア連邦)
• [Int'l Joint Research] Indiana University(米国)
• [Int'l Joint Research] KIAS(韓国)
• [Journal Article] Rigged Configurations and Unimodality (2021)
  • Author(s): Anatol N. Kirillov
  • Journal Title: Progr. Math.,Representation Theory, Mathematical Physics, and Integrable Systems Volume: 340 Pages: 453-496
  • DOI: 10.1007/978-3-030-78148-4_16
  • ISBN: 9783030781477, 9783030781484
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Double Grothendieck polynomials for Symplectic and Odd Orthogonal Grassmannians (2020)
  • Author(s): Hudson Thomas、Ikeda Takeshi、Matsumura Tomoo、Naruse Hiroshi
  • Journal Title: Journal of Algebra Volume: 546 Pages: 294-314
  • DOI: 10.1016/j.jalgebra.2019.11.002
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Skew hook formula for d-complete posets via equivariant K-theory (2019)
  • Author(s): Naruse Hiroshi、Okada Soichi
  • Journal Title: Algebraic Combinatorics Volume: 2-4 Issue: 4 Pages: 541-571
  • DOI: 10.5802/alco.54
  • Peer Reviewed / Open Access
• [Journal Article] A determinant formula associated with the elliptic hypergeometric integrals of type BCn (2019)
  • Author(s): Ito Masahiko、Noumi Masatoshi
  • Journal Title: Journal of Mathematical Physics Volume: 60 Issue: 7 Pages: 071705-071705
  • DOI: 10.1063/1.5094116
  • Peer Reviewed / Open Access
• [Journal Article] Elliptic extension of Gustafson's q-integral of type G2 (2019)
  • Author(s): Ito Masahiko、Noumi Masatoshi
  • Journal Title: Adv. Math. Volume: --
  • Open Access
• [Journal Article] Polynomial expressions of p-ary auction functions (2019)
  • Author(s): Shizuo Kaji, Toshiaki Maeno, Koji Nuida and Yasuhide Numata
  • Journal Title: Journal of Mathematical Cryptology Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Solomon-Terao algebra of hyperplane arrangements (2019)
  • Author(s): Takuro Abe, Toshiaki Maeno, Satoshi Murai and Yasuhide Numata
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 印刷中
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] ユニタリ鏡映群の同変シューベルト・カルキュラスに向けて (2019)
  • Author(s): 成瀬 弘
  • Journal Title: 数理解析研究所講究録 Volume: 2103 Pages: 137-146
• [Journal Article] Remarks on τ -functions for the difference Painleve equations of type E 8 (2018)
  • Author(s): Noumi, Masatoshi
  • Journal Title: Adv. Stud. Pure Math. Volume: 76
  • Peer Reviewed
• [Journal Article] Connection formula for the Jackson integral of type An and elliptic Lagrange interpolation (2018)
  • Author(s): Ito, Masahiko; Noumi, Masatoshi
  • Journal Title: SIGMA Symmetry Integrability Geom. Methods Appl. Volume: 14
• [Journal Article] Rigged configurations and Catalan, stretched parabolic Kostka numbers and polynomials: polynomiality, unimodality and log-concavity. (2018)
  • Author(s): A.N.Kirillov
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 76 Pages: 303-346
• [Journal Article] Elementary proof and application of the generating functions for generalized Hall-Littlewood functions (2018)
  • Author(s): Naruse Hiroshi
  • Journal Title: Journal of Algebra Volume: 516 Pages: 197-209
  • DOI: 10.1016/j.jalgebra.2018.09.010
  • Peer Reviewed
• [Journal Article] Universal Gysin formulas for the universal Hall-Littlewood functions (2018)
  • Author(s): Nakagawa Masaki、Naruse Hiroshi
  • Journal Title: Contemporary Mathematics Volume: 708 Pages: 201-244
  • DOI: 10.1090/conm/708/14267
  • Peer Reviewed
• [Journal Article] Rigged Configurations and Catalan, Stretched Parabolic Kostka Numbers and Polynomials : Polynomiality, Unimodality and Log-concavity. (2018)
  • Author(s): A.N. Kirillov
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 76 Pages: 303-346
  • Peer Reviewed
• [Journal Article] On Q-deformations of Postnikov-Shapiro algebras (2018)
  • Author(s): A.N. Kirillov and Gleb Nenashev
  • Journal Title: Seminar Lotharingien de Combinatoire Volume: 78B
• [Journal Article] Peterson isomorphism in K-theory and relativistic Toda lattice (2018)
  • Author(s): Takeshi Ikeda, Shinsuke Iwao and Toshiaki Maeno
  • Journal Title: Int. Math. Res. Notices Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Construction of double Grothendieck polynomials of classical types using Id-Coxeter algebras (2017)
  • Author(s): A.N. Kirillov and H. Naruse
  • Journal Title: Tokyo J. Math. Volume: 39-no.3 Pages: 695-728
  • Peer Reviewed
• [Journal Article] Degeneracy Loci Classes in K-theory - Determinantal and Pfaffian Formula - (2017)
  • Author(s): T. Hudson, T. Ikeda, T. Matsumura and H. Naruse
  • Journal Title: Advances in Mathematics Volume: 320 Pages: 115-156
