Organization
Graduate School of Mathematics Division of Mathematics Social Mathematics Professor
Graduate School
Graduate School of Mathematics
Undergraduate School
School of Science
Title
Professor
Contact information
OKADA, Soichi
Degree 1
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理学博士
Research Interests 2
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Representation Theory
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Combinatorics
Research Areas 1
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Others / Others / Algebra
Current Research Project and SDGs 4
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Enumerative combinatorics of plane partitions and alternating sign matrices
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Identities for determinants and Pfaffians, and their applications
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Combinatorics of Young diagrams and its applications to representation theory
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Algebras associated to posets and related combinatorics
Education 2
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The University of Tokyo Graduate School, Division of Science
- 1990
Country： Japan
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The University of Tokyo Faculty of Science
- 1985
Country： Japan
Professional Memberships 2
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日本数学会
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アメリカ数学会(American Mathematical Society)
Papers 31
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Symplectic Q-functions Reviewed International journal
Okada Soichi
JOURNAL OF COMBINATORIAL THEORY SERIES A Vol. 180 2021.5
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Pfallian formulas and Schur Q-function identities Reviewed International journal
Okada Soichi
ADVANCES IN MATHEMATICS Vol. 353 page： 446 - 470 2019.9
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Extended Schur's Q-functions and the full Kostant-Toda hierarchy on the Lie algebra of type D Reviewed International coauthorship
Yuji Kodama and Soichi Okada
Physica D: Nonlinear Phenomena Vol. 443 page： 133589 2023.1
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Birational rowmotion and Coxeter-motion on minuscule posets Reviewed International journal
Okada Soichi
ELECTRONIC JOURNAL OF COMBINATORICS Vol. 28 ( 1 ) 2021.1
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Intermediate symplectic characters and shifted plane partitions of shifted double staircase shape Reviewed
Combinatorial Theory Vol. 1 page： #10 2021
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A new q-Selberg integral, Schur functions, and Young books Reviewed International coauthorship International journal
Kim Jang Soo, Okada Soichi
RAMANUJAN JOURNAL Vol. 42 ( 1 ) page： 43 - 57 2017.1
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A new $q$-Selberg integral, Schur functions, and Young books Reviewed
J. Kim, S. Okada
J.~Kim and \underline{\underline{S.~Okada}}, Vol. 42 ( 1 ) page： 43-57 2017
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On the expansion coefficients of KP tau function Reviewed
A. Nakayashiki, S. Okada, Y. Shigyo
J. Integrable Syst. Vol. 2 ( 1 ) page： xyx007 2017
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Pieri rules for classical groups and equinumeration between generalized oscillating tableaux and semistandard tableaux Reviewed
S. Okada
Electron. J. Combin. Vol. 23 ( 4 ) page： 4.43 2016
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Identities for determinants and Pfaffians, and their applications Invited Reviewed
M. Ishikawa and S. Okada
Sugaku Expositions Vol. 27 page： 85--116 2014
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A compound determinant identity for rectangular matrices and determinants of Schur functions Reviewed
Masao Ishikawa, Masahiko Ito, Soichi OKADA
Advances in Applied Mathematics Vol. 51 page： 635--654 2013
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An application of the Cauchy-Sylvester theorem on compound determinants to a BC_n Jackson integral Reviewed
Masahiko ITO, Soichi OKADA
Partitions, q-Series, and Modular Forms page： 145-157 2012
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Spinor representations and symmetric functions
岡田 聡一
表現論シンポジウム講演集 page： 129-141 2011.11
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(q,t)-Deformations of multivariate hook product formulae Reviewed
Soichi OKADA
J. Algebraic Combin. Vol. 32 page： 399-416 2010.11
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行列式・パフィアンに関する等式とその表現論，組合せ論への応用 Invited Reviewed
