Numata Yasuhide

Faculty of Science　Mathematics　MathematicsProfessor
Last Updated :2025/01/11

■Researcher basic information

Degree

  • Hokkaido University, Mar. 2007

Researchmap personal page

Home Page URL

Researcher number

  • 00455685

ORCID ID

J-Global ID

Research Field

  • Natural sciences, Algebra

■Research activity information

Papers

  • On the action of the toggle group of the Dynkin diagram of type A
    Yasuhide Numata, Yuiko Yamanouchi
    Algebraic Combinatorics, 5, 1, 149, 161, Cellule MathDoc/CEDRAM, 28 Feb. 2022, [Peer-reviewed]
    Scientific journal
  • Solomon-Terao algebra of hyperplane arrangements
    Abe, Takuro, Maeno, Toshiaki, Mural, Satoshi, Numata, Yasuhide
    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, 71, 4, 1027, 1047, MATH SOC JAPAN, Oct. 2019, [Peer-reviewed]
    English, Scientific journal, We introduce a new algebra associated with a hyperplane arrangement A, called the Solomon-Terao algebra ST(A, eta), where eta is a homogeneous polynomial. It is shown by Solomon and Terao that ST(A, eta) is Artinian when eta is generic. This algebra can be considered as a generalization of coinvariant algebras in the setting of hyperplane arrangements. The class of Solomon-Terao algebras contains cohomology rings of regular nilpotent Hessenberg varieties. We show that ST(A, eta) is a complete intersection if and only if A is free. We also give a factorization formula of the Hilbert polynomials of ST(A, eta) when A is free, and pose several related questions, problems and conjectures.
  • Polynomial expressions of p-ary auction functions
    Shizuo Kaji, Toshiaki Maeno, Koji Nuida, Yasuhide Numata
    Journal of Mathematical Cryptology, 13, 2, 69, 80, Walter de Gruyter {GmbH}, 01 Jun. 2019, [Peer-reviewed]
    English, Scientific journal, One of the common ways to design secure multi-party computation is twofold: to realize secure fundamental operations and to decompose a target function to be securely computed into them. In the setting of fully homomorphic encryption, as well as some kinds of secret sharing, the fundamental operations are additions and multiplications in the base field such as the field F-2 with two elements. Then the second decomposition part, which we study in this paper, is (in theory) equivalent to expressing the target function as a polynomial. It is known that any function over the finite prime field F-p has a unique polynomial expression of degree at most p - 1 with respect to each input variable; however, there has been little study done concerning such minimal-degree polynomial expressions for practical functions. This paper aims at triggering intensive studies on this subject, by focusing on polynomial expressions of some auction-related functions such as the maximum/minimum and the index of the maximum/minimum value among input values.
  • A topos associated with a colored category
    Kuribayashi, Katsuhiko, Numata, Yasuhiude
    JOURNAL OF COMBINATORIAL THEORY SERIES A, 156, 142, 163, ACADEMIC PRESS INC ELSEVIER SCIENCE, May 2018, [Peer-reviewed]
    English, Scientific journal, We show that a functor category whose domain is a colored category is a topos. The topos structure enables us to introduce cohomology of colored categories including quasi-schemoids. If the given colored category arises from an association scheme, then the cohomology coincides with the group cohomology of the factor scheme by the thin residue. Moreover, it is shown that the cohomology of a colored category relates to the standard representation of an association scheme via the Leray spectral sequence. (C) 2018 Elsevier Inc. All rights reserved.
  • Toward a combinatorial formula for an irregular conformal block of rank one
    名古屋 創, 沼田 泰英
    Josai Mathematical Monographs, 10, 10, 81, 95, 城西大学大学院理学研究科, Mar. 2017, [Peer-reviewed]
