Researcher Profile and Settings

Master

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

researchmap

Profile and Settings

Degree

• Ph.D.（Penn. State Univ.）

Profile and Settings

• Name (Japanese)

  Saito

• Name (Kana)

  Mutsumi

• Name

  200901045266929228

Achievement

Research Interests

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

Research Areas

• Natural sciences / Algebra

Published Papers

• Logarithmic A-hypergeometric series

  Mutsumi Saito

  International Journal of Mathematics 31 (13) 2050110 - 2050110 0129-167X 2020/12 [Refereed]
   

  The method of Frobenius is a standard technique to construct series solutions of an ordinary linear differential equation around a regular singular point. In the classical case, when the roots of the indicial polynomial are separated by an integer, logarithmic solutions can be constructed by means of perturbation of a root. The method for a regular [Formula: see text]-hypergeometric system is a theme of the book by Saito, Sturmfels and Takayama. Whereas they perturbed a parameter vector to obtain logarithmic [Formula: see text]-hypergeometric series solutions, we adopt a different perturbation in this paper.
• 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]
  
• Limits of Jordan Lie Subalgebras

  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.
• First syzygies of irreducible A-hypergeometric quotients

  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.
• The freeness and minimal free resolutions of modules of differential operators of a generic hyperplane arrangement

  Norihiro Nakashima, Go Okuyama, Mutsumi Saito

  JOURNAL OF ALGEBRA 351 (1) 294 - 318 0021-8693 2012/02 [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.
• Irreducible quotients of A-hypergeometric systems

  Mutsumi Saito

  COMPOSITIO MATHEMATICA 147 (2) 613 - 632 0010-437X 2011/03 [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.
• THE SPECTRUM OF THE GRADED RING OF DIFFERENTIAL OPERATORS OF A SCORED SEMIGROUP ALGEBRA

  Mutsumi Saito

  COMMUNICATIONS IN ALGEBRA 38 (3) 829 - 847 0092-7872 2010 [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.
• CRITICAL MODULES OF THE RING OF DIFFERENTIAL OPERATORS OF AN AFFINE SEMIGROUP ALGEBRA

  Mutsumi Saito

  COMMUNICATIONS IN ALGEBRA 38 (2) 618 - 631 0092-7872 2010 [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.
• NOETHERIAN PROPERTIES OF RINGS OF DIFFERENTIAL OPERATORS OF AFFINE SEMIGROUP ALGEBRAS

  Mutsumi Saito, Ken Takahashi

  OSAKA JOURNAL OF MATHEMATICS 46 (2) 529 - 556 0030-6126 2009/06 [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.
• Primitive ideals of the ring of differential operators on an affine toric variety

  Mutsumi Saito

  TOHOKU MATHEMATICAL JOURNAL 59 (1) 119 - 144 0040-8735 2007/03 [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.
• Finite generation of rings of differential operators of semigroup algebras

  M Saito, WN Traves

  JOURNAL OF ALGEBRA 278 (1) 76 - 103 0021-8693 2004/08 [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.
• Logarithm-free A-hypergeometric series

  M Saito

  DUKE MATHEMATICAL JOURNAL 115 (1) 53 - 73 0012-7094 2002/10 [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.
• Isomorphism classes of A-hypergeometric systems

  M Saito

  COMPOSITIO MATHEMATICA 128 (3) 323 - 338 0010-437X 2001/09 [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 [Refereed][Not invited]
  
• Hypergeometric polynomials and integer programming

  M Saito, B Sturmfels, N Takayama

  COMPOSITIO MATHEMATICA 115 (2) 185 - 204 0010-437X 1999/01 [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.
• Grobner deformations of regular holonomic systems

  M Saito, B Sturmfels, N Takayama

  PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 74 (7) 111 - 113 0386-2194 1998/09 [Refereed][Not invited]
  
• Symmetry algebras of normal A-hypergeometric systems

  Mutsumi Saito

  Hokkaido Mathematical Journal 25 (3) 591 - 619 0385-4035 1996 [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 [Refereed][Not invited]
  
• NORMALITY OF AFFINE TORIC VARIETIES ASSOCIATED WITH HERMITIAN SYMMETRICAL SPACES

  M SAITO

  JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN 46 (4) 699 - 724 0025-5645 1994/10 [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 [Refereed][Not invited]
  
• PARAMETER SHIFT IN NORMAL GENERALIZED HYPERGEOMETRIC SYSTEMS

  M SAITO

  TOHOKU MATHEMATICAL JOURNAL 44 (4) 523 - 534 0040-8735 1992/12 [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.
• A LOCALIZATION THEOREM FOR D-MODULES

  M SAITO

  TOHOKU MATHEMATICAL JOURNAL 43 (2) 213 - 234 0040-8735 1991/06 [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

Association Memberships

• 日本数学会   Mathematical Society of Japan   

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 Projects

• レゾナントパラメータを持つA-超幾何微分方程式系の代数的・組み合わせ論的研究

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2022/04 -2026/03 

  Author : 齋藤 睦
• 線型群の作用のコンパクト化とその応用

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2018/04 -2023/03 

  Author : 齋藤 睦

   

  本研究の目的は，線型群の表現のコンパクト化を考え，その構造を明らかにすることにより線型作用の極限を統一的に扱うことを目指すことである。数学の多くの対象においてその線型変形の極限を考察することが良くあるので，この研究は多方面での応用が期待できる。本研究期間においては，応用面では主に超幾何微分方程式系の変形に適用することを目指している。
  線型群の表現のコンパクト化そのものについては，顕著な前進はなかったが，超幾何微分方程式系の理論については，前年度に引き続きフロベニウスの方法による解の構成について進展があった。確定特異点型の常微分方程式の古典的理論におけるフロベニウスの方法とは，決定方程式がジェネリックのときの級数解の指数を別変数と見て微分する方法のことで，Logを含む級数解を構成できる。前年度にA-超幾何微分方程式系でも同様の考え方で一般的な定理を証明することができた。
  引き続き，2021年度では，北海道科学大学の奥山豪氏との共同研究として，フロベニウスの方法の研究を続けた。まず，前年度に得られた結果を精密化した。さらに，或る緩い条件を満たすKer(A)の基底に付随して多項式環のイデアルを複数定義し，それらイデアルの関係を条件として，フロベニウスの方法により解空間の基底を構成できることを示した。さらに，青本-ゲルファント系やLauricellaのC型関数に付随するA-超幾何微分方程式系について，パラメータが0のときにフロベニウスの方法により具体的に基本解を構成した。以上の成果について，「Logarithmic A-hypergeometric series II」という奥山氏との共著論文にまとめ，arXivに挙げ，論文雑誌に投稿中である。
• Research on monoids consisting of limits of linear actions on a projective manifold

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

  Date (from‐to) : 2015/04 -2019/03 

  Author : SAITO Mutsumi

   

  As a compactification of the projective linear group PGL(V), I have proposed PM(V). It is a compact topological space including PGL(V) as a dense open subset, and it is a monoid acting on the projective space P(V). In addition, I have related it to the well-known compactification -- the wonderful compactification of PGL(V). In collaboration with Hiroyasu Takeda, I have made a description of the process of confluence of hypergeometric systems a la Gel’fand, as a limit under the adjoint action of a principal nilpotent p-tuple generalizing a principal nilpotent element.
• Quantization of singular nilpotent orbits of reductive Lie groups and realization of unitary representations