石川 雅雄，岡田 聡一
数学 Vol. 62 ( 1 ) page： 85-114 2010.1
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Instanton calculus and chiral one-point functions in supersymmetric gauge theories Reviewed
S. Fujii, H. Kanno, S. Moriyama and S. Okada
Adv. Theor. Math. Phys. Vol. 12 page： 1401-1428 2008
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An elliptic generalization of Schur's Pfaffian identity Reviewed
Soichi OKADA
Adv. Math. Vol. 204 page： 530-538 2006
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Generalizations of Cauchy's determinant and Schur's Pfaffian Reviewed
Masao ISHIKAWA, Soichi OKADA, Hiroyuki TAGAWA, Jiang ZENG
Adv. Appl. Math. Vol. 36 page： 251-287 2006
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*Enumeration of Symmetry Classes of Alternating Sign Matrices and Characters of Classical Groups Reviewed
Soichi OKADA
J. Algebraic Combin. Vol. 23 page： 43-69 2006
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A Pfaffian--Hafnian Analogue of Borchardt's Identity Reviewed
Masao ISHIKAWA, Soichi OKADA, Hiroyuki KAWAMUKO
Electron. J. Combin. Vol. 12 page： N9 2005
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*Applications of minor summation formulas to rectangular-shaped representations of classical groups Reviewed
Journal of Algebra Vol. 205 page： 337 1998
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The number of rhombus tilings of a "punctured" hexagon and the minor summation formula Reviewed
Advances in Applied Mathematics Vol. 21 page： 381 1998
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*Special polynomials associated with the Painleve equations Reviewed
Masatoshi NOUMI, Soichi OKADA, Kazuo OKAMOTO, Hiroshi UMEMURA
Proceedings of the Taniguchi Symposium 1997 Integrable Systems and Algebraic Geometry page： 349-372 1998
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Applications of minor-summation formula : I Littlewood's formulas Reviewed
Journal of Algebra Vol. 183 page： 193 1996
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*Algebras associated to the Young-Fibonacci lattice Reviewed
Transaction of the American Mathematical Society Vol. 346 ( 2 ) page： 549 1994
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Robinson-Schecsted-type algorithms for the spinor representations of the orthogonal Lie algebra so(2n, C) Reviewed
Journal of Algebra Vol. 158 ( 1 ) page： 155 1993
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Alternating sign matrices and some deformations of Weyl's denominator formulas Reviewed
Journal of Algebraic Combinatorics Vol. 2 ( 2 ) page： 155 1993
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A Robinson-Schensted-type algorithm for SO(2n,C) Reviewed
Journal of Algebra Vol. 143 page： 2 1991
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Partially strict shifted plane partitions Reviewed
Jounal of Combinatirial Theory Series A Vol. 53 page： 1 1990
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Wreath products by the symmetric groups and product posets of Young's Lattices Reviewed
Journal of Combinatorial Theory Series A Vol. 55 page： 1 1990
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On the generating funtions for certain clesses of plane partitions Reviewed
Jounal of Combinatorial Theory, Series A Vol. 51 page： 1 1989
Books 2
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古典群の表現論と組合せ論（上）
岡田聡一（ Role： Sole author）
培風館 2006.3
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古典群の表現論と組合せ論（下）
岡田聡一（ Role： Sole author）
培風館 2006.3
Presentations 27
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Applications of the minor-summation formula to combinatorics and representation theory Invited International conference
Soichi Okada
Enumerative Combinatorics 2022.12.14
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Symplectic Q-functions Invited International conference
Soichi Okada
Integrable Probability, Combinatorics and Representation Theory 2022.9.29
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Proof of birational file homomesy for minuscule posets Invited International conference
Soichi Okada
Dynamical Algebraic Combinatorics 2020.10.23
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Symplectic Q-functions Invited International conference
Soichi Okada
Algebraic Combinatorics Online Workshop 2020.4.29
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Birational rowmotion and Coxeter-motion on minuscule posets Invited International conference
Soichi Okada
AMS Spring Central and Western Joint Sectional Meeting, Special Session on Dynamical Systems and Algebraic Combinatorics