    English, In this note, we give a combinatorial formula for a particular three-point irregular conformal block of rank one using the Littlewood-Richardson numbers and propose a conjectural formula for the general threepoint irregular conformal block of rank one.
  • Sperner property and finite-dimensional Gorenstein algebras associated to matroids
    Toshiaki Maeno, Yasuhide Numata
    Journal of Commutative Algebra, 8, 4, 549, 570, Rocky Mountain Mathematics Consortium, 2016, [Peer-reviewed]
    English, Scientific journal, We prove the Lefschetz property for a certain class of finite-dimensional Gorenstein algebras associated to matroids. Our result implies the Sperner property of the vector space lattice. More generally, it is shown that the modular geometric lattice has the Sperner property. We also discuss the Gröbner fan of the defining ideal of our Gorenstein algebra.
  • On graphs whose Hosoya indices are primitive Pythagorean triples
    Kadoi, Tomoe, Numata, Yasuhide
    NOTES ON NUMBER THEORY AND DISCRETE MATHEMATICS, 22, 1, 59, 80, BULGARIAN ACAD SCIENCE, 2016, [Peer-reviewed]
    English, Scientific journal, We discuss families of triples of graphs whose Hosoya indices are primitive Pythagorean triples. Hosoya gave a method to construct such families of caterpillars, i.e., trees whose vertices are within distance 1 of a central path. He also pointed out a common structure to the families. In this paper, we show the uniqueness of the common structure.
  • A Mathematical Problem for Security Analysis of Hash Functions and Pseudorandom Generators.
    Koji Nuida, Takuro Abe, Shizuo Kaji, Toshiaki Maeno, Yasuhide Numata
    Int. J. Found. Comput. Sci., 26, 2, 169, 194, 2015, [Peer-reviewed]
    Scientific journal
  • The holonomic gradient method for the distribution function of the largest root of a Wishart matrix
    Hiroki Hashiguchi, Yasuhide Numata, Nobuki Takayama, Akimichi Takemura
    JOURNAL OF MULTIVARIATE ANALYSIS, 117, 296, 312, ELSEVIER INC, May 2013, [Peer-reviewed]
    English, Scientific journal, We apply the holonomic gradient method introduced by Nakayama et al. (2011) [23] to the evaluation of the exact distribution function of the largest root of a Wishart matrix, which involves a hypergeometric function F-1(1) of a matrix argument. Numerical evaluation of the hypergeometric function has been one of the longstanding problems in multivariate distribution theory. The holonomic gradient method offers a totally new approach, which is complementary to the infinite series expansion around the origin in terms of zonal polynomials. It allows us to move away from the origin by the use of partial differential equations satisfied by the hypergeometric function. From the numerical viewpoint we show that the method works well up to dimension 10. From the theoretical viewpoint the method offers many challenging problems both to statistics and D-module theory. (C) 2013 Elsevier Inc. All rights reserved.
  • Cohomology rings and the strong lefschetz property
    Tadahito Harima, Toshiaki Maeno, Hideaki Morita, Yasuhide Numata, Akihito Wachi, Junzo Watanabe
    Lecture Notes in Mathematics, 2080, 189, 199, Springer Verlag, 2013
    English, In book, The Lefschetz property originates in the Hard Lefschetz Theorem for compact Kähler manifolds, so it is natural that some results discussed in the former chapters have geometric backgrounds. For example, Corollary 4.17 on the flat extension can be understood from the cohomology ring of projective space bundles in a geometric setting.
  • Invariant theory and lefschetz properties
    Tadahito Harima, Toshiaki Maeno, Hideaki Morita, Yasuhide Numata, Akihito Wachi, Junzo Watanabe
    Lecture Notes in Mathematics, 2080, 201, 209, Springer Verlag, 2013
    English, In book, In this chapter we discuss topics of invariant theory such as coinvariant algebras of reflection groups. In particular the coinvariant algebras of real reflection groups have the SLP, and the set of Lefschetz elements is explicitly determined in most cases.
