研究者データベース

齋藤 睦(サイトウ ムツミ)
理学研究院 数学部門 数学分野
教授

基本情報

所属

  • 理学研究院 数学部門 数学分野

職名

  • 教授

学位

  • Ph.D.(Penn. State Univ.)

J-Global ID

研究キーワード

  • 代数解析学   環論   Algebraic Analysis   Representation Theory   Ring Theory   

研究分野

  • 自然科学一般 / 代数学

所属学協会

  • 日本数学会   Mathematical Society of Japan   

研究活動情報

論文

  • Confluent hypergeometric systems associated with principal nilpotent p-tuples
    Mutsumi SAITO, Hiroyasu TAKEDA
    International Journal of Mathematics 29 12 2018年10月 [査読有り][通常論文]
  • Projective linear monoids and hinges
    齋藤 睦
    http://arxiv.org/abs/1711.01397 2017年11月 [査読無し][通常論文]
  • Mutsumi Saito
    JOURNAL OF LIE THEORY 27 1 51 - 84 2017年 [査読有り][通常論文]
     
    Let g be a simple Lie algebra of rank n over C. We show that the n-dimensional abelian ideals of a Borel subalgebra of g are limits of Jordan Lie subalgebras. Combining this with a classical result by Kostant, we show that the g-module spanned by all n-dimensional abelian Lie subalgebras of g is actually spanned by the Jordan Lie subalgebras.
  • Mutsumi Saito
    JOURNAL OF PURE AND APPLIED ALGEBRA 217 1 31 - 44 2013年01月 [査読有り][通常論文]
     
    An A-hypergeometric system is not irreducible, if its parameter vector is resonant. In this paper, we present a way of computing a finite system of generators of the first syzygy module of an irreducible A-hypergeometric quotient. In particular, if the semigroup generated by A is simplicial and scored, then an explicit system of generators is given. (c) 2012 Elsevier B.V. All rights reserved.
  • Norihiro Nakashima, Go Okuyama, Mutsumi Saito
    JOURNAL OF ALGEBRA 351 1 294 - 318 2012年02月 [査読無し][通常論文]
     
    Let A be a generic hyperplane arrangement composed of r hyperplanes in an n-dimensional vector space, and S the polynomial ring in n variables. We consider the S-submodule D((m))(A) of the nth Weyl algebra of homogeneous differential operators of order m preserving the defining ideal of A. We prove that if n >= 3, r > n, m > r - n + 1, then D((m))(A) is free (Holm's conjecture). Combining this with some results by Holm, we see that D((m))(A) is free unless n >= 3, r > n, m < r - n + 1. In the remaining case, we construct a minimal free resolution of D((m))(A) by generalizing Yuzvinsky's construction for m = 1. In addition, we construct a minimal free resolution of the transpose of the m-jet module, which generalizes a result by Rose and Terao for m = 1. (C) 2011 Elsevier Inc. All rights reserved.
  • Mutsumi Saito
    COMPOSITIO MATHEMATICA 147 2 613 - 632 2011年03月 [査読無し][通常論文]
     
    Gel'fand, Kapranov and Zelevinsky proved, using the theory of perverse sheaves, that in the Cohen-Macaulay case an A-hypergeometric system is irreducible if its parameter vector is non-resonant. In this paper we prove, using the theory of the ring of differential operators on an affine toric variety, that in general an A-hypergeometric system is irreducible if and only if its parameter vector is non-resonant. In the course of the proof, we determine the irreducible quotients of an A-hypergeometric system.
  • Mutsumi Saito
    COMMUNICATIONS IN ALGEBRA 38 3 829 - 847 2010年 [査読無し][通常論文]
     
    We describe the set of Z(d)-graded prime ideals of the graded ring of the ring D of differential operators of a scored semigroup algebra. Moreover, we describe the characteristic varieties of Z(d)-graded critical D-modules of a certain type.
  • Mutsumi Saito
    COMMUNICATIONS IN ALGEBRA 38 2 618 - 631 2010年 [査読無し][通常論文]
     
