Researcher Profile and Settings

Master

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

researchmap

Profile and Settings

Affiliation

• Hokkaido University, Faculty of Science Department of Mathematics

Profile and Settings

• Name (Japanese)

  Numata

• Name (Kana)

  Yasuhide

• Name

  201901007000891029

Affiliation

• Hokkaido University, Faculty of Science Department of Mathematics

Achievement

Research Areas

• Natural sciences / Algebra

Published 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 2022/02/28 [Refereed]
  
• 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 0025-5645 2019/10 [Refereed][Not invited]
   

  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 1862-2976 2019/06/01 [Refereed][Not invited]
   

  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 0097-3165 2018/05 [Refereed][Not invited]
   

  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 城西大学大学院理学研究科 10 (10) 81 - 95 1344-7777 2017/03 [Refereed][Not invited]
   

  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 1939-2346 2016 [Refereed][Not invited]
   

  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 1310-5132 2016 [Refereed][Not invited]
   

  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 [Refereed][Not invited]
  
• 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 0047-259X 2013/05 [Refereed][Not invited]
   

  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 1617-9692 2013 

  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 1617-9692 2013 

  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 1617-9692 2013 

  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 1617-9692 2013 

  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 1617-9692 2013 

  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 1617-9692 2013 

  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 1617-9692 2013 

  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 1617-9692 2013 

  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 0075-8434 2013 [Refereed][Not invited]
  
• 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 0012-365X 2013/01 [Refereed][Not invited]
   

  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 0218-0006 2012/12 [Refereed][Not invited]
   

  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 0925-9899 2012/02 [Refereed][Not invited]
   

  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 1462-7264 2012 

  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 [Refereed][Not invited]
  
• 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 [Refereed][Not invited]
  
• 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 1386-923X 2011/08 [Refereed][Not invited]
   

  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 2011/07 [Refereed][Not invited]
  
• Statistical Analysis of Subject Selection Data in NCUEE Examination

  AOKI Satoshi, OTSU Tatsuo, TAKEMURA Akimichi, NUMATA Yasuhide

  Ouyou toukeigaku 応用統計学会 39 (2) 71 - 100 0285-0370 2010/12/25 [Not refereed][Not invited]
   

  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 Springer 62 (4) 645 - 672 0020-3157 2010/08 [Refereed][Not invited]
   

  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 0024-6107 2009/08 [Refereed][Not invited]
   

  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 [Refereed][Not invited]
  
• Tabloids and weighted sums of characters of certain modules of the symmetric groups

  Numata Yasuhide

  European Journal of Combinatorics Academic Press 29 (2) 480 - 492 0195-6698 2008/02 [Refereed][Not invited]
   

  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 0021-8693 2007/12 [Refereed][Not invited]
   

  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 Springer Netherlands 26 (1) 27 - 45 0925-9899 2007/08 [Refereed][Not invited]
   

  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.

MISC

• 超平面配置のSolomon-寺尾代数と応用 (変換群を核とする代数的位相幾何学)

  阿部 拓郎, 沼田 泰英, 前野 俊昭  数理解析研究所講究録  (2060)  154  -163  2018/04  [Not refereed][Not invited]
   

  本稿では, 準備中の論文[3]の報告として, 超平面配置に付随する新しい代数たるSolomon-寺尾代数を紹介し, その性質と応用を調べる. 特に正則幕零ヘッセンベルグ多様体のコホモロジー環との関係について述べる.
• Polynomial Expressions of p-ary Auction Functions

  Shizuo Kaji, Toshiaki Maeno, Koji Nuida, Yasuhide Numata  e-print arXiv  1703-  (07930)  2017/03  [Not refereed][Not invited]
  
• Polynomial Expressions of Carries in p-ary Arithmetics

  Shizuo Kaji, Toshiaki Maeno, Koji Nuida, Yasuhide Numata  e-print arXiv  1506-  (02742)  2015/06  [Not refereed][Not invited]
  
• Report of "Sage Days in Japan"

  Yokoyama Shun'ichi, Numata Yasuhide  RIMS Kokyuroku  1930-  (1930)  73  -79  2015/01  [Not refereed][Not invited]
  
• On Computation of the First Baues-Wirsching Cohomology of a Freely-Generated Small Category.