  • A generalization of lefschetz elements
    Tadahito Harima, Toshiaki Maeno, Hideaki Morita, Yasuhide Numata, Akihito Wachi, Junzo Watanabe
    Lecture Notes in Mathematics, 2080, 157, 170, Springer Verlag, 2013
    English, In book, In this chapter we would like to discuss a generalization of Lefschetz elements for an Artinian local ring to study the Jordan decomposition of a general element. The point of departure for us is Theorem 5.1 due to D. Rees. Several results from Chap. 6 (e.g., stable ideals, Borel fixed ideals, gin(I), etc) are needed at a few points in Chap. 5.
  • The strong lefschetz property and the schur–weyl duality
    Tadahito Harima, Toshiaki Maeno, Hideaki Morita, Yasuhide Numata, Akihito Wachi, Junzo Watanabe
    Lecture Notes in Mathematics, 2080, 211, 234, Springer Verlag, 2013
    English, In book, The purpose of this chapter is to illustrate a role played by the SLP in connection with the theory of Artinian rings and the Schur–Weyl duality. We assume that the reader is familiar with commutative algebra but perhaps without knowledge of representation theory, but we are hopeful that the expert in representation theory may also find the following sections of interest.
  • k-lefschetz properties
    Tadahito Harima, Toshiaki Maeno, Hideaki Morita, Yasuhide Numata, Akihito Wachi, Junzo Watanabe
    Lecture Notes in Mathematics, 2080, 171, 188, Springer Verlag, 2013
    English, In book, In this chapter we define the k-Lefschetz properties by generalizing the Lefschetz properties. The k-Lefschetz properties give us a way of computing generic initial ideals and graded Betti numbers of Artinian graded K-algebras.
  • Poset theory
    Tadahito Harima, Toshiaki Maeno, Hideaki Morita, Yasuhide Numata, Akihito Wachi, Junzo Watanabe
    Lecture Notes in Mathematics, 2080, 1, 38, Springer Verlag, 2013
    English, In book, This chapter was written to furnish a starting point for the study of Artinian rings in commutative algebra. We are primarily interested in the Sperner theory of finite posets.
  • Complete intersections with the SLP
    Tadahito Harima, Toshiaki Maeno, Hideaki Morita, Yasuhide Numata, Akihito Wachi, Junzo Watanabe
    Lecture Notes in Mathematics, 2080, 141, 156, Springer Verlag, 2013
    English, In book, The main result of this chapter is Theorem 4.10. This may be regarded as a generalization of Theorem 3.34 which states that the SLP is preserved by tensor products. Using the main theorem, we give some examples of complete intersections with the strong Lefschetz property.
  • Basics on the theory of local rings
    Tadahito Harima, Toshiaki Maeno, Hideaki Morita, Yasuhide Numata, Akihito Wachi, Junzo Watanabe
    Lecture Notes in Mathematics, 2080, 39, 95, Springer Verlag, 2013
    English, In book, The reader is assumed to have basic knowledge of the theory of commutative rings. Let R be a commutative ring with an identity element and let f1, f2..fm be elements of R.
  • The Lefschetz properties
    Tadahito Harima, Toshiaki Maeno, Hideaki Morita, Yasuhide Numata, Akihito Wachi, Junzo Watanabe
    Lecture Notes in Mathematics, 2080, 1, 252, Springer Verlag, 2013, [Peer-reviewed]
    English, Scientific journal
  • Separation of integer points by a hyperplane under some weak notions of discrete convexity
    Takuya Kashimura, Yasuhide Numata, Akimichi Takemura
    DISCRETE MATHEMATICS, 313, 1, 8, 18, ELSEVIER SCIENCE BV, Jan. 2013, [Peer-reviewed]
    English, Scientific journal, We give some sufficient conditions of separation of two sets of integer points by a hyperplane. Our conditions are related to the notion of convexity of sets of integer points and are weaker than existing notions. (C) 2012 Elsevier B.V. All rights reserved.
  • On Intersection Lattices of Hyperplane Arrangements Generated by Generic Points
    Hiroshi Koizumi, Yasuhide Numata, Akimichi Takemura
    ANNALS OF COMBINATORICS, 16, 4, 789, 813, SPRINGER BASEL AG, Dec. 2012, [Peer-reviewed]