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Evaluation of the $BC_n$ elliptic Selberg integral via the fundamental invariants (2017)
  • Author(s): M. Ito, M. Noumi
  • Journal Title: Proc. Amer. Math. Soc. Volume: 145 Issue: 2 Pages: 689-703
  • DOI: 10.1090/proc/13234
  • Peer Reviewed / Open Access
• [Journal Article] Geometric aspects of Painlev\'e equations (2017)
  • Author(s): K. Kajiwara, M. Noumi, Y. Yamada
  • Journal Title: J. Phys. A Math. Theor. Volume: 50 Issue: 7 Pages: 073001-073001
  • DOI: 10.1088/1751-8121/50/7/073001
  • Peer Reviewed / Open Access
• [Journal Article] Construction of double Grothendieck polynomials of classical types using Id-Coxeter algebras (2017)
  • Author(s): A.N. Kirillov and H. Naruse
  • Journal Title: Tokyo Journal of Mathematics Volume: 39
  • Peer Reviewed / Open Access
• [Journal Article] Rigged Configurations and Catalan, Stretched Parabolic Kostka Numbers and Polynomials : Polynomiality, Unimodality and Log-concavity (2017)
  • Author(s): A.N. Kirillov
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 印刷中
  • Peer Reviewed / Open Access
• [Journal Article] Geometric aspects of Painleve equations (2017)
  • Author(s): M.Noumi
  • Journal Title: Advanced Studies in Pure Mathematics Volume: -
  • Peer Reviewed / Open Access
• [Journal Article] Derivation of a BCn elliptic summation formmula via the fundamental invariants (2016)
  • Author(s): M. Ito and M. Noumi
  • Journal Title: Constr. Approx. Volume: 未定 Issue: 1 Pages: 33-46
  • DOI: 10.1007/s00365-016-9340-8
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Bethe's Quantum Numbers And Rigged Configurations (2016)
  • Author(s): R.Sakamoto and A.N.Kirillov
  • Journal Title: Nuclear Physics B Volume: 905 Pages: 359-372
  • DOI: 10.1016/j.nuclphysb.2016.02.020
  • Peer Reviewed / Open Access
• [Journal Article] On some quadratic algebras I+I/2 : Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and Reduced polynomials (2016)
  • Author(s): A.N. Kirillov
  • Journal Title: SIGMA Volume: 12
  • Peer Reviewed / Open Access
• [Journal Article] Notes on Schubert, Grothendieck and Key polynomials (2016)
  • Author(s): A.N. Kirillov
  • Journal Title: SIGMA Volume: 12
  • Peer Reviewed / Open Access
• [Journal Article] A generalization of the Sears--Slater transformation and elliptic Lagrange interpolation of type $BC_n$ (2016)
  • Author(s): M. Ito, M. Noumi
  • Journal Title: Adv. in Math. Volume: 229 Pages: 361-380
  • DOI: 10.1016/j.aim.2016.05.016
  • Peer Reviewed / Open Access
• [Journal Article] SOLUTIONS IN TERMS OF INTEGRALS OF MULTIVALUED FUNCTIONS FOR THE CLASSICAL HYPERGEOMETRIC EQUATIONS AND THE HYPERGEOMETRIC SYSTEM ON THE CONFIGURATION SPACE (2016)
  • Author(s): K. Mimachi, M. Noumi
  • Journal Title: Kyushu Journal of Mathematics Volume: 70 Issue: 2 Pages: 315-342
  • DOI: 10.2206/kyushujm.70.315
  • NAID: 130005421174
  • ISSN: 1340-6116, 1883-2032
  • Peer Reviewed / Open Access
• [Journal Article] Duality transformation formulas for multiple elliptic hypergeometric series of type BC (2016)
  • Author(s): Y. Komori, Y. Masuda and M. Noumi
  • Journal Title: Constr. Approx. Volume: 印刷中 Issue: 3 Pages: 483-516
  • DOI: 10.1007/s00365-015-9316-0
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Factorial P- and Q-Schur functions represent equivariant quantum Schubert classes (2016)
  • Author(s): T.Ikeda, L.Mihalcea and H.Naruse
  • Journal Title: Osaka J. of Mathematics Volume: 53
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Sperner property and finite-dimensional Gorenstein algebras associated to matroids (2016)
  • Author(s): Toshiaki Maeno and Yasuhide Numata
  • Journal Title: Journal of Commutative Algebra Volume: 8 Issue: 4 Pages: 549-570
  • DOI: 10.1216/jca-2016-8-4-549
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Presentation] Rigged Confugurations, past and present,I (2021)
  • Author(s): A.N.Kirillov
  • Organizer: International Workshop P-positivity in Matroid Theory and Related Topics
  • Int'l Joint Research
• [Presentation] Rigged Confugurations, past and present,II (2021)
  • Author(s): A.N.Kirillov
  • Organizer: International Workshop P-positivity in Matroid Theory and Related Topics
  • Int'l Joint Research
• [Presentation] FK algebras and posutuvuty (2021)
  • Author(s): A.N.Kirillov
  • Organizer: International Workshop P-positivity in Matroid Theory and Related Topics
  • Int'l Joint Research
• [Presentation] Log-concavity of graph chromatic polynomials. (2019)
  • Author(s): Anatol N. Kirillov
  • Organizer: Matroids and Tutte polynomials.