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

  Date (from‐to) : 2013/04 -2017/03 

  Author : Yamashita Hiroshi, SAITO Mutsumi, ABE Noriyuki

   

  In this research project, we aimed to give a good realization of irreducible unitary representations of reductive Lie groups corresponding to singular nilpotent orbits through geometric quantization of adjoint orbits. As a result, the embeddings of every singular quaternionic unitary representation of exceptional simple Lie groups of real rank 4 into real parabolically induced modules (the principal series) are specified, and we have shown the uniqueness of such embeddings. Moreover, geometric structure of singular quaternionic nilpotent orbits has been described in terms of lower rank Hermite symmetric pairs (tube type) or quaternionic symmetric pairs.
• Deformation of Lie subalgebras and systems of hypergeometric equations

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

  Date (from‐to) : 2012/04 -2015/03 

  Author : SAITO Mutsumi, YAMASHITA Hiroshi, ABE Noriyuki, OKUYAMA Go

   

  Related to confluence of the systems of hypergeometric equations a la Gelfand, deformation of Cartan subalgebras is studied. Let g be a complex simple Lie algebra of rank n. We have proved that an n-dimensional ideal of a Borel subalgebra is a limit of Jordan Lie subalgebras, which is the centralizer of a regular nilpotent element. Since a Jordan Lie subalgebra is a limit of Cartan subalgebras, an n-dimensional ideal of a Borel subalgebra is also a limit of Cartan subalgebras. Furthermore, combining with a classical result due to Kostant, we see that the g-module composed of all n-dimensional abelian subalgebras is spanned by Cartan subalgebras or Jordan Lie subalgebras.
• The synthetic study by real algebraic geometry, from sub-Riemannian geometry to tropical geometry

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

  Date (from‐to) : 2010/04 -2014/03 

  Author : GOO Ishikawa, YAMAGUCHI Keizo, IZUMIYA Shyuichi, SAITO Mutsumi, MACHIDA Yoshinori, TANABE Susumu, SAITO Sachiko, TAKAHASHI Masatomo, KITAGAWA Yumiko

   

  For singular surfaces associated to integral curves of exterior differential systems related to sub-Riemannian geometry and tropical geometry, we have realized the classification of singularities from real algebraic geometry, and we have completed their generic normal forms. From Legendre duality and control theory, we have classified singularities of tangential varieties to singularities of framed curves and surfaces, we have developed the notion of opening of mappings, and we have applied it to the classification for singularities of tangential varieties to general submanifolds. We have studied G2 sub-Riemannian geometry from non-linear control theory and the representation theory of real algebraic groups, and we have classified the related singularities. Moreover, we have developed the triality of D4 geometry and D4 singularity theory. We have completed the papers for all above themes, all of which already appeared or are under submission in international academic journals.
• Geometric invariants and model theory for singular unitary representations

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

  Date (from‐to) : 2010 -2012 

  Author : YAMASHITA Hiroshi, SAITO Mutsumi, SHIBUKAWA Youichi, ABE Noriyuki, NISHIYAMA Kyo

   

  In this research project, we investigated realization of singular irreducible unitary representations of Lie groups through geometric approach. First, we have established the Fock model version of Dvorsky-Sahi theory on an extension of the theta duality correspondence for singular unitary highest weight representations of reductive Lie groups, by decomposing tensor products of fundamental representations in terms of geometric invariants for representations in question. Second, the singular orbits in prehomogeneous vector spaces arising from quaternionic structure of exceptional simple Lie groups of real rank 4 have been described by using data on root systems, and we have proved that the singular quarternionic unitary representations, due to Gross and Wallach, can be realized by geometric quantization of the corresponding quarternionic nilpotent K-orbits.
• Study of hyergeometric systems with resonant parameters

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

  Date (from‐to) : 2009 -2011 

  Author : SAITO Mutsumi, JINZENJI Masao, OKUYAMA Go

   

  We have proved that an A-hypergeometric system is irreducible if and only if its parameter vector is nonresonant, using the theory of the ring of differential operators on an affine toric variety. In the course of the proof, we have determined the irreducible quotients of an A-hypergeometric system. We have presented 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 has been given.
• K3 surfaces and related algebraic varieties

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

  Date (from‐to) : 2008 -2011 

  Author : SHIMADA Ichiro, KIMURA Shun-ichi, ISHII Akira, TAKAHASHI Nobuyoshi, TAKAHASHI Hiroki, SUMIHIRO Hideyasu, HIRANOUCHI Toshiro, MATSUMOTO Makoto, ITO Hiroyuki, MUTSUMI Saito, OKA Mutsuo, KONDO Shigeyuki, MATSUMOTO Keiji, TERAO Hiroaki, ISHIKAWA Goo

   

  By writing various computer programs for the computational research of lattices and applying them to lattices of algebraic cycles on K3 surfaces(or related algebraic varieties), we obtained many geometric consequences. In particular, we classified the Zariski pairs of simple plane curves of degree 6 by introducing a notion of Z-splitting curves, and described their adjacency relations. We also presented an algorithm to determine the primitivity of a lattice of algebraic curves in the lattice of topological cycles for a given complex algebraic surface.
• Modern study on homogeneous spaces and unitary representations via duality

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

  Date (from‐to) : 2006 -2009 

  Author : YAMASHITA Hiroshi, YOSHIDA Tomoyuki, YAMAGUCHI Keizo, SAITO Mutsumi, SHIBUKAWA Youichi, TACHIZAWA Kazuya, KAWAZOE Takeshi, MATUMOTO Hisayosi, TANIGUCHI Kenji, WACHI Akihito, NISHIYAMA Kyo, MATSUKI Toshihiko, SEKIGUCHI Jiro, ISHI Hideyuki, SHIMENO Nobukazu, MORITA Hideaki, HIRAI Takeshi

   

  We have aimed at new development of representation theory of semisimple Lie groups and harmonic analysis on homogeneous spaces, by focusing our attention to various duality concerning representations and group orbits. Among other things, the Howe duality correspondence has been clearly understood in connection with generalized Whittaker models and isotropy representations for unitary highest weight modules. We have studied geometric invariants for discrete series and degenerate principal series representations. Significant achievements of this research projectinclude complex analytic continuation of the Matsuki duality on flag variety, classification of homomorphisms between generalized Verma modules, and numbers of contributions by other related researches.
• The Study of theoretical aspects and practical aspect on Grobner Bases

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

  Date (from‐to) : 2006 -2009 

  Author : HIBI Takayuki, SAITO Mutsumi, TAKEMURA Akimichi, YOKOYAMA Kazuhiro, OHSUGI Hidefumi, OAKU Toshinori, MATSUI Yasuko, SAITO Kyoji, TAKAYAMA Nobuki, AOKI Satoshi

   

  With taking the persistence and the internationalization into consideration, in collaboration with many researchers in various field including computational commutative algebra, computational algebraic analysis, computational algebraic statistics as well as algebraic algorithm, this research project strongly developed the study on the theoretical effectivity and practical effectivity of Grobner bases and succeeded in establishing the foundations of the modern theory of Grobner bases. At present, in order for our research group to be one of the strongest international footholds on the theoretical and practical research of Grobner bases, the CREST research project "Harmony of Grobner bases and the modern industry society," whose team leader is Takayuki Hibi, which is supported by JST (Japan Science and Technology Agency) follows this research project.
• 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

  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。
• Explicit Study of Algebraic Varieties

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

  Date (from‐to) : 2006 -2007 

  Author : SHIMADA Ichiro, OKA Mutsuo, ISHIKAAWA Goo, KONNO Kazuhiro, SAITO Mutsuni, TOKUNAGA Hiroo