    English, Scientific journal, We consider hyperplane arrangements generated by generic points and study their intersection lattices. These arrangements are known to be equivalent to discriminantal arrangements. We show a fundamental structure of the intersection lattices by decomposing the poset ideals as direct products of smaller lattices corresponding to smaller dimensions. Based on this decomposition we compute the Mobius functions of the lattices and the characteristic polynomials of the arrangements up to dimension six.
  • Exponents of 2-multiarrangements and multiplicity lattices
    Takuro Abe, Yasuhide Numata
    JOURNAL OF ALGEBRAIC COMBINATORICS, 35, 1, 1, 17, SPRINGER, Feb. 2012, [Peer-reviewed]
    English, Scientific journal, We introduce a concept of multiplicity lattices of 2-multiarrangements, determine the combinatorics and geometry of that lattice, and give a criterion and method to construct a basis for derivation modules effectively.
  • On the sperner property and gorenstein algebras associated to matroids　　　　　　　　　　　　　　　
    Toshiaki Maeno, Yasuhide Numata
    Discrete Mathematics and Theoretical Computer Science, 157, 168, 2012
    English, International conference proceedings, We introduce a certain class of algebras associated to matroids. We prove the Lefschetz property of the algebras for some special cases. Our result implies the Sperner property for the Boolean lattice and the vector space lattice. © 2012 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.
  • On computation of the characteristic polynomials of the discriminantal arrangements and the arrangements generated by generic points　　　　　　　　　　　　　　　
    Yasuhide Numata, A.Takemura
    Harmony of Grobner Bases and the Modern Industrial Society, (Takayuki Hibi, editor), World Scientific, 228, 252, 2012, [Peer-reviewed]
  • On a bijective proof of a factorization formula for Macdonald polynomials.
    François Descouens, Hideaki Morita, Yasuhide Numata
    Eur. J. Comb., 33, 6, 1257, 1264, 2012, [Peer-reviewed]
    Scientific journal
  • Strong Lefschetz Elements of the Coinvariant Rings of Finite Coxeter Groups
    Toshiaki Maeno, Yasuhide Numata, Akihito Wachi
    ALGEBRAS AND REPRESENTATION THEORY, 14, 4, 625, 638, SPRINGER, Aug. 2011, [Peer-reviewed]
    English, Scientific journal, For the coinvariant rings of finite Coxeter groups of types other than H(4), we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups.
  • Bumping algorithm for set-valued shifted tableaux
    Takeshi IKEDA, Hiroshi NARUSE, Yasuhide NUMATA
    FPSAC 2011 Reykjavik, Iceland, 527, 538, Jul. 2011, [Peer-reviewed]
    English, International conference proceedings
  • Statistical Analysis of Subject Selection Data in NCUEE Examination
    AOKI Satoshi, OTSU Tatsuo, TAKEMURA Akimichi, NUMATA Yasuhide
    Ouyou toukeigaku, 39, 2, 71, 100, Japanese Society of Applied Statistics, 25 Dec. 2010
    Japanese, In this paper we present statistical analysis of data on subject selection by examinees in NCUEE (National Center for University Entrance Examinations) examination in 2006. In NCUEE examinations, exmaminees can choose subjects depending on the university and the department they are applying. As seen from the well publicized news on skipping world history classes in some high schools, the pattern of subject selection is complicated and depends on many factors, incluing geography and sex. Analysis of influences of these factors is important in discussing the university entrance examination and the education in high shools in Japan. In this paper we deal with geographic factors by incorporating effect of individual cells into hierarchical models of contingency tables. We also estimate the influence of sex on selection of science subjects by conditional likelihood method. For confirming these effects we employ Markov chain Monte Carlo methods, in addition to asymptotic approximation.