  Yasuhiro Momose, Yasuhide Numata  Mathematical Software - ICMS 2014 - 4th International Congress, Seoul, South Korea, August 5-9, 2014. Proceedings  99  -105  2014  [Refereed][Not invited]
  
• A mathematical problem for security analysis of hash functions and pseudorandom generators

  Koji Nuida, Takuro Abe, Shizuo Kaji, Toshiaki Maeno, Yasuhide Numata  arXiv:CoRR  1206.0069-  2012  [Not refereed][Not invited]
  
• Sperner property, matroids and finite-dimensional Gorenstein algebras

  Toshiaki Maeno, Yasuhide Numata  TROPICAL GEOMETRY AND INTEGRABLE SYSTEMS  580-  73  -+  2012  [Refereed][Not invited]
   

  We discuss a combinatorial property of the vector space lattice and some polynomials associated to matroids. Stanley developed powerful methods based on Hard Lefschetz Theorem to handle combinatorial objects. It is known that proofs of the Sperner property of typical posets can be done by showing the Lefschetz property for related Artinian commutative graded algebras. We introduce certain finite-dimensional Gorenstein algebras associated to matroids to show the Sperner property for a class of ranked posets including vector space lattices. We also discuss the Grobner fans of the defining ideal of our Gorenstein algebras and some tropical hypersurfaces. https://books.google.co.in/books?hl=en&lr=&id=AZgCAQAAQBAJ&oi=fnd&pg=PA73&dq=Sperner+property,+matroids+and+finite-dimensional+Gorenstein+algebras&ots=FGpcbOXIkf&sig=b8Ba1p7u1aXAl9RxyjAkPEuyMuE#v=onepage&q=Sperner%20property%2C%20matroids%20and%20finite-dimensional%20Gorenstein%20algebras&f=false
• パラメータ固定ハッシュ関数の理論的安全性評価に関する一考察―関数の近似可能性の観点から―

  縫田光司, 阿部拓郎, 鍛冶静雄, 沼田泰英, 前野俊昭  情報処理学会シンポジウムシリーズ(CD-ROM)  2011-  (3)  ROMBUNNO.2C2,NUIDA  2011/10/12  [Not refereed][Not invited]
  
• On security evaluation of hash functions from a viewpoint of function approximation

  縫田 光司, 阿部 拓郎, 鍛冶 静雄, 沼田 泰英, 前野 俊昭  コンピュータセキュリティシンポジウム2011 論文集  2011-  (3)  331  -336  2011/10/12  [Not refereed][Not invited]
  
• On moments of the noncentral Wishart distributions and weighted generating functions of matchings (Combinatorial Representation Theory and its Applications)

  Kuriki Satoshi, Numata Yasuhide  RIMS Kokyuroku  1738-  (1738)  142  -154  2011/04  [Not refereed][Not invited]
  
• A Mathematical Problem for Security Analysis of Hash Functions and Pseudorandom Generators

  Koji Nuida, Takuro Abe, Shizuo Kaji, Toshiaki Maeno, Yasuhide Numata  ADVANCES IN INFORMATION AND COMPUTER SECURITY  7038-  144  -+  2011  [Refereed][Not invited]
   

  The aim of this paper is to emphasize the significance of a certain mathematical problem in research on information security. We point out that the mathematical problem, which we refer to as "Function Density Problem," has connections to the following two major cryptographic topics; security analysis of hash functions in the real world (like SHA-1), and construction of pseudorandom generators with some enhanced security property. We also provide a first example to show how a study of Function Density Problem can contribute to the progress of the above-mentioned two topics. Other potential applications of Function Density Problem to information security are also discussed.
• 計算代数統計の進展―日比プロジェクト応用系の研究成果

  沼田泰英  越境する数学 CREST研究報告会 平成23年  42  2011  [Not refereed][Not invited]
  