   

  (1) By a joint work with Due Tai Pho, we proved the unirationality of supersingular K3 surfaces in characteristic 5 with Artin invariant 〓 3. (2) We classified all possible configurations of rational double points on complex normal algebraic K3 surfaces, and discussed the same classification for supersingular K3 surfaces in sufficiently large characteristics. (3) We investigated the set of isomorphism classes of transcendental lattices of complex algebraic varieties obained from a single algebraic variety defined over a number field by various embeddings of the base field into the complex number field, and produced many examples of non-homeomorphic complex algebraic varieties that are conjugate under the automorphism of the complex number field. In particular, we investigate the case of singular. K3 surfaces by means of the class field theory of imaginary quadratic fields. As an application, we obtained a lower bound of the degree of the base field of singular K3 surfaces. (4) Suppose that we obtain a supersingular K3 surface X (P) by reduction at a finite place P of a base field of a singular K3 surface X defined over a number field. We investigated the orthogonal complement of the Neron-Severi lattice of X in that of X (P), and proved an analogous result as the case of the transvendental lattices. (5) We proved a Lefschetz hyperplane section theorem for the topological fundamental group of the complement of the duial variety in the Grassmannian variety, and invetigate the Zariski-van Kampen type relation of that fundamental group with the barid group of a punctured Riemann surface.
• 計算可換代数と計算代数幾何についての国際研究集会の企画調査

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2006 -2006 

  Author : 日比 孝之, 齋藤 睦, 竹村 彰通, 横山 和弘, 大杉 英史, 大阿久 俊則

   

  研究代表者らは、従来から、組合せ論と計算可換代数に関する国際研究集会を継続的に開催し、当該分野の国際的な研究活動を円滑に推進させる重要な役割を担ってきた。研究代表者らは、類似の趣旨の国際研究集会「計算可換代数と計算代数幾何」を2008年6月、札幌に於いて開催する準備を進めている。目下の所、当該国際研究集会の根幹となる研究領域は、D加群とアルゴリズム、ジェネリックイニシャルイデアルの理論と実践、シテジーとヒルベルト函数、応用数学・情報数学における計算代数幾何的な展開、グレブナー基底の効率的計算、とするのが有力である。当該企画調査では、それらの研究領域の妥当性を慎重に審議した。具体的には、研究代表者と研究分担者が5回の連絡会議を開催し、会議を担当する分担者が研究領域の調査結果を報告し、海外招待講演者の候補者などを議論した。以下、連絡会議の開催時期、開催場所、審議する研究領域、会議を担当した分担者氏名を記載する。 ●第1回(2006年6月、京都)「D加群とアルゴリズム」 齋藤睦 大阿久俊則 高山信毅 ●第2回(2006年7月、東京)「ジェネリックイニシャルイデアル」 横山和弘 大杉英史 大阿久俊則 ●第3回(2006年9月、札幌)「シチジーとヒルベルト函数」 齋藤睦 大杉英史 高山信毅 (Juergen Herzog) ●第4回(2006年11月、京都)「情報数学における計算代数的な展開」 竹村彰通 阪田省二郎 松井泰子 青木敏 ●第5回(2006年1月、京都)「グレブナー基底の効率的計算」 横山和弘 松井泰子 高山信毅
• Computational Analysis of Hypergeometric Differential Equations

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

  Date (from‐to) : 2003 -2006 

  Author : TAKAYAMA Nobuki, NORO Masayuki, FUKUYAMA Katsushi, MASUDA Tetsu, OAKU Toshinori, SAITO Mutsumi

   

  We have obtained the following results 1. We constructed vol(A)-linearly independent convergent series solutions for A-hypergeometric differential-difference equations. 2. We prove that any local Grober fan is a polyhedral fan. As an application of this fact, we give an algorithm of computing local BS polynomials, that of computing local tropical varieties, and discuss a relation of slopes and local Grobner fan. 3. We gave a tangent cone algorithm to study D-modules locally. We have more results. As to these, refer to Japanese version of this research report and papers.
• グレブナー基底の理論的有効性と実践的有効性に関する共同研究の企画調査

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2005 -2005 

  Author : 日比 孝之, 齋藤 睦, 竹村 彰通, 大阿久 俊則, 横山 和弘, 大杉 英史

   

  研究代表者と研究分担者が研究連絡会議を開催し、グレブナー基底の理論的有効性と実践的有効性に関する共同研究の企画調査を実施する旨の研究計画に従い、5月(東京大学)、8月(立教大学)、9月(東京大学)、11月(大阪大学)と研究連絡会議を開催し、それぞれ、竹村彰通が計算代数統計の、大阿久俊則が計算代数解析の、大杉英史が計算可換代数と計算代数幾何の、横山和弘と松井泰子がグレブナー基底の計算の効率化の研究領域に関する調査結果を報告した。なお、8月の研究連絡会議は、日本学術振興会の国際研究集会(立教大学、2005年8月22日〜26日)Theoretical Effectivity and Practical Effectivity of Grobner Basesに付随して開催し、拡大研究連絡会議と称し、Juergen Herzog(ドイツ)、Henry P.Wynn(イギリス)、Uli Walther(米国)を招聘した。海外招聘者は、グレブナー基底の理論的有効性と実践的有効性に関し、それぞれの専門領域における欧米諸国の研究の現状を報告し、組織的な国際共同研究に発展させる際の具体的な指針についての貴重な意見を述べた。 企画調査の結果、今後、我が国において、少なくとも5年以上の長期に亘って継続されるグレブナー基底の理論と実践に関する共同研究を推進するための総括的な方針を擁立することができ、永続的な国際共同研究に発展することを視野に入れ、実際の研究活動を展開する準備が整った。企画調査の結果を踏まえ、その実際の研究活動とし、京都大学数理解析研究所、平成18年度、プロジェクト研究「グレブナー基底の理論的有効性と実践的有効性」が始まる。
• Study of universal Grobner bases of zero-dimensional lattice ideals

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

  Date (from‐to) : 2003 -2005 

  Author : HIBI Takayuki, SAITO Mutsumi, OHSUGI Hidefumi, MATSUI Yasuko, TAKAYAMA Yukihide, TERAI Naoki

   

  The lattice ideal of dimension zero appears in the research of both pure mathematics and applied mathematics. The original purpose of the present research project was to establish the algebraic theory of universal Groebner bases of lattice ideal of dimension zero and to study of its theoretical effectivity to commutative algebra and algebraic geometry as well as its practical effectivity to integer programming, coding theory together with algebraic statistics. First, we investigated the universal Groebner basis of the lattice ideal of dimension zero arising from the toric ideal of a finite graph and succeeded in describing its structure in terms of the finite graph. Second, in the study of a problem on integer programming arising from a finite graph for which Gomory's relaxation can be applied, we developed the technique to decide the estimation of the computational complexity of finding an optimal solution by using combinatorics on finite graphs. Third, we achieved the study of finding an explicit expression of the corner polyhedron of a lattice ideal of dimension zero in terms of the Minkowski sum of a bounded convex polytope and a convex cone, and obtained some results on the combinatorics of the polyhedral structure of the corner polyhedron. Finally, we developed the algebraic study on the Markov basis of the contingency table in algebraic statistics and presented a statistic model arising from a complete multipartite graph. These research results will contribute to the development of the algebraic study of integer programming. In addition, we organized two international meetings related with computational commutative algebra and Groebner bases.
• Isotropy representations associated with Harish-Chandra modules and nilpotent orbit theory

