4 2012 - 3 2016

Grant number：24540020

Grant amount：\5070000 （ Direct Cost: \3900000 、 Indirect Cost：\1170000 ）

We derived the determinant of the matrices related to the character tables of the symmetric groups. We also gave a (q,t) analogue of a well-known partition identity. In connection with the affine Lie algebras, we found an interesting Schur function identities, which arose from the basic representation of the affine Lie algebra.

researchmap

2011 - 2013

Grant number：23540243

Grant amount：\5070000 （ Direct Cost: \3900000 、 Indirect Cost：\1170000 ）

We mainly studied the detailed structure of the fundamental solution to the Schroedinger equation on compact symmetric spaces from the point of view of number theory and representation theory. One of our main results is as follows. Under certain assumptions on the vector potential and on compact symmetric spaces, the singular support of the fundamental solution to the magnetic Schroedinger equation becomes a lower dimensional subset of the compact symmetric space, which is given in terms of generalized Gauss sums at a rational time. On the other hand, at an irrational time, the singular support of the fundamental solution coincides with the whole symmetric space.

researchmap

2009 - 2011

Grant number：21540016

Grant amount：\4420000 （ Direct Cost: \3400000 、 Indirect Cost：\1020000 ）

The importance of the "Brauer-Schur functions", which had been introduced by myself, is now being recognized in the area of KP hierarchy, its reductions and representations of affine Lie algebras. I published a short note entitled "A note on Brauer-Schur functions", which is contained in the proceedings of the international conference on Mathematical physics held in Tianjin, China.

researchmap

2007 - 2010

Grant number：19340032

Grant amount：\14040000 （ Direct Cost: \10800000 、 Indirect Cost：\3240000 ）

Towards developing harmonic analysis on huge groups, we did integrated studies of probability theory and group representations. Harmonic analysis is a discipline which seeks deep mathematical structures by looking at symmetries of the objects and develops analysis relying on them. In this study, we are led to huge groups describing the symmetries because our objects are so big as to have an infinite degree of freedom. Main results among the ones we obtained are (i) classification and explicit formulas of the characters which are building blocks of harmonic analysis, and (ii) a series of theorems which construct a bridge between asymptotic behavior of representations of groups and probabilistic limit theorems.

researchmap

2007 - 2009

Grant number：19654005

Grant amount：\3200000 （ Direct Cost: \3200000 ）

昨年度に基礎を確立した複素および1進の反復積分の関数等式の導出法(Wojtkowiak氏との共同研究)を延長して,具体的な実例計算をさらに検証した.とりわけ古典的な高次対数関数について知られている分布関係式(distribution relation)の1進版を導出することに成功した.分布関係式は,様々な特殊値を代入することで,高次対数関数の特殊値の間に成立する様々な関係式を組織的に生み出す重要なものであり,1進の場合にも並行してガロア群上の関数族(1-コチェイン)を統御する要となることが期待されるが,前年度までに得られた関数等式との整合性についても検証を行った.8月にケンブリッジのニュートン数理科学研究所で行われた研究集会"Anabelian Geometry"において口頭発表を行った.このときの講演に参加していたH.Gangl氏,P.Deligne氏から今後の研究指針を考える上で有用になると思われるコメントを頂戴することが出来た.また分布関係式の低次項の解消問題に関連して,有理的な道に沿った解析接続の概念にっいて考察を進める必要が生じた.こうしたテーマに関連して研究分担者の鳥居氏には,有理ホモトピー論に関する情報収集を担当していただき,また研究分担者の鈴木氏には,量子代数やKZ方程式との関連で組みひも群の数理についての情報収集を担当していただいた.以上の研究成果の一部は,共同研究者のWojtkowiak氏と協力して,"On distribution formula of complex and 1-adic polylogarithms"という仮題の草稿におおよその骨子をまとめたが,まだ完成に至っていない.周辺にやり残した問題(楕円ポリログ版など)もあり,これらについて一定の目処をつけてから公表までの工程を相談する予定になっている.