  • Graph presentations for moments of noncentral Wishart distributions and their applications (Special issue: Algebraic methods in computational statistics)
    Kuriki, Satoshi, Numata, Yasuhide
    Annals of the Institute of Statistical Mathematics, 62, 4, 645, 672, SPRINGER HEIDELBERG, Aug. 2010, [Peer-reviewed]
    English, We provide formulas for the moments of the real and complex noncentral Wishart distributions of general degrees. The obtained formulas for the real and complex cases are described in terms of the undirected and directed graphs, respectively. By considering degenerate cases, we give explicit formulas for the moments of bivariate chi-square distributions and 2 x 2 Wishart distributions by enumerating the graphs. Noting that the Laguerre polynomials can be considered to be moments of a noncentral chi-square distributions formally, we demonstrate a combinatorial interpretation of the coefficients of the Laguerre polynomials.
  • Signed-eliminable graphs and free multiplicities on the braid arrangement
    Takuro Abe, Koji Nuida, Yasuhide Numata
    JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 80, 121, 134, OXFORD UNIV PRESS, Aug. 2009, [Peer-reviewed]
    English, Scientific journal, We define specific multiplicities on the braid arrangement by using signed graphs. To consider their freeness, we introduce the notion of signed-eliminable graphs as a generalization of Stanley's classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of certain deformations of the braid arrangement in terms of directed graphs.
  • An edge-signed generalization of chordal graphs, free multiplicities on braid arrangements, and their characterizations
    Takuro Abe, Koji Nuida, Yasuhide Numata
    Proceedings of 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (poster), 1, 12, 2009, [Peer-reviewed]
    English, International conference proceedings
  • Tabloids and weighted sums of characters of certain modules of the symmetric groups
    Numata Yasuhide
    European Journal of Combinatorics, 29, 2, 480, 492, Academic Press, Feb. 2008, [Peer-reviewed]
    English, We consider certain modules of the symmetric groups whose basis elements are called tabloids. Some of these modules are isomorphic to subspaces of the cohomology rings of subvarieties of flag varieties as modules of the symmetric groups. We give a combinatorial description for some weighted sums of their characters, i.e., we introduce combinatorial objects called (rho, l)-tableaux and rewrite weighted sums of characters as the numbers of these combinatorial objects. We also consider the meaning of these combinatorial objects, i.e., we construct a correspondence between (rho, l)-tableaux and tabloids whose images are eigenvectors of the action of an element of cycle type p in quotient modules. (c) 2007 Elsevier Ltd. All rights reserved.
  • The strong Lefschetz property of the coinvariant ring of the Coxeter group of type H-4
    Yasuhide Numata, Akihito Wachi
    JOURNAL OF ALGEBRA, 318, 2, 1032, 1038, ACADEMIC PRESS INC ELSEVIER SCIENCE, Dec. 2007, [Peer-reviewed]
    English, Scientific journal, We prove that the coinvariant ring of the irreducible Coxeter group of type H-4 has the strong Lefschetz property. (C) 2007 Elsevier Inc. All rights reserved.
  • Pieri's formula for generalized Schur polynomials
    Numata Yasuhide
    Journal of Algebraic Combinatorics, 26, 1, 27, 45, Springer Netherlands, Aug. 2007, [Peer-reviewed]
    English, Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted- Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized Schur operators to generalize the Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur operators are generalizations of semi-standard Young tableaux. We define a generalization of Schur polynomials as expansion coefficients of generalized Schur operators. We show that the commutation relation of generalized Schur operators implies Pieri's formula for generalized Schur polynomials.