    Let D be the ring of differential operators of an affine semigroup algebra. Regarding the Krull dimension of finitely generated Z(d)-graded D-modules, we characterize critical Z(d)-graded D-modules. Moreover, we explicitly describe cyclic ones.
  • Mutsumi Saito, Ken Takahashi
    OSAKA JOURNAL OF MATHEMATICS 46 2 529 - 556 2009年06月 [査読無し][通常論文]
     
    We consider the Noetherian properties of the ring of differential operators of an affine semigroup algebra. First we show that it is always right Noetherian. Next we give a condition, based on the data of the difference between the semigroup and its scored closure, for the ring of differential operators being anti-isomorphic to another ring of differential operators. Using this, we prove that the ring of differential operators is left Noetherian if the condition is satisfied. Moreover we give some other conditions for the ring of differential operators being left Noetherian. Finally conjecture necessary and sufficient conditions for the ring of differential operators being left Noetherian.
  • Mutsumi Saito
    TOHOKU MATHEMATICAL JOURNAL 59 1 119 - 144 2007年03月 [査読無し][通常論文]
     
    We show that the classification of A-hypergeometric systems and that of multi-graded simple modules (up to shift) over the ring of differential operators on an affine toric variety are the same. We then show that the set of multi-homogeneous primitive ideals of the ring of differential operators is finite. Furthermore, we give conditions for the algebra being simple.
  • M Saito, WN Traves
    JOURNAL OF ALGEBRA 278 1 76 - 103 2004年08月 [査読無し][通常論文]
     
    We prove that the ring of differential operators of any semigroup algebra is finitely generated. In contrast, we also show that the graded ring of the order filtration on the ring of differential operators of a semigroup algebra is finitely generated if and only if the semigroup is scored. (C) 2004 Elsevier Inc. All rights reserved.
  • M Saito
    DUKE MATHEMATICAL JOURNAL 115 1 53 - 73 2002年10月 [査読無し][通常論文]
     
    We give a dimension formula for the space of logarithm-free series solutions to an A-hypergeornetric (or a Gel'fand-Kapranov-Zelevinskii (GKZ) hypergeometric) system. In the case where the convex hull spanned by A is a simplex, we give a rank formula for the system, characterize the exceptional set, and prove the equivalence of the Cohen-Macaulayness of the toric variety defined by A with the emptiness of the exceptional set. Furthermore, we classify A-hypergeometric systems as analytic D-modules.
  • M Saito
    COMPOSITIO MATHEMATICA 128 3 323 - 338 2001年09月 [査読無し][通常論文]
     
    Given a finite set A of integral vectors and a parameter vector, Gel'fand, Kapranov, and Zelevinskii defined a system of differential equations, called an A-hypergeometric (or a GKZ hypergeometric) system. Classifying the parameters according to the D-isomorphism classes of their corresponding A-hypergeometric systems is one of the most fundamental problems in the theory. In this paper we give a combinatorial answer for the problem under the assumption that the finite set A lies in a hyperplane off the origin, and illustrate it in two particularly simple cases: the normal case and the monomial curve case.
  • Differential algebras on semigroup algebras
    M. Saito, W. Traves
    Contemporary Mathematics 286 207 - 226 2001年 [査読無し][通常論文]
  • M Saito, B Sturmfels, N Takayama
    COMPOSITIO MATHEMATICA 115 2 185 - 204 1999年01月 [査読無し][通常論文]
     
    We examine connections between A-hypergeometric differential equations and the theory of integer programming. In the first part, we develop a 'hypergeometric sensitivity analysis' for small variations of constraint constants with creation operators and b-functions. In the second part, we study the indicial polynomial (b-function) along the hyperplane x(i) = 0 via a correspondence between the optimal value of an integer programming problem and the roots of the indicial polynomial. Grobner bases are used to prove theorems and give counter examples.
  • M Saito, B Sturmfels, N Takayama
    PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 74 7 111 - 113 1998年09月 [査読無し][通常論文]
  • Mutsumi Saito
    Hokkaido Mathematical Journal 25 3 591 - 619 1996年 [査読無し][通常論文]
     