researchmap

2007 - 2008

Grant number：19540031

Grant amount：\4420000 （ Direct Cost: \3400000 、 Indirect Cost：\1020000 ）

researchmap

2005 - 2006

Grant number：17540026

Grant amount：\3200000 （ Direct Cost: \3200000 ）

I focused on the applications of the representation theory of the symmetric groups to certain nonlinear systems of differential equations. More precisely I investigated the Cartan matrices of the symmetric groups which play an important role in modular representation theory. It has been known that the coefficients of Q-functions appearing in the expansion of 2-reduced Schur functions are non-negative integers. These are called the Stembridge coefficients. I noticed that the matrices of Stembridge coefficients are "similar" to the decomposition matrices for the 2-modular representations of the symmetric groups. I proved that they are transformed to each other by simple column operations, and that the elementary divisors of the Cartan matrices and those of the so-called "Gartan matrices" coincide. Next I introduced the "compound basis" for the space of the symmetric functions and expanded (non-reduced) Schur functions in terms of our new basis. I found that the appearing coefficients are all integers. This compound basis arose naturally, at least for me, from representation theory of certain affine Lie algebras, which I have been studying for many years. At the present moment our basis is obtained only for the case of characteristic 2, but it is plausible that this exists for any characteristic p. A natural problem occurs: What is the transition matrix between the two bases, i.e., Schur function basis and our compound basis ? In a joint work with Mizukawa and Aokage, it is proved that the determinant of this transition matrix is a power of 2. This is a non-trivial fact.

researchmap

2004 - 2006

Grant number：16540154

Grant amount：\3600000 （ Direct Cost: \3600000 ）

The main purpose of the present research is to study asymptotic behavior of various characteristic quantities of representations of symmetric groups and other similar discrete groups as the sizes of the groups grow, and to investigate the limiting pictures from the viewpoint of scaling limits in probability theory and statistical mechanics. Features to be noted in this research include using methods of limit theorems in quantum probability and making much of relations to free probability and random matrices. The following are several concrete results.
1. We studied the spectral distributions of Laplacians with respect to the Gibbs states in zero temperature and infinite volume limit as graphs grow with their degrees and temperatures keeping certain scaling balances. We computed the asymptotic behavior in details under the formulation of quantum central limit theorem by using creation and annihilation operators on interacting Fock spaces.
2. Through combinatorial hard analysis of moments of the Jucys-Murphy element, we studied universal understanding of concentration phenomena in various statistical ensembles consisting of Young diagrams, including those which come from irreducible decomposition of a representation of the symmetric group such as the Littlewood-Richardson coefficients. Many of them are closely related to some properties of random walks on a certain modified Young graph. Here also we applied methods of quantum probability effectively.
3. Under cooperation with T. Hirai and E. Hirai, we constructed a nice factor representation which expresses any character of a wreath product of a compact group with the infinite symmetric group as its matrix element. This representation reflects directly the characterizing parameters for the character beyond a general representation of Gelfand-Raikov.

researchmap

2003 - 2006

Grant number：15340010

Grant amount：\10700000 （ Direct Cost: \10700000 ）

As a joint work with Osamu Iyama (Nagoya Univ.), we have defined the mutation as an action of braid group on the set of indecomposable objects in a general triangulated category. We have developed a general theory of mutation and have applied it to the classification problem of rigid Cohen-Macaulay modules. In particular, we succeeded to describe the perfect classification of rigid Cohen-Macaulay modules over a Veronese subring of dimension 3 and of degree 3. Through this consideration of the mutation, we are able to obtain further examples as well where we can classify the rigid Cohen-Macaulay modules. Actually the study was done by considering the maximal orthogonal subcategories in the stable category of Cohen-Macaulay modules. An investigation of the similar problem in a derived category is now in progress.
We also made a study on the deformation and the degeneration of modules. And we succeeded to construct a noncommutative parameter space of the universal deformations of a module. By this, we are now able to understand well the classical obstruction theory of modules. This noncommutative deformation theory is also in progress.