• ウィシャート分布のモーメント公式と多変量非心ガンマ分布の無限分解可能性

  栗木哲, 沼田泰英  統計関連学会連合大会講演報告集  2010-  245  2010/09  [Not refereed][Not invited]
  
• On an edge-signed generalization of chordal graphs and free multiplicities on braid arrangements (Representation Theory and Combinatorics)

  Numata Yasuhide  RIMS Kokyuroku  1689-  (1689)  78  -88  2010/05  [Not refereed][Not invited]
  
• 非心ウィシャート分布のモーメントのグラフ表現とその応用

  栗木哲, 沼田泰英  統計関連学会連合大会講演報告集  2009-  277  2009/09  [Not refereed][Not invited]
  
• On a bijective proof of a factorization formula for Macdonald polynomials at roots of unity (Expansion of combinatorial representation theory)

  Numata Yasuhide  RIMS Kokyuroku  1647-  (1647)  69  -80  2009/05  [Not refereed][Not invited]
  
• An extended Schur's lemma and its application

  沼田 泰英  Wakhok  (9)  5  -16  2009/03  [Not refereed][Not invited]
  
• GENERALIZED SCHUR OPERATORS ON PLANAR BINARY TREES (Combinatorial Representation Theory and Related Topics)

  NUMATA YASUHIDE  RIMS Kokyuroku Bessatsu  8-  (8)  85  -97  2008/05  [Not refereed][Not invited]
  
• Bicolor-eliminable multiplicities on the braid arrangement

  阿部 拓郎, 縫田 光司, 沼田 泰英  CCYR4テクニカルレポート（北大講究録）  2008  [Not refereed][Not invited]
  
• Pieri's formula for generalized Schur polynomials(The world of Combinatorial Representation Theory)

  NUMATA Yasuhide  RIMS Kokyuroku  1497-  (1497)  1  -14  2006/06  [Not refereed][Not invited]
  
• ロビンソン-シェンステッド対応の一般化 (表現論における組合せ論的手法とその応用 研究集会報告集)

  沼田 泰英  数理解析研究所講究録  1438-  (1438)  186  -197  2005/07  [Not refereed][Not invited]
  

Books etc

• 組合せ論的表現論と表現論的組合せ論

  沼田 泰英, 京都大学数理解析研究所 
  京都大学数理解析研究所 2016
• The Lefschetz properties

  張間 忠人, 前野 俊昭, 森田 英章, 沼田 泰英, 和地 輝仁, 渡邊 純三 
  Springer 2013 (ISBN: 9783642382055)

Research Projects

• Discrete structures related to hyperplane arrangements, generalization, deepening, and applications

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2023/04 -2028/03 

  Author : 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
• Fusion of algebra, geometry and combinatorics based on the roots of Poincare polynomials of hyperplane arrangements

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Research (Exploratory)

  Date (from‐to) : 2020/07 -2023/03 

  Author : 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。
• Combinatorics related to representation theory and enumeration of paths

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

  Date (from‐to) : 2018/04 -2023/03 

  Author : 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.
• Research on the coinvariant ring theory for hyperplane arrangements and the new developments of its representation and geometry

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

  Date (from‐to) : 2016/04 -2021/03 

  Author : 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。
• Application fo topological data analisys to geographic information

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

  Date (from‐to) : 2016/07 -2019/03 

  Author : 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.
• Enumerative combinatorics on matchings and its application

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)

  Date (from‐to) : 2013/04 -2017/03 

  Author : 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.
• Studies on association schemoids with insights gained from cohomology theory of small categories

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research

  Date (from‐to) : 2013/04 -2016/03 

  Author : 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.
• 教具の作成を通した数学教育の試み

  日産科学振興財団：理科/環境教育助成

  Date (from‐to) : 2008/11 -2009/11
• THE RING OF DIFFERENTIAL OPERATORS ON AN AFFINE TORIC VARIETYAND ITS APPLICATIONS

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

  Date (from‐to) : 2006 -2008 

  Author : 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。