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

  Date (from‐to) : 2002 -2005 

  Author : YAMASHITA Hiroshi, YOSHIDA Tomoyuki, SAITO Mutsumi, NISHIYAMA Kyo, OCHIAI Hiroyuki, SEKIGUCHI Jiro

   

  The principal purpose of this project is to develop nilpotent orbit theory for Harish-Chandra modules corresponding to irreducible admissible representations of real semisimple Lie groups, in order to get deep understanding on generalized Whittaker models in relation to various nilpotent invariants of representations. We focused our attention to the isotropy representations which give the multiplicities in the associated cycles of Harish-Chandra modules. (1)The isotropy representations have been described explicitly for all singular unitary highest weight modules of simple Hermitian Lie algebras by using the projection to the PRV-components. As a result we have proved the irreducibility of such isotropy representations. A new proof of the Howe duality correspondence for reductive dual pairs is obtained via isotropy representations. For EVII, we have shown that the isotropy representations for certain unitary highest weight modules of non-scalar type give the Dvorski-Sahi correspondence. This allows relating the isotropy representations and harmonic analysis on certain compact symmetric spaces of real rank one. (2)General machinery has been established to describe the isotropy representations of Harish-Chandra modules with irreducible associated varieties, by using the principal symbols of differential operators of gradient-type. Applying this machinery, we have revealed an explicit relationship between the isotropy representations for discrete series and the generic fiber of the moment map for the conormal bundle of closed orbits on the flag variety. (3)To identify the generic fiber of the moment map, we have studied the Richardson orbits associated with symmetric pairs. It has been an open problem whether the parabolic subgroups for the Richardson orbits act transitively on the set of Richardson elements in the symmetric part of its nilradical. In this project, we have got a progress to this problem, by giving nice sufficient conditions for the transitivity, and also a counterexample for the Lie groups of type A
• 無限次元リー群およびリー代数に対する表現論の新たな展開

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2002 -2004 

  Author : 山下 博, 山口 佳三, 齋藤 睦, 澁川 陽一, 和地 輝仁

   

  本課題研究は,無限次元半単純リー代数(群)や量子群に対して,有限次元群の場合の既約許容表現,すなわちハリシュ-チャンドラ加群に相当する新しい表現の族を構成・分類するために,「代数的量子化」の理論が無限次元の場合にいかに展開できるか,その可能性を探ること主目標としている.今年度は,昨年度に行った試行的研究を推し進め,代数的ディラク作用素及びディラクコホモロジーを用いて,共役類の量子化と許容表現の構成を検討した.また,リー代数の作用に関する不変式論,トーリック多様体上の微分作用素環,量子群の研究を併せて行った.その研究経過と得られた知見について,以下に報告する. 研究代表者山下は,A型のリー代数に対して,簡約あるいはべき零な部分リー代数から定まるディラク作用素の構成とコホモロジー空間の構造を検討した.その過程で,表現論国際会議(平成16年8月開催,於新疆大学)に参加し,有限次元の場合の専門家であるJing-Song Huang(香港科技大)およびPavle Pandzic(Zagreb大)と研究打合せを重点的に行った.無限次元リー代数に理論が拡張できる部分と障害となる部分が明らかになり,今後研究を発展させるために重要な手がかりが得られたと考えている.また,ディラク作用素と基盤研究(B)(課題番号14340001)で実施中の離散系列に対する等方表現との間の関係を調べた.研究分担者和地は,対称対に関する不変微分作用素を定める普遍包絡代数の元の研究を行い,デュアルペアと関わる明示的公式を得た. 無限次元リー代数の表現の研究に資するため,研究分担者齋藤は,アフィン半群環上の微分作用素環やA-超幾何系において基本的な加群の圏Oを扱い,各々の圏Oにおいてヴァーマ的対象や単純対象と,それらの間の基本的な関手について考察した.研究分担者澁川は,力学的ヤン・バクスター方程式の集合論的解を構成した.
• グレブナー基底の理論的有効性と実践的有効性についての国際研究集会の企画調査

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2003 -2003 

  Author : 日比 孝之, 松井 泰子, 大杉 英史, 齊藤 睦, 寺井 直樹, 高山 幸秀

   

  当該企画調査では,Oberwoltach型の中規模準備会議を開催し,当該分野の研究の動向を詳細に分析し,研究目的に列挙した研究領域(トーリックイデアルとグレブナー基底,整数計画とGomory relaxation,0次元ラティスイデアルの普遍グレブナー基底,単項式イデアルの極小自由分解,幾何学的なBuchbergerアルゴリズムの高速化)の妥当性を慎重に審議した。その準備会議の概要を列挙する。[1]可換代数におけるアルゴリズム的手法(責任者:寺井直樹/於:大阪大学/平成15年7月)斉次代数の極小自由分解とベッチ数列,トーリック環の正則度と重複度などを題材とし,可換代数におけるアルゴリズム的手法を議論した。[2]有限グラフと0次元ラティスイデアル(責任者:大杉英史/於:立教大学/平成15年11月)有限グラフの隣接行列から生起する0次元ラティスイデアルを可換代数と組合せ論の両面から具象的に探求し,未解決問題を集約した。[3]グレブナー基底と応用数学(責任者:大杉英史/於:立教大学/平成16年1月)整数計画における代数的手法の有効性,Gomory relaxationと算術次数,符号理論と統計数学におけるトーリックイデアルとグレブナー基底の有効性について研究した。[4]可換代数と代数幾何(責任者:日比隆之/於:大阪大学/平成16年3月)いわゆるaffine algebraic geometryとその周辺領域,多項式環の組合せ論についての国際会議である。海外からの参加者はJurgen Herzogら7名であった。
• Moduli Spaces and Special Functions

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

  Date (from‐to) : 2001 -2003 

  Author : MATSUMOTO Keiji, SHIMADA Ichiro, SAITO Mutsumi, MAEDA Yoshitaka

   

  The head of investigator MATSUMOTO Keiji constructed period maps and automorphic forms derived from the inverses of period maps for certain families of algebraic varieties by using Prym varieties of algebraic curves. In fact, it was shown that the period map for the family of smooth cubic surfaces could be expressed in terms of periods of the Prym varieties for curves of genus 10. Automorphic forms on the 4-dimensional complex ball giving the inverse of this period map were expressed by theta constants associated to the Prym varieties. For the family of the 4-fold coverings of the complex projective line branching at eight points, the period map from this family to the 5-dimensional complex ball was constructed by using the Prym varieties of these curves. Automorphic forms on the 5-dimensional complex ball giving the inverse of this period map were expressed by theta constants associated to the Prym varieties. SAITO Mutsumi showed that the ring of differential operators on affine tone varieties and the algebra of symmetries of the system of A-hypergeometric differential equations were anti-isomorphic, and classified systems of A-hypergeometric differential equations combinatonally under these symmetries. He studied the condition that the graded ring gr(D(R_A)) was finitely generated, and gave the composition factors of the ring R_A of functions on any tone variety as a D(R_A)-module. SHIMADA Ichiro showed that if the singularity of each singular fiber was not bad for an algebraic fiber space, the boundary homomorphism from the second homotopy group of the base space to the fundamental group of any general fiber could be constructed. He showed that the fundamental group of the complement of a resultant hypersurface was commutative. He also showed that any supersingular K3 surface could be expressed as a branched double cover of the projective plane.
• 凸多面体を巡る組合せ数学の代数的諸相についての国際研究集会の企画調査