Other Activities and Achievements

Books and other publications

Research Themes

  • Discrete structures related to hyperplane arrangements, generalization, deepening, and applications
    Grants-in-Aid for Scientific Research
    01 Apr. 2023 - 31 Mar. 2028
    吉永 正彦, 阿部 拓郎, 石川 昌治, 島田 伊知朗, 辻栄 周平, 徳永 浩雄, 沼田 泰英, 東谷 章弘
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (A), Osaka University, 23H00081
  • Fusion of algebra, geometry and combinatorics based on the roots of Poincare polynomials of hyperplane arrangements
    Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Research (Exploratory)
    30 Jul. 2020 - 31 Mar. 2023
    阿部 拓郎, 沼田 泰英, 鍛冶 静雄
    研究計画二年目となる2021年度もコロナ禍の真っただ中であり、限定的な対面打ち合わせしかできない中、オンラインツールなどを積極的に用いて研究計画を推進した。その結果得られた、辻栄周平氏とTan Nhat Tran氏との国際共同研究について説明する。
    グラフとは点と辺からなるシンプルな研究対象であるが、これと対応するグラフ配置の研究は、超平面配置の研究開始以来深く調べられていた。特にStanleyによる、コーダルグラフであることとグラフ配置の自由性の同値性は極めて重要な結果であり、この場合根はグラフに完全除去順序を入れた場合のある種の辺の本数として理解することができる。他方この一般化として、辺に向きを付け頂点に重さを付けた、有向グラフから定まるグラフ配置の研究が近年注目を集めている。その中でも重要な配置として、Shi配置とIsh配置と呼ばれる配置がある。これらの間をつなぐ超平面配置レベルでの自然な変形が存在しており、これらの特性多項式がShi配置Ish配置と同じであることが知られていた。Shi配置、Ish配置どちらも自由であるため、これらの変形も自由であるかどうかが問題となっていた。まずこれらの自由性を示し、更にそれらがグラフの全く新しい変形理論から自然に理解可能であることが分かった。これは頂点に乗った重さとある頂点に入る辺とを交換する操作で、この操作で「特性多項式が保たれること」がわかり、かつある仮定の下で「自由配置に対応するグラフにこの変形を施したものも自由である」ことがわかった。これは根のグラフ理論的理解に対するブレイクスルーであり、この範疇に含まれる配置をさらに研究することで、整数根への理解がさらに進むと期待される。本結果はプレプリントとして公開済みである。
    Japan Society for the Promotion of Science, Grant-in-Aid for Challenging Research (Exploratory), Kyushu University, 20K20880
  • Combinatorics related to representation theory and enumeration of paths
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    01 Apr. 2018 - 31 Mar. 2023
    沼田 泰英
    本研究では, ヤング図形やその一般化といった表現論に関連する組合せ論的対象について, 数え上げ組合せ論的見地からの研究を行います. 特に, 既知の数え上げ公式などについて, 広い意味でのLattice path methodによる解釈を与え, 公理化をすることにより, 統一的な証明やより広い対象への一般化を目標としています.
    当該年度においては, Hook Length formulaとよばれる数え上げ公式について着目し, 特に, その公式の全単射による証明を与えるにあたって鍵となる Hillman-Grassl アルゴリズムと呼ばれるアルゴリズムについての研究を進めました. 特に, ヤング図形やその類似物であるd-complete poset と呼ばれる対象の一部にケースバイケースの方法で与えられている一連のアルゴリズムに関して統一的な記述を与えることを目標に研究を進めました.
    対象となっているアルゴリズムを走らせるために十分な条件を公理として課した半順序集合においては, 広い意味でのLattice path methodを用いることで, Hillman-Grassl アルゴリズムの類似のアルゴリズムを構成することが出来ました. また, Swivel と呼ばれるクラスのd-complete posetを含まないようなd-complete posetのうち既約なものについては, 与えた公理を満たすような実現があることを, 具体的に実現を構成することで示すことが出来ました. Swivelを含まないd-complete posetで既約ではないものについては, これらを組み合わせることで実現を与えることが出来ます.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Shinshu University, 18K03206
  • Research on the coinvariant ring theory for hyperplane arrangements and the new developments of its representation and geometry
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
    01 Apr. 2016 - 31 Mar. 2021
    阿部 拓郎, 沼田 泰英, 榎本 直也, 吉永 正彦, 村井 聡
    本年度はSolomon-寺尾代数理論を発展させるための結果をいくつか残すことができた。その中でも特に重要な結果は、複素鏡映群に対応する複素鏡映配置の多重配置の代数構造の解明である。実鏡映配置、特にワイル配置の場合は斎藤恭司氏や寺尾宏明氏らにより深く研究がなされていたが、複素鏡映配置については統一的な研究はなされていなかった。その大きな理由は、実の場合に極めて強力な解析ツールであった斎藤の原始微分が、複素鏡映の場合にきちんと定式化されていない点にあった。近年この点が、Gerhard Roehrle氏らの研究により大きく発展したことを受け、吉永正彦氏、Roehrle氏及びChristian Stump氏らと、well-generatedな複素鏡映配置に良い重複度を載せた場合の自由性について研究を行った。これは二つのパートからなる。まず第一に、各超平面にその複素鏡映の位数だけ重複度を載せたものの周辺にある多重複素鏡映配置の自由性を特徴づけることに成功した。第二に、well-generatedな場合の複素鏡映配置に対する、斎藤のHodge分解の構成に成功した。これらの結果から、実の場合の結果で齋藤のHodge分解などにおいて、本質的になにが重要であるかが明確になった。
    更にこの結果から、実鏡映配置を基礎として、イデアル配置のSolomon-寺尾代数と正則冪零ヘッセンベルグ多様体との間にコホモロジーのレベルで関係がついたことを踏まえれば、この複素鏡映配置に関する結果を用いることで、同様の関係が存在するクラスが複素レベルで発見されることが期待される。また、ワイル配置に対するSolomon-寺尾図式の幾何学的表現論的理解を、複素鏡映配置のレベルまであげるための基礎的な情報としても、本結果は重要な意味を持つ。
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Kyushu University, 16H03924