researchmap

2004 - 2005

Grant number：15654007

Grant amount：\3200000 （ Direct Cost: \3200000 ）

昨年度に赤澤尋樹氏が行ったグラスマン代数のある商代数にあらわれる斜交表現の不変式のなす環の三叉グラフの生成関係式をつかった表示の研究に基づいて,研究代表者が過去に米国のガロウファリディス教授との共同研究のなかで,より立ち入った議論を必要とする部分について簡単なまとめを行った.
長年の懸案であるモーデル型の楕円曲線の1パラメータ族についてGrothendieck-Teichmueller理論で有用なものの探索を引き続いて精力的に行ったが,残念ながらまだ十分に議論が展開できていない.一方でグロタンディーク・デッサンに付随する代数曲線の不変量の計算法について,主に三角群に付随して注目に値する実例について保型関数との関連で議論を進め,いくつか注目に値する知見を深めることが出来た.
6月には,カリフォルニア工芸州立大学の加藤五郎氏の岡山来訪を実現し,物質,空間,時間の前層化に基づく氏による非常に前衛的な理論に関する講演"Extension Type Yoneda Lemma for Relativistic t-Topos"を通して数理物理におけるカテゴリー理論の可能性について知見を深めた.
研究分担者には、それぞれの専門の立場から研究課題に関連する内容の発展について協力を頂いていた.特に田中克己助教授には,国内出張を通して,情報収集に協力していただいた.
また本研究課題と関連する必要な図書備品・消耗品の購入を通して,各分担者の研究活動を支援した.

researchmap

2002 - 2005

Grant number：14340017

Grant amount：\10000000 （ Direct Cost: \10000000 ）

We studied a measure function that describes the meta-abelian quotient of the monodromy representation associated with the universal family of elliptic curves and its relation with generalized Dedekind sums.
In particular, we showed a congruence formula that describes moment integrals of the measure function along variation of weights. Equations in the Grothendieck-Teichmueller group satisfied by the Galois image were investigated.
Using genus zero non-Galois covers, we found a new type equation. Utilizing the Magnus-Gassner type representation, another new type equation was found to hold in the topological matrix ring in two variables.
In a collaboration with H.Tsunogai, using a characterization of the lemniscate elliptic curve as a Grothendieck dessin, we studied the behavior of Galois parameters of the Grothendieck-Teichmueller group, and described the decomposition of the standard parameter into a product of mutually transposed harmonic parameters in terms of adelic beta functions. In a collaboration with P.Lochak and L.Schneps, we replaced a toplogical path from the standard tangential basepoint to the five cyclic point by a composition of algebraic paths that are transformed by the Galois group with Grothendieck-Teichmueller parameters. Then, we succeeded in interpreting the five cyclic decomposition of the standard parameter in the fundamental group of the moduli spaces of the 5-pointed projective lines. Comparing the method of Ihara-Matsumoto with a paper by Gerritzen-Herrlich-Put about stable compactification of moduli spaces of n-pointed projective lines, we obtained a natural interpretation of tangential base points on those moduli spaces. Through discussions with Wojtkowiak at Nice University, a new direction of investigation and perspectives about 1-adic itereated integrals was obtained.

researchmap

2003 - 2004

Grant number：15540030

Grant amount：\2900000 （ Direct Cost: \2900000 ）

This is an effort for understanding the role of Schur functions and Schur's Q functions, the projective analogue of Schur functions, in representation theory. To be more precise, we proved the following theorem. Schur functions associated with the rectangular Young diagrams occur as weight vectors of the basic representation of the affine Lie algebra of type D^{(2)}_2. And also, they turn out to be the homogeneous tau functions of the nonlinear Schroedinger hierarchy. The key idea for proving the above is to write down the representation spaces and operators in terms of fermions, and derive polynomials via the boson-fermion correspondence. We have succeeded in verifying the similar phenomena for the case of the affine Lie algebra of type A^{(2)}_2. In 2004 we considered the following problem. Clarify the nature of the coefficients in the 2 reduced Schur functions when expanded in terms of Schur's Q functions. Through some experimental computations in small rank cases, I had been convinced that these coefficients are of great interests, both from representation theoretical and combinatorial points of view. Finally we realized that these coefficients are nothing but the so-called Stembridge numbers. As a result we could relate these numbers with the representation theory of affine Lie algebras. Looking carefully at the table of these numbers, we found a simple formula for the elementary divisors of the Cartan matrices of the symmetric groups. More than 10 years ago, Olsson in Copenhagen gave a formula for those, which is expressed in terms of a generating function and is rather complicated. Our version is more direct and combinatorial.