  • Invited
• [Presentation] Generalized RSK correspondence (2019)
  • Author(s): Anatol N. Kirillov
  • Organizer: Toric Topology 2019
• [Presentation] Generalized colored hook formula (2019)
  • Author(s): H.Naruse
  • Organizer: New interactions between Geometry and Combinatorics
  • Int'l Joint Research / Invited
• [Presentation] Formal eigenfunctions for the elliptic Ruijsenaars difference operators (2019)
  • Author(s): M. Noumi
  • Organizer: Elliptic integrable systems, special functions and quantum field theory
  • Int'l Joint Research
• [Presentation] Darboux and Moutard transformations -- What I learned from Jon (2019)
  • Author(s): M. Noumi
  • Organizer: Integrable systems, special functions and combinatorics (ISLAND V)
  • Int'l Joint Research
• [Presentation] Elliptic hypergeometric integrals (2019)
  • Author(s): M. Noumi
  • Organizer: Representation Theory, Probability, and Symmetric Functions
  • Int'l Joint Research
• [Presentation] Some combinatorial properties of dual factorial Schur functions (2019)
  • Author(s): Hiroshi Naruse
  • Organizer: Workshop on Algebraic and Enumerative Combinatorics
  • Int'l Joint Research / Invited
• [Presentation] Hook formula and equivariant K-theory (2019)
  • Author(s): Hiroshi Naruse
  • Organizer: Crystals and Their Generalizations
  • Int'l Joint Research / Invited
• [Presentation] 逆平面分割の母関数と同変K-理論 (2019)
  • Author(s): 成瀬 弘、岡田 聡一
  • Organizer: 日本数学会年会
• [Presentation] Elliptic hypergeometric functions and elliptic difference Painleve equation (2018)
  • Author(s): M. Noumi
  • Organizer: XIX International Congress on Mathmatical Physics
  • Int'l Joint Research / Invited
• [Presentation] Elliptic hypergeometric integrals (2018)
  • Author(s): M. Noumi
  • Organizer: Summer Research Institute on q-Series
  • Int'l Joint Research / Invited
• [Presentation] Elliptic hypergeometric integrals and associated determinant formulas (2018)
  • Author(s): M. Noumi
  • Organizer: 13th International Conference on Symmetry and Integrability of Difference Equations (SIDE 13)
  • Int'l Joint Research / Invited
• [Presentation] The Lefschetz property for Artinian Gorenstein algebras (2018)
  • Author(s): Toshiaki Maeno
  • Organizer: Algebraic Geometry in Positive Characteristic and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] On generalized cohomology theories of nilpotent Hessenberg varieties (2018)
  • Author(s): A.N.Kirillov
  • Organizer: Meeting on Hessenberg varieties
  • Int'l Joint Research / Invited
• [Presentation] Equivariant K-theory and hook formula for skew shape on d-complete set (2018)
  • Author(s): Hiroshi Naruse
  • Organizer: Algebraic and Enumerative Combinatorics in Okayama
  • Int'l Joint Research / Invited
• [Presentation] Discrete Painleve equations and special functions (2017)
  • Author(s): M.Noum
  • Organizer: ESI workshop: ''Elliptic Hypergeometic Functions in Combinatorics, Integrable Systems and Physics''
  • Place of Presentation: Erwin Schrodinger Institute（オーストリア）
  • Year and Date: 2017-03-21
  • Int'l Joint Research / Invited
• [Presentation] Cactus group and Gelfund-Tsetlin group (2017)
  • Author(s): A.N.Kirillov
  • Organizer: Spring School on Representation Theory
  • Place of Presentation: Tokyo University, Department of Mathematics （東京都目黒区）
  • Year and Date: 2017-03-13
• [Presentation] Dilogarithm Identities (2017)
  • Author(s): A.N.Kirillov
  • Organizer: Special Lecture at National Research University High Economy School, Moscow, Russia
  • Int'l Joint Research / Invited
• [Presentation] Bethe Ansatz and Combinatorics (2017)
  • Author(s): A.N.Kirillov