  • Application fo topological data analisys to geographic information
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    19 Jul. 2016 - 31 Mar. 2019
    Numata Yasuhide
    The purpose of this research project was to try to apply methods of topologiacal data analysis to data of geographic information systems. We had some seminars to discuss with researchers in the related areas. In the seminar, we discuss and study the related algebraic topology and methods of topological data analysis. Moreover we calculated the persistence homologies of data of positions of public facilities, e.g., bus stops in some cities. We tried to classify the data sets by differences of persistence homologies.
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Shinshu University, 16KT0131
  • Enumerative combinatorics on matchings and its application
    Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)
    01 Apr. 2013 - 31 Mar. 2017
    NUMATA Yasuhide
    We studied on matchings ingraphs mainly, and obatained some results on enumerative combinatorics by this research.
    We showed the uniqueness of the family of triples whose mathing numbers are pitagorian tripes in some condition. We also obtained some results on Young diagrams and Young tableaux.
    Japan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B), Shinshu University, 25800009
  • Studies on association schemoids with insights gained from cohomology theory of small categories
    Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research
    01 Apr. 2013 - 31 Mar. 2016
    KURIBAYASHI Katsuhiko, MATSUO Kentaro, MOMOSE Yasuhiro, HANAKI Akihide, NUMATA Yasuhide
    We have proposed the notion of (association) schemoids generalizing that of association schemes, which are widely used in algebraic combinatorics, from a small categorical point of view. In our study, the equivalence between the categories of groupoids and that of thin schemoids is established. Moreover, in order to develop homotopy theory for schemoids, we define a homotopy relation on the category of quasi-schemoids and study its fundamental properties. In consequence, the 2-category of small categories is embedded into the 2-category of quasi-schemoids. As for categorical representation theory for schemoids, we have proved Mitchell's embedding theorem for a tame schemoid. The result allows us to give a cofibrantly generated model category structure to the category of chain complexes over a functor category with a schemoid as the domain. We show that every Hamming scheme of binary codes is Morita equivalent to the association scheme arising from the cyclic group of order two.
    Japan Society for the Promotion of Science, Grant-in-Aid for Challenging Exploratory Research, Shinshu University, 25610002
  • 教具の作成を通した数学教育の試み　　　　　　　　　　　　　　　
    理科/環境教育助成
    Nov. 2008 - Nov. 2009
    日産科学振興財団, 稚内北星学園大学, Principal investigator
  • THE RING OF DIFFERENTIAL OPERATORS ON AN AFFINE TORIC VARIETYAND ITS APPLICATIONS
    Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    2006 - 2008
    SAITO Mutsumi, YAMASHITA Hirosh, YANAGAWA Kohji, SHIMADA Ichiro, NUMATA Yasuhide, YANAGAWA Kohji, SHIMADA Ichiro, NUMATA Yasuhide
    アフィントーリック多様体上の(アフィン半群環の)微分作用素環D の構造及びその(微分作用素の)階数による次数環 Gr(D) の構造の研究に関しての構造の研究に関して大きな進展があった。まず, いつもDは右ネターであることを示した。次に左ネター性についてであるが, 左ネターであるための或る十分条件、或る必要条件を与え, さらに、必要十分条件を予想した。また、クリティカル D-加群の特徴付けを行い, 単項生成の場合の分類を行った。さらに、Gr(D) がネター環のとき、 Gr(D) の素イデアルを記述した。
    Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Hokkaido University, 18540002

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