researchmap

2001 - 2004

Grant number：13440001

Grant amount：\12800000 （ Direct Cost: \12800000 ）

1.Asai-Yoshida's conjecture on the number of group homomorphisms has a connection with p-adic anallysis (Yoshida, Takegahara, K.Conrad, etc.). Furthermore, Yoshida solved the conjecture for the homomorphisms from the fundamental group of a compact Riemann surface whose importence in quantum field theory was pointed by M.Mulase.
2.Yoshida and Oda accomplished a fundamental theory of crossed Burnside rings of finite groups.
3.Yoshida, Sasaki, Oda study cohomology theory of finite groups, especially Hochschild cohomology, crossed Mackey functors and quantum doubles of group algebra.
4.Yoshida, Bannai and Keisuke Shiromoto sutudies combinatorics, in particular, distance regular graphs, designs and code theory, especially an application of homological algebra to the theory of codes on a ring.
5.Koshitani studies modular representation thoery of finite groups and gave an affirmative answer to the Broue conjecture for groups with some special defect groups.
6.Nakamura and Yamashita studied some related areas (algebraic geometry and representation theory) and gave some interested results, especially a relation with finite simple groups.
7.We invited three mathematicians from abroad.
・2001 FAN Yun(Wuhan U.) Talk on Broue conjecture(Kyushu Univ.)
・2003 Keith CONRAD(Connecticut U.) Talk of p-adic analysis, number theory(Hokkaido U., Kyoto U.)
・2004 Segre BOUC(CNRS) Talk on Dade groups and Burnside rings (Hokaido U., Kyoto U.)
A large number of results of this research has been printed and published. The remainder results will be serially published. The investigator gave some talks related to this research in some conferences--
"Generating functions and related topics" (Sapporo 2001), "20-th Symposium on Algebraic Combinatorics (Kyoto 2002), "Extended Group Seminar" (Sapporo 2002,2004, and so on.

researchmap

2001 - 2003

Grant number：13640215

Grant amount：\2900000 （ Direct Cost: \2900000 ）

We studied Nelson's model derived from the Pauli-Fierz model through several physical approximations. The Pauli-Fierz model describes an electron coupled with the quantized radiation field in nonrelativistic quantum electrodynamics, when we regard the electron as a nonrelativistic particle. We proved that the Nelson model has infrared catastrophe when its Hamiltonian has the Coulomb potential appearing in the structure of atoms. Developing the proof and using the Carleman operator, we clarified and characterized a mathematical mechanism which causes infrared safe or infrared divergence. The Carleman operator is derived from the so-called pull-through formula, and we gave the exact operator-theoretical proof for the formula, which is the fast to succeed in it. By this proof, we can investigate mathematical properties of the domain of the Carleman operator and pull-through formula, which resulted in our results. Because Nelson's model has infrared catastrophe by our results, we find another representation in which the model has a ground state. This representation describes the actual physical phenomenon. So, we removed both, infrared and ultraviolet cutoffs, and proved the Nelson model without both cutoffs has a ground state in the representation.
We studied the norm resolvent convergence for the Hamiltonian describing relativistic particle coupled with the Aharonov-Bohm field in 2-dim. space. We investigated which self-adjoint extension has the most suitable representation to the actual physics among several self-adjoint extensions corresponding to the boundary conditions around singularities.

researchmap

2000 - 2003

Grant number：12304001

Grant amount：\29930000 （ Direct Cost: \24800000 、 Indirect Cost：\5130000 ）

Certain compactification of moduli space of abelian varieties was studied as well as moduli spaces of G-orbits for a finite subgroup G of SL(2,C) and SL(3,C). The main issues we have in mind are as follows (a) Study of a resolution of singularity of the quotient C^3/G as a moduli space (b) study of Kempf stability and compactification of moduli spaces (c) A canonical ompactification SQ_<g,N> of the moduli A_<g,N> over Z[1/N] of abelian varieties and related moduli.
There were remarkable progresses on each subject during this project. The main results are as follows : first there was a remarkable progress in the study on Hilbert schemes of G-orbits. We copuld give a new explanation to the phenomenon of McKay correspondence which was discovered over twenty years, and extending it to the three dimensional case, we obtained a lot of new resluts. The head investigator (Nakamura) proposed a generalization of McKay correspondence to the three or higher dimension, which was follows by many related results. In this sense this project payed a substantial role in the history of studying McKay correspondence. Among other things Nakamura showed that the Hilbert scheme of G-orbits is the canonical resolution of singularities of the quotient C^3/G. This is a new discovery which has never been observed, against the common sense in minimal model theory. Therefore this discovery has been accepted by specialists with surprise. Another substantial contribution of this project is that we constructed a new canonical compactification of moduli space A_<g,N> of abelian varieties This compactification is projective, it enjoys a desirable property as a compactification. From the stabdpoint of invariant theory, this compactification is ust that by stability. In this sense it is orthodox and is uniquely characterized by this property

