Researcher Database

Mutsumi Saito
Faculty of Science Mathematics Mathematics
Professor

Researcher Profile and Settings

Affiliation

  • Faculty of Science Mathematics Mathematics

Job Title

  • Professor

Degree

  • Ph.D.(Penn. State Univ.)

J-Global ID

Research Interests

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

Research Areas

  • Natural sciences / Algebra

Association Memberships

  • 日本数学会   Mathematical Society of Japan   

Research Activities

Published Papers

  • Confluent hypergeometric systems associated with principal nilpotent p-tuples
    Mutsumi SAITO, Hiroyasu TAKEDA
    International Journal of Mathematics 29 (12) 2018/10 [Refereed][Not invited]
  • Projective linear monoids and hinges
    齋藤 睦
    http://arxiv.org/abs/1711.01397 2017/11 [Not refereed][Not invited]
  • Mutsumi Saito
    JOURNAL OF LIE THEORY 27 (1) 51 - 84 0949-5932 2017 [Refereed][Not invited]
     
    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 0022-4049 2013/01 [Refereed][Not invited]
     
    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 0021-8693 2012/02 [Not refereed][Not invited]
     
    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 0010-437X 2011/03 [Not refereed][Not invited]
     
    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 0092-7872 2010 [Not refereed][Not invited]
     
    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 0092-7872 2010 [Not refereed][Not invited]
     
    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 0030-6126 2009/06 [Not refereed][Not invited]
     
    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 0040-8735 2007/03 [Not refereed][Not invited]
     
    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 0021-8693 2004/08 [Not refereed][Not invited]
     
    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 0012-7094 2002/10 [Not refereed][Not invited]
     
    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 0010-437X 2001/09 [Not refereed][Not invited]
     
    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 [Not refereed][Not invited]
  • M Saito, B Sturmfels, N Takayama
    COMPOSITIO MATHEMATICA 115 (2) 185 - 204 0010-437X 1999/01 [Not refereed][Not invited]
     
    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 0386-2194 1998/09 [Not refereed][Not invited]
  • Mutsumi Saito
    Hokkaido Mathematical Journal 25 (3) 591 - 619 0385-4035 1996 [Not refereed][Not invited]
     
    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 [Not refereed][Not invited]
  • Normality of affine toric varieties associated with Hermitian symmetric spaces
    M. Saito
    Journal of the Mathematical Society of Japan 46 669 - 724 1994 [Not refereed][Not invited]
  • 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 [Not refereed][Not invited]
  • M SAITO
    TOHOKU MATHEMATICAL JOURNAL 44 (4) 523 - 534 0040-8735 1992/12 [Not refereed][Not invited]
     
    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 0040-8735 1991/06 [Not refereed][Not invited]

Books etc

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

Works

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

Research Grants & Projects

  • Systems of hypergeometric equations and their related D-modules

Educational Activities

Teaching Experience

  • Linear Algebra I
    開講年度 : 2018
    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
  • Linear Algebra II
    開講年度 : 2018
    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
  • Exercises on Algebra A
    開講年度 : 2018
    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
  • Statistics
    開講年度 : 2018
    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

Campus Position History

  • 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

Position History

  • 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


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