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2002 -2002 

  Author : 日比 孝之, 枡田 幹也, 松井 泰子, 齋藤 睦, 大杉 英史, 寺井 直樹

   

  現在研究代表者らは国際研究集会「凸多面体を巡る組合せ数学の代数的諸相」を平成16年7月に札幌で開催する準備を進めているが,根幹となる研究領域を(1)グレブナー基底と組合せ数学(2)凸多面体の三角形分割と整数計画(3)外積代数とalgebraic shifting(4)単項式イデアルの極小自由分解(5)斉次整域の正則度と重複度,とする原案が有力である.当該企画調査では,当該分野の昨今の研究動向などに関する周到な調査を遂行し,上記項目を研究集会の研究領域の根幹とすることの妥当性を吟味するため,Oberwolfach型の中規模国内準備会議を2回開催した.すなわち,「グレブナー基底の理論的有効性と実践的有効性」(責任者:大杉英史/於:京都大学/平成14年7月)と「ジェネリックイニシャルイデアルの研究」(責任者:寺井直樹/於:大阪大学/平成14年12月)である.前者においては凸多面体の三角形分割と整数計画問題,符号理論と暗号理論などにおけるグレブナー基底の果たす役割について多角的に研究した.後者においては多項式環と外積代数のジェネリックイニシャルイデアルの相互関係を可換代数と組合せ論の両面から具象的に探究した.その他,平成14年6月にイタリアで開催された研究集会「可換代数の昨今の潮流」に研究代表者と研究分担者の一部が参加し,当該分野の研究の進展状況を把握した.当該企画調査の結論として,上記の根幹となる研究領域はすべて妥当であると判断され,個々の研究領域において当該国際研究集会に相応しい話題を選別する作業を推進した.
• Classification problem of hypergeometric differential systems

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

  Date (from‐to) : 2000 -2001 

  Author : SAITO Mutsumi, SHIBUKAWA Youichi, MATSUMOTO Keiji, YAMASHITA Hiroshi, WACHI Akihito, YAMADA Hiro-fumi

   

  With support of many examples with a computer, and by communication with world-wide experts in several fields, we obtained the following results. Mutsumi Saito generalized the classification theorem of A-hypergeometric systems to the cases when A is inhomogeneous and/or when we work in the analytic category. He also gave a dimension formula for the log-free series solutions when A is homogeneous, and a rank formula and the proof of the equivalence of Cohen-Macaulayness with the condition that the ranks are the same at all parameters, when A is homogeneous, and the convex hull of A is a simplex. Hiroshi Yamashita obtained some results useful to know when an isotropy representation is irreducible. Furthermore he systematically constructed nonzero quotient representations of isotropy representations attached to discrete series. Keiji Matsumoto clarified a pairing between twisted cohomology groups associated with generalized Airy functions. Writting a base of twisted cohomology groups by Young diagrams, he showed that for the base, the pairing can be explicitly written by skew-Schur polynomials. Youichi Shibukawa solved the classification problem for R operators. For the simplest affine Lie algebra A_1^<(1)> , using two of its realizations, Hiro-Fumi Yamada discovered the weight vectors are written by a modular version of Schur functions and Schur's Q-functions respectively. Akihito Wachi has studied the structure of generalized Verma modules, in particular, their irreducibility, emphasizing their relations with invariant functions.
• Infinite dimensional quaternionic representations and nilpotent orbits

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

  Date (from‐to) : 2000 -2001 

  Author : YAMASHITA Hiroshi, NISHIYAMA Kyo, SHIBUKAWA Youichi, SAITO Mutsumi, WACHI Akihito, OHTA Takuya

   

  The associated variety of an irreducible Harish-Chandra module gives a fundamental nilpotent invariant for the corresponding irreducible admissible representation of a real reductive group. Moreover, the multiplicity in the Harish-Chandra module of an irreducible component of the associated variety can be regarded as the dimension of a certain finite-dimensional representation, called the isotropy representation. The head investigator, Yamashita, has already shown that, in many cases, the isotropy representation can be described, in principle, by means of the principal symbol of a differential operator of gradient-type whose kernel realizes the dual Harish-Chandra module. In this research project, we have begun a systematic study of the isotropy representations attached to Harish-Chandra modules with irreducible associated varieties, including quaternionic representations, discrete series and unitary highest weight modules. The results are summarized as follows: We developed a general theory for the isotropy representations, starting from the Vogan theory on associated cycles. In particular, a criterion for the irreducibility of an isotropy representation is presented. Also, we looked at when the isotropy representation can be described in terms of a differential operator of gradient-type. As for the discrete series, a nonzero quotient of the isotropy representation has been constructed in a unified manner. It seems that this quotient representation is large enough in the whole isotropy module. We have shown that this is the case if the theta-stable parabolic subgroup canonically determined from the discrete series in question admits a Richardson nilpotent orbit with respect to the complexified symmetric pair. The isotropy representation is explicitly described for every singular unitary highest weight module of Hermitian Lie algebras BI, DI and EVII. This allows us to deduce that the isotropy modules are irreducible for all singular unitary highest weight modules of arbitrary simple Hermitian Lie algebra. Principal contribution by the investigators : Saito developed his research on A-hypergeometric system, which is closely related to a realization of unitary highest weight modules. He has established a formula for the rank of a homogeneous A-hypergeometric system. Wachi constructed an analogue of the Capelli identity for generalized Verma modules of scalar type. Nishiyama and Ohta gave a correspondence of nilpotent orbits associated to a symmetric pair, by menas of the moment map with respect to a reductive dual pair.
• Combinatorial Representation Theory of Affine Lie Alqebras and Symmetric Groups

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

  Date (from‐to) : 1999 -2000 

  Author : YAMADA Hiro-fumi

   

  My first attempt was to describe the weight basis of the basic representations of several typical affine Lie algebras. In particular, for the simplest affine Lie algebra A^<(1)>_1, I considered two realizations of the basic representation and found that the modular version of the Schur functions and Schur's Q-functions occur as weight basis, respectively. Analysing these two realizations, I found an interesting phenomenon for the elementary divisors of the spin decomposition matrices for the symmetric group. Namely the elemntary divisors of the spin decomposition matrices for prime 2 are all powers of 2. Though this fact actually can be proved by a general theory of modular representations, I could give a simple proof of this by utilizing representations of the affine Lie algebra A^<(1)>_1. Studying the zonal polynomials, which are a specialization of the Jack polynomials, I found an interesting fact in the character tables of the symmetric group. Later I recognizes that this fact had been found more than 50 years ago by Littlewood, whose proof is a bit complicated. I gave a simple proof of this fact as well as its spin version with Hiroshi Mizukawa, a graduate student. The main tools for the proof are again Schur functions and Schur's Q-functions. In the joint work with Takeshi Ikeda I could obtain all the homogeneous polynomial solutions for the nonlinear Schrodinger hierarchy. The schur functions indexed by the rectangular Young diagrams play an essential role in this theory.
• Algebro-analytic and/or representation-theoretic study of hypergeometric differential systems

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

  Date (from‐to) : 1998 -1999 

  Author : SAITO Mutsumi, SHIBUKAWA Youichi, YAMASHITA Hiroshi, YAMADA Hiro-Fumi

   