researchmap

2001 - 2002

Grant number：13640024

Grant amount：\3400000 （ Direct Cost: \3400000 ）

Toward the complete classification of Cohen-Macaulay modules over a commutative local ring, we made new progress in solving the problems on degenerations of Cohen-Macaulay modules and the problems on the family of modules of G-dimension 0.
(1) Degeneration of Cohen-Macaulay modules :
One can define a partial order on the set of isomorphism classes of modules by using the degeneration relation. This order is related to the Horn order that has been defined by Bongartz for modules over finite dimensional algebras. One may conjecture that the order would be generated by the degenerations of Auslander-Reiten sequences whenever the cateogry of Cohen-Macaulay modules is of fintie representation type. This conjecture claims that such an order defined geometrically could be related to the combinatorial nature of Auslander-Reiten quiver. I actually gave a complete proof of this conjecture in the case that the local ring has dimension 2. I also proved this if the local ring is an integral domain of dimension 1. These results are published in Journal of Algebra (2002).
(2) Modules of G-dimension 0 :
As one of the generalizations of classification theory of Cohen-Macaulay modules over a Gorenstein ring, it is important to consider the modules of G-dimension 0 over a general local ring. It had been thought that the category of G-dimension 0 could have similar properties to the category of Cohen-Macaulay modules. However, I made a lot of examples that disprove it. In particular, if the cube of maximal ideal of the local ring is zero, then I succeeded to give a necessary and sufficient condition for the ring to have a nontrivial module of G-dimension 0. Actually I gave a way of construction of such indecomposable modules with continuous parameter. Using this construction I have shown that the family of modules of G-dimension 0 may not be a contravariantly finite subcategory in the cateogory of finitely generated modules. These results were reported in the Workshop of NATO Scientific Program in Romania (2002), and to be published from Kluwer Press.

researchmap

2000 - 2001

Grant number：12640149

Grant amount：\3500000 （ Direct Cost: \3500000 ）

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.

researchmap

2000 - 2001

Grant number：12640001

Grant amount：\3600000 （ Direct Cost: \3600000 ）

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.

researchmap

1999 - 2001

Grant number：11440012

Grant amount：\10600000 （ Direct Cost: \10600000 ）

In the 2001 fiscal year, we obtained the following two major research results :
1. We showed that there exists a canonical basis for the module of vector fields tangent to a Coveter arrangement in a multiple order. We also showed the basis has deep relations with the flat structure of the orbit space.
2. We determined the structure of the differential module generated by reciprocals of linear forms. In particular, we obtained a formula for its Poincare series (to appear in J. Algebra).

researchmap

1999 - 2000

Grant number：11640001

Grant amount：\3600000 （ Direct Cost: \3600000 ）

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.