    The structure of the symmetry algebras of normal A-hypergeometric systems is studied and determined in terms of generators and relations. An irreducible component of the semisimple part of their symmetry Lie algebras is proved to be either of A-type or of C-type. This result generalizes Hrabowski’s theorem [Hr]. © 1996 by the University of Notre Dame. All rights reserved.
  • Contiguity relations for the Lauricella functions
    M. Saito
    Funkcialaj Ekvacioj 38 37 - 58 1995年 [査読無し][通常論文]
  • Normality of affine toric varieties associated with Hermitian symmetric spaces
    M. Saito
    Journal of the Mathematical Society of Japan 46 669 - 724 1994年 [査読無し][通常論文]
  • Restrictions of A-hypergeometric systems and connection formulas of the hypergeometric function of prism type
    M. Saito, N. Takayama
    International journal of Mathematics 5 537 - 560 1994年 [査読無し][通常論文]
  • M SAITO
    TOHOKU MATHEMATICAL JOURNAL 44 4 523 - 534 1992年12月 [査読無し][通常論文]
     
    We treat the problem of shifting parameters of the generalized hypergeometric systems defined by Gelfand when their associated toric varieties are normal. In this context we define and determine the Bernstein-Sato polynomials for the natural morphisms of shifting parameters. We also give some examples.
  • M SAITO
    TOHOKU MATHEMATICAL JOURNAL 43 2 213 - 234 1991年06月 [査読無し][通常論文]

書籍

  • グレブナー基底の現在
    日比孝之他 (担当:共著)
    数学書房 2006年
  • D-modules and microlocal calculus
    M. Kashiwara (担当:単訳)
    American Mathematical Society 2003年
  • Groebner deformations of hypergeometric differential equations
    M. Saito, B. Sturmfels, N. Takayama (担当:共著)
    Springer-Verlag 2000年

作品等

  • 表現論,微分方程式系とその周辺
    2007年
  • Representation Theory, Systems of Differential Equations and their Related Topics
    2007年
  • 「2003年度表現論シンポジウム」
    2003年
  • 群の表現論と等質空間上の解析学
    1995年

共同研究・競争的資金等の研究課題

  • Systems of hypergeometric equations and their related D-modules

教育活動情報

主要な担当授業

  • 線形代数学Ⅰ
    開講年度 : 2018年
    課程区分 : 学士課程
    開講学部 : 全学教育
    キーワード : 行列, 連立1次方程式, 基本変形, 階数, 行列式, 逆行列
  • 線形代数学Ⅱ
    開講年度 : 2018年
    課程区分 : 学士課程
    開講学部 : 全学教育
    キーワード : ベクトル空間, 線形写像, 線形独立, 基底, 固有値, 固有ベクトル, 対角化
  • 代数学演習
    開講年度 : 2018年
    課程区分 : 学士課程
    開講学部 : 理学部
    キーワード : 環と加群,一意分解整域,単項イデアル整域, 単項イデアル整域上の有限生成加群,単因子
  • 統計学
    開講年度 : 2018年
    課程区分 : 学士課程
    開講学部 : 理学部
    キーワード : 確率変数,確率分布,仮説検定,統計的推論,統計ソフトウェア,生物統計学; 分割表,MCMC法,マルコフ基底,トーリックイデアル,グレブナー基底; 解析的集合,特異点,特異点解消,学習理論,カルバック・ライブラー情報量

大学運営

学内役職歴

  • 2015年4月1日 - 2017年3月31日 大学院理学研究院副研究院長
  • 2017年4月1日 - 2019年3月31日 大学院理学研究院副研究院長
  • 2019年4月1日 - 2021年3月31日 教育研究評議会評議員
  • 2019年4月1日 - 2021年3月31日 大学院理学研究院副研究院長

委員歴

  • 2013年03月 - 2015年02月   日本数学会   全国区代議員


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