  With support of many examples by a computer, and by communication with world-wide experts in several fields, we obtained the following results. Mutsumi Saito has studied A-hypergeometric systems. He, in collaboration with Bernd Sturmfels and Nobuki Takayama, found and studied an unexpected relationship between A-hypergeometric systems and integer programmings, and showed the invariance of the rank of a regular holonomic system under Grobner deformations, and obtained three sufficient conditions for the rank of an A-hypergeometric system to equal the volume of the convex hull spanned by A. He classified parameters according to D-isomorphism classes of their corresponding A-hypergeometric systems. Hiro-Fumi Yamada has studied the relationship between Q-functions and affine Lie algebras. He showed a Q-function expressed as a polynomial of power sum symmetric functions is a weight vector for the basic representation of a certain affine Lie algebra realized on the polynomial ring, and illustrated the corresponding weight by Young diagrams. He also found an unexpected relation of Schur's S-functions and Q-functions. Hiroshi Yamashita has studied Harish-Chandra modules. He specified the embedding of Borel-de Siebenthal discrete series into the principal series representations. He also described the associated cycles of some important representations, such as discrete series and unitary highest weight representations, by using the principal symbols of invariant differential operators of gradient type whose kernels realize their dual Harish-Chandra modules. Youichi Shibukawa has worked on Ruijsenaars-Schneider dynamical integrable system. Related to its Lax presentation, he, in collaboration with Nariya Kawazumi, obtained all meromorphic solutions to the Bruschi-Calogero differential equation.
• Grobner asymptotic expansion for regular holononic systems

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

  Date (from‐to) : 1998 -1999 

  Author : IKEDA Hiroshi, SAITO Mutsumi, TAKAYAMA Nobuki, TAKANO Kyoichi

   

  We establish a method to analyze asymptotic behavior of regular holonopic systems at infinity. The first order approximation is governed by the initial system. In case of GKZ hypergeometric systems, the correspondiny systems are essentially monomial ideals and hence can be analyzed by nsins combinatorial methods for them. Our research project develops to the following new directions. (1)Bayer and Sturmfels showed recently that monomial ideals can be studied through graph theory an stairs. Their method can be applied to study GKZ hypergeometric systems. (2)Our method to determine asymptotic behavior will be a foundation to study the rational solutions and the global solutions. Some hypergeometric systems are special solutions of Painleve systems. There will be an exciting interaction between studies on Painleve systems and hypergeometric systems on the rational solutions, isomorphism problem and the global solutions. (3)It is an important problem to determine the asymptotic behaviors around an irregular singular point. It, however, is still open.
• Differential operators of gradient type on symmetric spaces and representations of Lie algebras

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

  Date (from‐to) : 1997 -1999 

  Author : YAMASHITA Hiroshi, SHIBUKAWA Youichi, SAITO Mutsumi, YAMADA Hiro-fumi, NISHIYAMA Kyo, HIRAI Takeshi

   

  The purpose of this project is to study the embeddings of irreducible Harish-Chandra modules into various induced representations of a semisimple Lie group, by using the invariant differential operators of gradient type on certain homogeneous vector bundles over the Riemannian symmetric space. The kernel of such a differential operator realizes the maximal globalization of the dual Harish-Chandra module, and the determination of the embeddings in question is reduced to specifying the equivariant functions in this kernel space. First, the generalized Gelfand-Graev representations form a family of induced modules parametrized by the nilpotent orbits. Concerning the Harish-Chandra modules with highest weights for a simple Lie group of Hermitian type, the generalized Whittaker models associated with the holomorphic nilpotent orbits are specified. Namely, it is shown that each highest weight module embeds, with nonzero and finite multiplicity, into the generalized Gelfand-Graev representation attached to the unique open orbit in its associated variety. As for the unitary highest weight module, the space of the embeddings can be completely described in terms of the principal symbol of the differential operator of gradient type. Second, we consider a simple Lie group of quaternionic type. The 0th n-homology spaces, or equivalently, the embeddings into the principal series, of the Borelde Siebenthal discrete series are described, by using the Schmid differential operator of gradient type. We find in particular that the n-homology space has exactly two exponents if the real rank of the group is not one. Third, the relationship between the multiplicities in the associated cycles and the differential operators of gradient type are clarified for certain Harish-Chandra modules with irreducible associated varieties. The multiplicity can be written down by means of the principal symbol of a gradient type differential operator.
• Combinatorial aspects of representations of groups and algebras

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

  Date (from‐to) : 1997 -1998 

  Author : YAMADA Hirofumi, NAKAJIMA Tatsuhiro, TERAO Hiroaki, SGIBUKAWA Youichi, SAITO Mutsumi, YAMASHITA Hiroshi

   

  I focused on a relationship of Schur's Q-functions and affine Lie algebras. First I found that the Q-functions, expressed as polynomials of power sum symmetric functions, form a weight basis for the basic representation of certain affine Lie algebras, realized on a polynomial ring. Q-functions are parametrized by the strict partitions. Using some combinatorics of Young diagrams, I determined the weight of the given Q-function. This procedure was applied to the simplest affine lie algebra $A^{(1)}_1$ to find an identity satisfied by Schur functions and Q-functions indexed by some specific partitions. At first this identity seemed funny : However this was proved to be true by making use of decomposition matrices of the spin representations of the symmetric group. By virtue of this fact, I turned to a study of the decomposition matrices themselves. As a first result I proved that the determinant of the decomposition matrix of the spin representations is equal to a power of two when the characteristic equals two. Another feature of my research is the so called "higher Specht polynomials" for the complex reflection group G(r, p, n). The group G(r, p, n) acts on the polynomial ring of n variables. The "coinvariant ring" is the quotient by the ideal which is generated by invariants over the group. It is known that the action of G(r, p, n) on this coinvariant ring is isomorphic to the regular representation. The higher Specht polynomials appear naturally as basis vectors of each irreducible component.
• エルミート対称空間上のねじれD-加群の研究

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1996 -1996 

  Author : 齋藤 睦

   

  青本-Gelfand超幾何微分方程式系の表現論的一般化を谷崎俊之氏は、コンパクトエルミート対称空間上のあるねじれ微分作用素環D_λのある加群として提案した。 A型とC型でねじれ具合が極小軌道に対応する場合には、いわゆるA-超幾何微分方程式系になり、これらに関しては、特性サイクル、回の級数表示、積分表示などが既にGelfandをはじめとする人々によって得られている。しかし、他の場合については非常に少しのことしか知られていない。特に、非自明な解が存在するかどうかさえ知られていない。今回私は、B型とD型の一つについてねじり具合が極小軌道に対応した場合である特別な状況のとき、非自明な級数解を構成し、谷崎氏提案の微分方程式系がA型とD型以外にも非自明なものが存在することを示した。 また、北海道大学の大学院生の和地輝仁氏と共同で、A型とC型の場合に、カペリの恒等式の類似と言える恒等式を発見、証明した。これを用いてこの場合に一般Verma加群のJantzenよる既約判定法と菅修一氏によるb-関数の零点による判定法を直接結び付けた。(これは、Wallachの結果の別証である。)また現在、この恒等式を使って、谷崎氏提案の微分方程式系の研究に応用できないかどうか考察中である。 さらに今年度では、A-超幾何微分方程式系の理論と整数計画法の関連について考察した。この問題に関しては、カリフォルニア州立大学バ-クレイ校のB.Sturmfels氏、神戸大学の高山信毅氏との共著論文を投稿中である。この共著論文では、A-超幾何微分方程式系のb-函数の理論を応用して、実行可能解の母函数を求めるためのアルゴリズムを得、また、A-超幾何微分方程式系の決定多項式を計算して、最適解の満たすべき性質を求めた。
• 実半単純リー群の表現とベき零軌道のケーリ-型変換