researchmap

1998 - 2000

Grant number：10440004

Grant amount：\12900000 （ Direct Cost: \12900000 ）

In this research project, we obtained the following results.
1. Okada obtained explicit branching rules for the tensor products and restrictions of the irreducible representations of the classical groups corresponding to nearly-rectangular shaped Young diagrams. And he proved that the partition functions of the square ice model related to the alternating sign matrices with symmetry can be expressed in terms of the irreducible characters of the classical groups.
2. Kashiwara described the crystal bases for the quantum group U_q (gl(m, n))(with G.Benkart and S.Kang). Also, in the study of D-modules on the flag varieties, he proved the Kazhdan-Lusztig conjecture for the affine Lie algebras at a non-critical level (with T.Tanisaki), and showed that the duality for D-modules on the flag variety corresponds to that of Harish-Chandra modules (with D.Bartlet).
3. Koike described, in terms of generalized Brauer diagrams, the structure of the centralizer algebra of the spin groups on the tensor product of the basic spin representation and the tensor powers of the natural representation. Also he gave a realization of irreducible representations of the spin groups in the above tensor products.
4. Terada gave an geometric interpretation to the Robinson-Schensted correpondence between Brauer diagrams and up-down tableaux. And he constructed an Robinson-Schensted-type bijection for the Weil representation of sp (2n)(with T.Roby).
5. Yamada described weight vectors in the basic representations of some affine Lie algebras in terms of symmetric functions (with T.Nakajima), and found an interesting facts on the spin decomposition matrices of the symmetric groups. And He found the Littlewood's multiple formula for the spin irreducible characters of the symmetric groups (with H.Mizukawa).

researchmap

1997 - 2000

Grant number：09304003

Grant amount：\25700000 （ Direct Cost: \25700000 ）

With this grant, we have studied the representations of Lie groups, quantum algebras, Hecke algebras, etc. in the point of view of mathematical Physics and Combinatorics.
In the first year (19997), the activities are closely tied with RIMS project 1997 (Representation theory). In this project, we focused our activity on the study of the representations of real Lie groups from the geometrical view point of homogeneous spaces. With around 40 participants from abroad, the international science exchange was performed successfully. These results are published in Advanced Studies in Pure Mathematics, vol.26.
In 1998, the activities are closely tied with RIMS project 1998 (Combinatorial methods in representation theory). We had around 25 participants from abroad. Main theme was the representation of quantum groups and affine Hecke algebras.
In 1999, we focused on mathematical physics, and held the international symposium "Physical Combina-torics" at International Institute for advanced study and Research Institute for mathematical Sciences. We had advances in the representation theory of quantum algebras, the structure of solutions to Kniznik-Zamolodhikov equations and its q-analogy, conformal field theory. These activities were published in "Physical Combinatorics, Progress in Math. vol.191, Birkhauser.
In the last year 2000, we organized the international symposium "Math-Odyssey2001" in order to summarize our four years activities. The main theme was algebro-analytic methods in representation theory and mathematical physics. The proceedings will be published by the publisher Birkhauser.

researchmap

1998 - 1999

Grant number：10640146

Grant amount：\3000000 （ Direct Cost: \3000000 ）

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.

researchmap

1997 - 1999

Grant number：09440002

Grant amount：\13900000 （ Direct Cost: \13900000 ）

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.

researchmap

1997 - 1998

Grant number：09874001

Grant amount：\1700000 （ Direct Cost: \1700000 ）

高次元カテゴリーに関係したいくつかの成果が得られた。主なものをあげる。
1. カテゴリーの母関数。カテゴリーεの母関数とは、形式的無限和ε(t)=Σ_<X∈ε>t^X/|Aut(X)|のことである。ここでt^Xは、対象Xの同型類に対応した不定元である。これについて、以下の結果を得た。
(1) Joyalによる種の理論との関係。
(2) εが強いKrull-Schmidtカテゴリー(つまり直和分解の一意性が成立する)なら、指数関数型恒等式
ε(∪)=exp(Con(ε)(∪))
が成り立つ。
(3) 逆に、指数関数型恒等式が成り立てば、若干の条件の下でεが強いKrull-Schmidtカテゴリーになる。これらの結果は、北大のプレプリント(1998#416,432、いずれもJ.Algebraに投稿中)に見られる:T.Yoshida,Categorical aspects of generating functions(I) :Exponential formulas,(II)Operations on categories and functors.
2. クロスバーンサイド環。Gを有限群、Sを有限G-モノイドとする。クロスG集合とは、Sへの重み関数をともなう有限G-集合のことである。クロスG-集合については、テンソル積が定義できる。したがって、クロスG-集合のグロタンディエック環(クロスバーンサイド環)が定義できる。これについて、以下の結果が得られた。
(1) クロスG-集合とDrinreldによるquarltum doubleとの関係。
(2) クロスバーンサイド環の基本定理。
(3) ベキ等元公式とその応用。
これらの結果は、次の論文に公表予定(J.Algebraに投稿)である:
F.Oda-T.Yoshida,Crossed Burnside rings(I),(II).そのほか、マッキー関手や、群論の古典的問題に関係したいくつかの結果が得られており、順次論文として公表予定である。

researchmap

1997 - 1998

Grant number：09640001

Grant amount：\2400000 （ Direct Cost: \2400000 ）

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.

