Project/Area Number  07454028 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
解析学

Research Institution  Kobe University 
Principal Investigator 
TAKANO Kyoichi Department of Science, Kobe University Professor, 理学部, 教授 (10011678)

CoInvestigator(Kenkyūbuntansha) 
NOUMI Masatoshi Department of Science, Kobe University Professor, 理学部, 教授 (80164672)
HIGUCHI Yasunari Department of Science, Kobe University Professor, 理学部, 教授 (60112075)
SASAKI Takeshi Department of Science, Kobe University Professor, 理学部, 教授 (00022682)
KABEYA Yoshitsugu Department of Science, Kobe University Assitant, 理学部, 助手 (70252757)
AIZAWA Sadakazu Department of Science, Kobe University Professor, 理学部, 教授 (20030760)

Project Fiscal Year 
1995 – 1996

Project Status 
Completed(Fiscal Year 1996)

Budget Amount *help 
¥5,400,000 (Direct Cost : ¥5,400,000)
Fiscal Year 1996 : ¥1,900,000 (Direct Cost : ¥1,900,000)
Fiscal Year 1995 : ¥3,500,000 (Direct Cost : ¥3,500,000)

Keywords  confluent hypergeometric function / space of initial values of Painleve / irreducibility of Painleve / classical function / quantum group / qhypergeometric function / percolation / projective homogeneous space / 合流型超幾何関数 / パンルヴェ初期値空間 / パンルヴェ系の既約性 / 古典関数 / 量子群 / q超幾何関数 / パーコレーション / 射影等質空間 / 射影等質曲面 / ホロノミック系 / パンルヴェ系 / 初期値空間 / シンプレクティック構造 / 超幾何微分方程式 / q差分方程式 / リー環的対称性 
Research Abstract 
1.Hypergeometric differential equations : (1) We obtained the system of differential equations satisfied by integrals associated with a nondegenerated quadratic hypersurface and n hyperplanes in the (k1)dim. complex projective space and we studied certain symmetries of the system.(2) We made clear geometrically the processes of confluence of general confluent hypergeometric functions on Grassmann manifolds. 2.Painleve systems : (1) We found processes of conflunce among the spaces of initial conditions of Painleve systems. The processes are compatible with the well known ones.(2) We proved the irreducibility of the second and the fourth Painleve functions exept the known classical functions, by determining ivariant devisors of Hamiltonian vector fields associated with the Painleve systems. 3.Quantum groups and qfunctions : (1) We realized a family of quantum complex projective spaces as one of quantum homogeneous spaces associated with a family of coideals, and we expressed the zonal spherical functions in terms of AskeyWilson polynomials. (2) We solved afflrmatively the integrality conjecture of Macdonald for the (q, t)Kostka coefficients, by constructing raising operators for Macdonald's symmetric polynomials. 4.Percolation problem : We obtained the order of the spectral gap in the case where + and  spins are randomly distributed on the boundary. The order is the same as that in the case where there are on spins no the boundary. 5.We showed that the affine geometry of surfaces in the 3dim. projective space works aiso in the case where some invariants are degenerated, and we classified projective homogeoenus surfaces.