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1996 -1996 

  Author : 山下 博, 平井 武, 本多 尚文, 山田 裕史, 齊藤 睦

   

  1.実半単純リー群Gの表現,より正確には、表現を微分して得られる展開環U(g)上のHarish-Chandra加群Hの随伴多様体ν(H)は、Riemann対称対(G,K)を複素化して得られる対(G_C, K_C)の接空間pにおけるべき零K_軌道からなる。研究代表者は、「各K_軌道O⊂ν(H)からケーリ-型変換と偏極化をとおしてH上に局所自由に作用するべき零部分環(群)n_oの存在」を示した昨年度からの研究を押しすすめ、Hが規約最高ウェイト表現の場合に、対応するべき零部分環n_oの具体的記述を与えた。この一連の研究結果をとりまとめた論文を日本数学会および数理解析研究所共同研究集会で口頭発表し、学会雑誌へ投稿した(京大行者明彦氏との共著)。 2.半単純リー群Gの極小べき零共役類に付随した極小ユニタリ表現H_mは、既約ユニタリ表現の分類問題とも深く関わる重要な表現である。(1)の成果をふまえて、G=SU(n,n)の極小表現Hmの一般化されたホイッタッカー模型を、HmをG/K上で実現するG_-不変な2階偏微分方程式系を用いて決定した(論文準備中)。さらに、極小表現のフォック模型を使って、U(n_o)-加群としてのHmの構造を明らかにした。この結果を任意の最高ウェイト加群に拡張することを目標とした研究を現在実施中である。 3.各研究分担者は、ホロノミックな不確定特異点型微分方程式系(本多)、多変数超幾何方程式(齊藤)、あるいは各種の群の表現に対するシューア・ワイルの相互律の研究(平井・山田)を各自押しすすめると同時に、これらののテーマが深く関わる上記2の研究実施の過程で、個人的な討論やセミナーをとおして本研究に常時参加した。
• Classical Problems in Group Theory

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

  Date (from‐to) : 1996 -1996 

  Author : YOSHIDA Tomoyuki, BANNAI Eiichi, TUJISITA Toru, YAMAMADA Hirofumi, NAKAMURA Iku, SAITO Mutsumi

   

  The purpose of this reserch was the study of classical problems for discrete groups and its applications. In this research, the investigators obtained the following results. These results will be arranged and published in order. 1. On the crossed Burnside ring of a finite group, (a) we discovered its relation with the quantum double of the group algebra ; (b) we proved the fundamental theorem (an embedding into the products of some group alegabras) ; (c) we obtained an idempotent formula and applied it to the classical problems. We have arranged them as a preprint (Crossed G-sets and crossed Burnside rings) and gave lectures on them in some conferences (Seattle, Yamagata, Kusatsu). 2. On a relationship between our classical problems and Topological Quantum Field Theory (TQFT), we checked that Dijkgraaf-Witten invariants are, in some cases, almost algebraic integers. For example, the invariant for a 3-torus is surely a rational integer. Furthermore, we have a weak result for cyclic gauge group case ; however, in this case, the original conjecture had to be revised. These statements will be found in the proceeding of Symposium on Algebra held in Yamagata. 3. We obtained many important results on Schur functions, especially a deep connection with affine Lie algebras. These results was expressed in a conference on combinatorics held in Mineapolis. 4. Investigators have a lot of results in some other area which related with our project : ring theory, real algebraic geometry, theory of monoidal categories (Kumamoto), a relationship between dynamical system and intuitional logic (Sapporo). 5. Using the funds for equipment, we purchased a workstation and a personal computer, which were used to run some formula manipulation programs (GAP,Mathematica).
• 一般超幾何微分方程式系の代数的性質の研究

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1994 -1994 

  Author : 齋藤 睦

   

  超幾何微分方程式系の一般化の一つであるα-超幾何微分方程式系は、非常に対称性の豊かな方程式系である。私は以前、このα-超幾何微分方程式系の隣接関係式に関連して、α-超幾何微分方程式系のb-函数-(これは佐藤幹夫氏によって導入された通常のb-函数の類似である。)を定義し、方程式系が正規と呼ばれる場合に計算した。これまでの研究により、このb-函数がα-超幾何微分方程式系の対称性を統制する非常に重要なものであることが判っていた。 今年度、北海道大学大学院生の近藤昌晴氏と共同で、合流型α-超幾何微分方程式系のb-函数を定義し、方程式系が正規の場合に計算した。その結果は、以前に行った合流型でない場合の素直な一般化となっている。 また、神戸大学の高山信毅氏との共同研究では、αが一般プリズム△_1×△_の場合に級数解の接続公式を導いた。αが一般プリズムの場合は、α-超幾何函数はロ-リチェラ函数F_Dであるが、既にF_Dの接続公式は、松本圭司氏、佐々木武氏、高山信毅氏、吉田正章氏により、ブレイド群のI-コサイクルとして実現され、また、青山和彦氏、加藤芳文氏、三町勝久氏によりF_Dのq-類似の接続公式も対称群の1-コサイクルとして表わされていた。我々の結果も同様の結果ではあるが、証明方法は新しく、より一般のα-超幾何函数の接続公式への適用が期待される。我々の証明方法の基本的なアイデアは次の通りである。まず、b-函数の理論を応用するとパラメータが特殊でない場合に、△_1×△_-超幾何微分方程式系をその特異部分空間に制限することにより、△_1×△_-超幾何函数の接続公式が△_1×△_-超幾何函数の接続公式から導かれることが判る。更に、△_1×△_のセカンダリ-ポリトープを考慮し、底空間を適当に単連結空間に分解すれば、級数解に関する上述の帰納的計算がうまくいき、最後には△_1×△_1-超幾何函数、つまり、ガウスの超幾何函数の接続公式に帰着されるという訳である。
• 可積分測地流

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1994 -1994 

  Author : 清原 一吉, 齋藤 睦, 泉屋 周一, 西森 敏之

   

  この研究の目標としたものの内、複素射影空間をモデルとするような、可積分測地流を持つ多様体のクラスとしてKahler-Liouville多様体を定義し、その性質を詳細に調べた。結果は「On a class of Kahler manifolds whose grodesic flows ane integrable」という題の論文にまとめつつある。その内容の概略は次の通りである。まず定義は実のLiouville多様体の形式的なHermite版となっている。そこには2次の(エルミートな)第一積分が複素次元個与えられているだけだが、適当な非退化性の条件の下で、同じ個数の無限小同型が自動的にでてくることがわかり、これにより測地流は可積分となっている。その内部構造を大まかに表すものとして有限半順序集合が不変量として付随している。多様体がcompactのとき、上記の無限小同型により問題の複素多様体はいわゆるトーリック多様体になることがわかる。特に代数多様体であり、又ample divisonをもつので射影的でもある。トーリック多様体としての構造を決定するfanの様子も上記の有限半順序集合を用いて簡明に記述される。compact Kahler-Liouville多様体(+適当な条件)の複素多様体としての様子はこれで完全にわかる。雑な云い方をすれば、有限半順序集合の各点にさまざまな次元の複素射影空間が付随しており、それらから半順序に従って次々にbundleを作ってできるものである。計量及び2次の第一積分たちもこの構成法(分解)に適合しており、それらの自由度は結局複素射影空間上での自由度に帰着していることがわかる。複素射影空間上の構造は部分多様体である実射影空間上のLiouville多様体の構造から決まっており、それがいつ複素化できるかも完全にわかる。そのようなものはやはり一変数関数の自由度がある。
• 単体的複体と凸多面体の離散構造の代数的諸相