researchmap

1996 - 1996

Grant number：08640001

Grant amount：\2500000 （ Direct Cost: \2500000 ）

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_<C->軌道からなる。研究代表者は、「各K_<C->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。

researchmap

1996 - 1996

Grant number：08454001

Grant amount：\5200000 （ Direct Cost: \5200000 ）

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).

researchmap

1996 - 1996

Grant number：08211257

Grant amount：\1000000 （ Direct Cost: \1000000 ）

複素鏡映群の大きな系列の一つであるG(r,p,n)についてその"高次Specht多項式"の構成を行った.
古典的Weyl群を包括する複素鏡映群としてG(r,p,n)というシリーズがある.この群はn変数の多項式環pに自然に作用する.そのとき基本不変式で生成されるイデアルJによる剰余環R=P/JはG=G(r,p,n)の"余不変式環"とよばれ表現としては正則表現と同値であることが知られている.余不変式環RはGがWeyl群のとき,対応する代数群の旗多様体のコホモロジー環と同型であることから幾何学的にも非常に重要な対象である.そこでRの各既約成分をその基底もふくめて具体的に多項式として記述するということが問題となる.私は投稿中の共著論文"Higher Specht polynomials for complex reflection group G(r,p,n)"においてこの問題を解決した.G(r,p,n)の既約表現Young図形で統制さるのでその組合せ論を援用して,初等的に計算可能な形で基底が記述される.今後のG(r,p,n)のモジュラー表現などの研究において大きな役割を果たすものと信ずる.
なおこの結果は96年の日本数学会秋季総合分科会において口頭発表された.また97年にウイーンで開催される国際会議"Formal Power Series and Algebraic Combinatorics"でも発表予定である.

researchmap

1993 - 1993

Grant number：05640210

Grant amount：\2000000 （ Direct Cost: \2000000 ）

代表者の行った研究を中心に述べる。これは2つあり,対流項をもつ準線形放物型方程式の解の爆発問題と半線形波動方程式の散乱問題である。前者は都立航空高工専の鈴木龍一等との共同研究であり,まず解の爆発と大域存在を分ける非線形項と初期データの条件を定め,次に爆発解の爆発時刻での漸近挙動,特に一点爆発が起る場合について,対流項が与える影響をしらべた。後者は北海道大理学部の久保田幸次との共同研究であり,空間2時元の問題に対してオプティマルな結果を与えた。小さな初期データに対して大域解の存在条件は知られていたが,その解が散乱状況にあり,自由な波動方程式の解との対応で散乱作用素の存在を示すことが未解決で残されていた。この論文ではそれを肯定的に解決した。これらの他にも代表者は非線形波動現象に関する新しい結果をいくつか得ており,口答発表をしている。これらは漸次論文として発表する予定である。
分担者の行った研究は多く,全てを研究発表のリストにあげることができない。またその内容についてくわしく述べる余裕はないが,例えば酒井良の論文では2次元求積領域の境界の正規性をポテンシャル論を用いて示しており,西岡國雄の論文ではP-放物型方程式に対し,初期データと係数が共鳴を起す場合に解の漸近挙動をしらべている。
リストにはあげてないが山田裕史の50ページになる論文も発表されており,(Japanese Jiurnal of Math),全体として科学研究費(一般C)の上記研究課題は達成できたと考えている。

researchmap

1993 - 1993

Grant number：05640054

Grant amount：\2100000 （ Direct Cost: \2100000 ）

昨年における研究計画にあげた課題において、今年は特に代数的サイクルに関連する部分、合流型超幾何関の行列式について、進展を見た。代数的サイクルについてはゲルファント,カプラノフ,ゼレビンスキーによる超幾何関数の級数表示から由来する関係式に関して、これらの例について特に研究を行なった。それらの関係式が代数的対応により由来することがわかった。現在この方向での一般化について、トーラス埋め込みの言葉で整理中である。2つ目の合流型超幾何関数の行列式については斉藤恭司氏によって研究されていたオシラトリー積分と関連をもち、さらに、ヘシアン行列式によりexplicitな表示を持つことが示された。オシラトリー積分についても、今後の研究課題である。