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On the existence of generalized parking spaces for complex reflection groups Invited International conference
Soichi Okada
2015 Combinatorics Workshop
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Generalized Cauchy determinant and Schur Pfaffian, and their applications Invited International conference
Soichi Okada
Lattice Models: Exact Methods and Combinatorics
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Restricted Littlewood formulae via minor summation formula International conference
Soichi OKADA
19th International Linear Algebra Society Conference
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Hook product formulae International conference
Soichi OKADA
Asian Mathematical Conference 2013
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Symmetric functions and branching rules for spinor representations International conference
Soichi OKADA
2012 Shanghai Conference on Algebraic Combinatorics
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Spinor representations and symmetric functions International conference
Soichi OKADA
Representation Theory of Chevalley Groups and Related Topics
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Spinor representations and symmetric functions International conference
岡田 聡一
2011年度表現論シンポジウム
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Two-parameter Deformations of Multivariate Hook Product Formulae International conference
Soichi OKADA
International Conference on Asymptotics and Special Functions
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Universal characters for spinor representations International conference
Combinatorics and Mathematical Physics
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Symmetric functions and spin representations International conference
Workshop on Whittaker Functions, Crystal Bases, and Quantum Groups
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Two parameter deformation of multivariate hook product formulae International conference
Workshop on Two-dimensional Lattice Models
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A compound determinant identity and its application International conference
Workshop on Combinatorics and Statistical Physics
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Multivariate generalisation of Hankel determinants of Catalan numbers and middle binomial coefficients International conference
Workshop on Statistical-Mechanics and Quantum-Field Theory Methods in Combinatorial Enumeration
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Trace generating functions of plane partitions
組合せ論的表現論の拡がり
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Aspects of rational universal characters International conference
International Conference Formal Power Series and Algebraic Combinatorics
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An identity for the power-sums of contents and its application to gauge theory, International conference
Workshop on Combinatorial Problems Raised by Statistical Mechanics
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Special Polynomials associated with the Painleve Equations International conference
Workshop of Combinatorics on q-series and Partitions
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Cauchy 型の行列式，Pfaffian と Littlewood-Richardson 係数
代数的組合せ論とその周辺
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Generalizations of Cauchy's determinant and Schur's Pfaffian International conference
17th International Conference on Formal Power Series and Algebraic Combinatorics
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Variations on Cauchy's determinant and Schur's Pfaffian International conference
2nd Joint AMS-DMV-OMG Meeting, Special Session on Algebraic Combinatorics
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交代符号行列の数え上げと Cauchy 型行列式，Pfaffian
第 49 回代数学シンポジウム
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Alternating sign matrices, square ice models and characters of classical groups International conference
Combinatorics and Intergrable Models
Research Project for Joint Research, Competitive Funding, etc. 1
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交代符号行列，対称関数の組合せ論とその表現論，可積分系への応用
2006
KAKENHI (Grants-in-Aid for Scientific Research) 6
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Expansion of the research on alternating sign matrices, plane partitions and tilings in the aspect of distributive lattice
Grant number：20K03558 2020.4 - 2023.3
Authorship：Coinvestigator(s) Grant type：Competitive
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対称関数の代数的組合せ論とその表現論，数え上げ組合せ論への展開
Grant number：18K03208 2018.4 - 2021.3