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1994 -1994 

  Author : 日比 孝之, 澁川 陽一, 齋藤 睦

   

  近年,現代数学の様々な分野において離散構造の重要性が認識されてきた.古典的な組合せ論の研究対象である単体的複体,半順序集合や凸多面体に限っても,その面,鎖の個数や格子点の数え上げは可換代数や代数幾何と深い接点を持つことが判明し,更に,凸多面体の三角形分割の組合せ論は超幾何函数の理論などとの相互関係を保ちながら急激に進展している.このような現状において,当該研究の目的は(1)凸多面体の離散構造の研究を代数的側面から刺激し進展させること,及び(2)単体的複体に付随する可換代数の代数的不変量を組合せ論的に記述することであった.目的(1)について,当該年度は,整凸多面体P⊂R^Nに含まれる格子点の個数i(P,n)の母函数から定義されるδ-列の組合せ論的特徴付けを探究した.我々は,函数i(P,n)をHilbert函数とする可換整域A(P)を定義しその代数的振舞からPのδ-列の組合せ論的諸性質を研究した.更に,可換整域A(P)が次数1の元で生成されるならば,Pのδ-列はいわゆる上限定理型の不等式を満たすことに着目し,A(P)が次数1の元で生成されるための必要十分条件をPの組合せ論で記述することを試み,部分的な成果を得た.目的(2)について,当該年度は,単体的複体Δに付随するStanley-Reisner環k[Δ]のBetti数列を組合せ論的に記述する研究を遂行した.我々は,k[Δ]の有限自由分解が純となるような単体的複体Δを組合せ論的に分類することに挑戦したが,その際,計算代数の成果とグレブナ-基底の基礎理論を使って,計算機実験をしたことが有益であった.更に,Δが有限半順序集合Xの順序複体のとき,k[Δ]のBetti数列をXのメビウス函数を使って表示する方法を模索し,modular束Xの順序複体のCohen-Macaulay型を計算するための効果的な公式を発見した.
• 複素解析的葉層構造の特異点の研究

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1994 -1994 

  Author : 諏訪 立雄, 齋藤 睦, 山口 佳三, 中居 功, 石川 剛郎, 泉屋 周一

   

  研究代表者を中心に、主として、複素解析的特異葉層構造の不変部分多様体に関する留数についての研究、及びそこでも用いられたCech-Rhamコホモロジー群およびstratifyされた空間上の積分理論の応用について研究を行った。 前者については、フランスのD.Lehmannとの共同研究において、複素2次元正則ベクトル場の非特異不変曲線に関する、Camacho-Sadの指数の一般化を考察し、複素解析的特異葉層構造の不変部分多様体に関する留数について、不変部分多様体が特異点を持つ場合にも留数を定義し、留数定理を証明した。またこの留数の計算法を求め、これが、いわば“相対的Grothendieck留数"で表わされることを示した。これらの結果は共著論文としてまとめられ、Journal of Differential Geometryに掲載される予定である。 後者に関しては、Cech-de Rhamコホモロジー群およびstratifyされた空間上の積分理論がを用いて代数的位相幾何学(ホモロジー理論,Poincare および Alexander-Lefschetzの双対性等)を記述し、代数幾何学,複素解析幾何学における基本的諸事実(特に、交点理論、分岐理論、特性類に関するもの、例えば、Thom類の特徴付け、因子の交点理論、Poincare-Hopfの定理、Milnor数の公式、Riemann-Hurwitzの公的、埋め込みに対するGrothendieck-Riemann-Rochの定理等)を特性類の留数の立場から統一的に証明し、さらにこれらを精密化、一般化した。この方法の特徴的なことは、(1)種々の公式が特異点集合に局所化された形で得られること、(2)Hardな解析を用いずに自然に計算が出来ること等である。
• 代数的表現論の開発

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1990 -1990 

  Author : 堀田 良之, 長谷川 浩司, 齋藤 睦, 清水 勇二, 石田 正典, 小田 忠雄

   

  代数群の大域的指標を統制する偏微分方程式系として,HarishーChandro方程式系というホロノミ-系があり,ここ数年来その構造を研究してきた。本年度は,これとは異なる動機で成立したものであるが共通の側面をもつ青本・Gelfand型のホロノミ-系について幾分かの成果を得た。 青本・Gelfandのホロノミ-系は,本来Gauss以来長い間研究されてきた超幾何型微分方程式(函数)の一般化(多変数化)を目ろんだもので,いくつかの定式化があるが末だその全貌は明らかではない。 まず,このホロノミ-型微分方程式系は,代数群がベクトル空間に線型作用しているとき構成されるわけであるが,特に変換群がト-ラスであって相似変換を含むとき“一般超幾何型"と呼ばれている。この場合このホロノミ-系のフ-リエ変換を考えると,そのサポ-トは有限個の軌道からなり,かつ斉次的である。さらにD加群的考察によって,このホロノミ-系は群の(無限小)指標に関して“捩れ同変"(新しい概念)であることが判明した。このことを手がかりにすると,このフ-リエ変換されたホロノミ-系は確定特異点型(Fuchs型)であることが証明される。従って,斉次性によって,元の青本・Gelfand型のホロノミ-系(一般超幾何型も確定特異点型であることが結論される。 この定理は,今後この方程式系を考察・応用する場合欠かせない基礎事実となるであろう。 その他,この方程式系の特性多様体の構造について,ト-リック多様体の側面からの幾何学的・組合せ論的研究が,小田・石田らによってなされている。
• D加群と表現論

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1989 -1989 

  Author : 堀田 良之, 長谷川 浩司, 齋藤 睦, 佐武 一郎, 石田 正典, 小田 忠雄

   

  研究代表者の堀田は、従来からのテ-マである指標の代数解析的研究すなわち指標D加群を追求する中で、青木・Gelfandの微分方程式系のD加群的取扱いを試みた。その結果、この方程式は、ト-ラスの埋込みによって、オイラ一方程式系を転移し、それをフ-リエ変換したものが主要部を占めることが分かった。このことから、この方程式系の正則性(確定特異点型であること)についての知見が得られる。 関連して、小田は、ト-リック多様体の代数幾何の研究の中で、組合せ幾何学との関連で、上のGelFand学派の結果の新しい応用を見出した。さらに、石田は、ト-リック多様体のある種の不変量の計算法を具体化し、特にカスプ特異点について新しい知見を得た。これらの研究は青木・Gelfand方程式の特性多様体の構造を解明するために大きく役立つものと思われる。 代数群の数論からの研究を行った佐武は、有理構造をもつ対称領域が有利点をもつ条件、およびその商空間の志村モデルの定義体との関係等を明らかにした。 D加群れの表現論への応用を試みた齋藤は、ト-ラスの作用がある多様体上のホロノミ-D加群の局所化定理を得た。これによって、指標をレフシェツ型不動点定理から計算する方法が拡大されたことになる。 可解格子模型とアフィン・リ一環の表現論の関連を追求してきた長谷川は、柏原・三輪が構成したBroken Zn-Symmetric modelと呼ばれる模型、すなわちYang-Baxter方程式の解を、Baxterの8頂点解に関連した代数(Sklyanin代数)を用いて体系的に導出することに成功した。
• Systems of hypergeometric equations and their related D-modules