  • Organizer: Colloquium Talk at National Research University High Economy School, Moscow, Russia
  • Int'l Joint Research / Invited
• [Presentation] Quadratic algebras, Integral Systems and Combinatorics (2017)
  • Author(s): A.N.Kirillov
  • Organizer: Special Lecture at National Research University High Economy School, Moscow, Russia
  • Int'l Joint Research / Invited
• [Presentation] Quadratic algebras and universal Dunkl operators. (2017)
  • Author(s): A.N.Kirillov
  • Organizer: Seminar talk at Kavli Institute of Physics and Mathematics Of the Universe (IPMU)
  • Int'l Joint Research / Invited
• [Presentation] Elliptic difference Painlev\'{e} equation and elliptic hypergeometric integrals (2017)
  • Author(s): M. Noumi
  • Organizer: XXVth International Conference on Integrable
  • Int'l Joint Research / Invited
• [Presentation] Robinson-Schensted-Knuth correspondence and Rigged Configurations (2016)
  • Author(s): A.N.Kirillov
  • Organizer: Lecture course "Robinson-Schensted-Knuth correspondence and Rigged Configurations"
  • Place of Presentation: The Taiwan National University, Taiwan
  • Year and Date: 2016-12-13
  • Int'l Joint Research / Invited
• [Presentation] On some quadratic algebras (2016)
  • Author(s): A.N.Kirillov
  • Organizer: Lecture course "On some quadratic algebras"
  • Place of Presentation: The Stockholm University, Sweden
  • Year and Date: 2016-10-13
  • Int'l Joint Research / Invited
• [Presentation] Elliptic hypergeometric integrals and elliptic interpolation functions (2016)
  • Author(s): M.Noum
  • Organizer: BIRS workshop: ''Paineleve Equations and Discrete Dynamics''
  • Place of Presentation: Banff International Reseach Station, Banff, Canada
  • Year and Date: 2016-10-03
  • Int'l Joint Research / Invited
• [Presentation] Flat connections for q-KZ equations associated with affine Birman--Murakami--Wenzl algebras (2016)
  • Author(s): A.N.Kirillov
  • Organizer: Classical and quantum integrable systems and supersymmetry
  • Place of Presentation: The Chern Mathematical Institute, Tianjin, China
  • Year and Date: 2016-09-19
  • Int'l Joint Research / Invited
• [Presentation] Gustafson-Rkha 和公式の楕円化 (2016)
  • Author(s): 伊藤雅彦・野海正俊
  • Organizer: 日本数学会2016年度秋季総合分科会
  • Place of Presentation: 関西大学（大阪府吹田市）
  • Year and Date: 2016-09-15
• [Presentation] On braided Hopf lgebras (2016)
  • Author(s): A.N.Kirillov
  • Organizer: Hopf algebras and Tensor Categories
  • Place of Presentation: Tsukuba University, Japan（茨城県つくば市）
  • Year and Date: 2016-09-11
  • Invited
• [Presentation] 楕円超幾何積分と楕円補間函数 (2016)
  • Author(s): 野海正俊
  • Organizer: RIMS 研究集会「可積分系数理の現状と展望」
  • Place of Presentation: 京都大学北部総合研究科棟・益川ホール（京都府京都市）
  • Year and Date: 2016-09-05
  • Invited
• [Presentation] Macdonald 多項式 -- アフィン Hecke 環からのアプローチ (2016)
  • Author(s): 野海正俊
  • Organizer: 超幾何学校 2016
  • Place of Presentation: 神戸大学理学部（兵庫県神戸市）
  • Year and Date: 2016-08-31
  • Invited
• [Presentation] Bethe subalgebras in affine Birman--Murakami--Wenzl algebras (2016)
  • Author(s): A.N.Kirillov
  • Organizer: MS Seminar at Kavli IPMU
  • Place of Presentation: Tokyo University, the Kavli Institute (IPMU) （千葉県柏市）
  • Year and Date: 2016-04-20
  • Invited
• [Book] ベーテ仮設の数理 (2021)
  • Author(s): 坂本玲峰, アナトール N. キリロフ
  • Total Pages: 320
  • Publisher: 森北出版
  • ISBN: 9784627082410
• [Book] 多変数超幾何函数 -- ゲルファント講義 1989 (2016)
  • Author(s): 野海正俊・梅田亨・若山正人
  • Publisher: 日本評論社
• [Remarks] P-positivity in Matroid Theory and Related Topics
  • URL: https://www.kurims.kyoto-u.ac.jp/~kirillov/workshop2021.html
• [Funded Workshop] International Workshop P-positivity in Matroid Theory and Related Topics (2021)