岡田 聡一
Authorship：Principal investigator Grant type：Competitive
Grant amount：\4420000 （ Direct Cost: \3400000 、 Indirect Cost：\1020000 ）
この研究では，対称関数の代数的組合せ論（特に対称関数の関係式）の組合せ論，表現論への展開を目指し，(A) 古典型ルート系に付随した Schur Q 関数，(B) 平面分割の母関数，(C) d-complete な半順序集合の組合せ論，の 3 つのパートに分けて研究を進めた．
今年度の研究のパート (A) では，Hamel, King によって組合せ論的に定義された直交群，斜交群に対する factorial Q 関数に代数的な表示式を与えた．パート (B) では，ある種の平面分割の数え上げ問題に関する共同研究を開始し，このプロジェクトに関連して，非退化とは限らない交代双線型形式を保つ群 (intermediate symplectic group) のある種の直既約表現の指標に対して，Weyl 型の指標公式（行列式の比としての表示）を見出した．パート (C) では，d-complete な半順序集合の基本的な例であるミニスキュール半順序集合に付随した双有理版 rowmotion と呼ばれる力学系において，順序イデアルの元の個数の双有理版 homomesy（およびその精密化）に（具体的な表示式や計算機を用いない）統一的な証明を与えることに成功した．また，土谷との共同研究において，大杉・土谷によって導入された半順序集合に付随した 2 種類の凸多面体の間に区分線型な全単射を構成し，両者の Ehrhart 多項式が一致することに全単射による証明を与え，この全単射を用いて，これらの凸多面体の三角形分割を具体的に構成した．
パート (C) について，昨年度までの研究では双有理版 rowmotion における順序イデアルの元の個数の双有理版 homemesy を統一的な形で定式化できたものの，双有理版 rowmotion の具体的な表示式や計算機を用いた証明しか与えることができなかった．今年度の研究では，この homomesy に簡明な統一的証明を与えることに成功した．また，パート (B) についても新しい方向性を見出すことができた．
パート (B) では，得られた Weyl 型公式の組合せ論，表現論への応用を目指す．また，パート (C) では，双有理版 rowmotion における反鎖の元の個数の双有理版 homomesy の Lie 理論的な統一的定式化が課題として残されている． -
Finite-length analysis with computation complexity
Grant number：17H01280 2017.4 - 2022.3
Authorship：Coinvestigator(s) Grant type：Competitive
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The study of plane partitions, enumeration of tiliings, symmetric functions, Pfaffians and determinants
Grant number：16K05068 2016.4 - 2020.3
Ishikawa Masao
Authorship：Coinvestigator(s) Grant type：Competitive
The main result of our study in this period is the relation between the famous Selberg integral and the evaluation of Pfaffians.The q-analogue of the Selberg integral is conjectured by Richard Askey and proven by K. Kadell and L. Habsieger, independently. In our study we reduce the evaluation of certain hyperpfaffians to the Selberg type integrals and their q-analogues.In this study we exploit the minor summation formula for hyperpfaffians in the most general form, then find the de Bruijn formula for hyperpfaffians. This enable us to find many forms of applications for evaluations of hyperpfaffians, and we use the Vandermonde determinat to evaluate certain hyperpfaffians.We find several applications of the formula to several combinatorial numbers, the moments of the orthogonal polynomials and their q-analogues. We also study the finite distributive lattice structure of the set of half-turn symmetric alternating sign matrices and its relation to the type B positive root poset.
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Combinatorics of special polynomials associated to certain solutions of Painleve equations
Grant number：15K13425 2015.4 - 2018.3
Okada Soichi
Authorship：Principal investigator Grant type：Competitive
Grant amount：\3900000 （ Direct Cost: \3000000 、 Indirect Cost：\900000 ）
We study several aspects of integrable systems and symmetric functions, toward unraveling combinatorial structure of special polynomials associated to certain solutions of the Painleve-type equations. 1. We find Giambelli-type determinant identities, which characterize the expansion coefficients of the \tau-function of the KP hierarchy, and give new determinant expressions for Schur functions. 2. We find a relation between two generalizations of Schur Q-functions corresponding to rational Schur functions. Also we give several identities and positivity conjectures for Schur Q-functions associated to the root system of type C.
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Research on the difference systems associated with multivariable elliptic hypergeometric functions with Weyl group symmetry
Grant number：25400118 2013.4 - 2018.3
ITO Masahiko
Authorship：Collaborating Investigator(s) (not designated on Grant-in-Aid) Grant type：Competitive
The multivariable elliptic hypergeometric functions associated with root systems were studied via their systems of difference equations and Weyl group symmetry. Masahiko Ito (the leader of this research, Univ. of the Ryukyus) and Masatoshi Noumi (cooperating researcher of this research, Kobe univ.) provided a definition of the family of ``interpolation functions'' for the multivariable elliptic hypergeometric functions of type BCn. This is the main result of this research. As applications of the interpolation functions, several summation and translation formulae for the multivariable elliptic hypergeometric functions of type BCn were proved.
Teaching Experience (On-campus) 2
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Linear Algebra II
2011
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Linear Algebra I
2011