researchmap

1992 - 1992

Grant number：04640180

Grant amount：\1700000 （ Direct Cost: \1700000 ）

Σ=(X,σ)をコンパクト空間Xとその位相図型σよりなる位相力学系とする。本年度は富山が韓国ソウル大の集中講義において位相的に自由な作用をもつ力学系の基本性質を示した。そしてその土台の上に青木-富山はΣに附随すする位相変換群C^*環A(Σ)の性質とΣの基本集合との関係について本年の研究課題に大きな前進を見出した。Por(σ)をΣの周期点またL(σ)をΣの再帰点全体の集合とする。Xをコンパクト距離空間とする。 定理1A(Σ)がI型のC^*環であることとC(σ)=Per(σ)、すなわち再帰点はすベて周期点であることは同値である。 定理2A(Σ)がI型の部分を持たないことすなわちantiliminalなことと、集合L(σ)＼Per(σ)がXで稠密なことは同値である。
従来A(Σ)がいつI型のC^*環になるかについては色々な研究があったがそれらは測度力学系を規範としているために古典的な例も排助するような作用の仮定をおいたりまた結果ののべ方が軌道空間を用いるなど力学系の研究者からは意味の理解し難いものであった。これに対して上記の結果は作用について何も仮定をおかず又その結果の意味は解明である。基本集合についてはなお非遊走集合Ω(σ)とそれをめぐる集合(Ω(σ)＼C(σ)など)の問題が残っているがそれらは解明出来なかった。
上の結果のほかに富山は位相力学系の不変部分集合への分解とA(Σ)のC^*環的分解の関係を明らかにし、青木は閉多称体上の微分関相写像の空間において周期点がすべて双曲型であるような系の集合のC^1-位相についての内点の集合は構造安定な系と一致するという多年の懸案を解決した。
このほか高井はC^*力学系に附随するC^*環(クロス積)の位相安定指数についてコンパクト可換群の作用のときの最良評価式を得た。更に酒井山下山田は関連するそれぞれの分野において成果を挙げ論文が発表されている。(山田についてはJapanese J.Math.に発表される予定)

researchmap

1987 - 1987

Grant number：62540059

Grant amount：\2100000 （ Direct Cost: \2100000 ）

ユークリッド空間の組み合せ幾何学として, 特に低次元図形の埋め込みに関する研究を行った.
1.グラフの埋め込みに関して, 任期のtreeは, 隣接する2点間の距離は1より大きく, 隣接しない2点間の距離は1位かになるように, 必ず3元本空間内に埋め込むことが出来ることを証明した. (R〓dlおよび他のチエコの数学者の協力を得て, 次元をこれ以下に落とすことは出来ないことも解った. )
2.Erd〓s等の研究した単位距離グラフの拡張として, ユークリッド空間内の点集合VとVの3点の作る三角形の面積の関連を調べた. 5点以上の点の集合Vに対し, Vの中のどの3点もおなじ面積の三角形を作るならば, Vは正則な単体の頂点集合である. また, Vが十分多くの点を含み, Vの中の3点の作る三角形の面積が, r種類しかなければ, Vの中の2点間の距離はたかだか(r^3+r^2)種類しかない.
3.4点からなる距離空間(X,d)に対し, 距離関数dをd^c(0<c<1/2)に変えると(X,d^c)は必ずユークリッド空間に埋め込めることが知られている(Blumenthal). 同様な結果をn点からなる距離空間について証明した. すなわち, 任意の自然数nに対して, c(n)>0が存在して, 0<c<c(n)かつlXl=nならば, (X,d^c)はユークリッド空間に埋め込むことが出来る. 特にc(5)【greater than or equal】1/2)log_2(3/2), c(6)【greater than or equal】1/2)log_2(4/3)としてとることが出来る.
4.グラフが3次元凸多面体のグラフになるための必要十分条件を与えるSteinitzの定理について, 特別な場合(極大平面的グラフの場合)の簡単な証明を考察した.
そのほか, 研究課題の周辺領域に入る多くの結果を得たが, それ等については, 割愛する.

researchmap