1999 Fiscal Year Final Research Report Summary
Some Homological Properties in Geometric Invariant Theory
Project/Area Number 
10640038

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Josai University 
Principal Investigator 
NAKAJIMA Haruhisa Josai University, Faculty of Science, Prof., 理学部, 教授 (90145657)

CoInvestigator(Kenkyūbuntansha) 
MIKI Hiroo Kyoto Institute of Technology, Faculty of Engineering, Prof., 工芸学部, 教授 (90107368)
YOSHIZAWA Mitsuo Josai University, Faculty of Science, Prof., 理学部, 教授 (40118774)
ISHIBASHI Hiroyuki Josai University, Faculty of Science, Prof., 理学部, 教授 (90118513)
SEKIGUCHI Katsusuke Kokushikan University, Faculty of Engineering, Asso.Prof, 工学部, 助教授 (20146749)
OGAWA Yoshito Tohoku Institute of Technology, Faculty of Engineering, Asso.Prof., 工学部, 助教授 (60160777)

Project Period (FY) 
1998 – 1999

Keywords  Classical Group / covariants / coregular / Swan group / relative equidimensionality / transvection 
Research Abstract 
A representation (V,G) of a reductive algebraic group G over the complex number field C is said to be coregular, if V//G is nonsingular. For a semisimple G, irreducible coregular representations are determined by P.Littelemann, and for a simple G, all coregular representations are classified by V.L. Popov, G.W. Schwarz, O.M. Adomovich and E.O. Golovina. In this research, we have determined coregular representations of nonsemisimple reductive groups G with simple semisimple parts having enouch closed orbits. This is based on the decomposition of actions of algebraic tori on normal varieties into noblowingup actions of codimension one and blowingup actions of codimension 2. The Chow groups preserve under quotient morphisms in the latter actions. Moreover, in the relaion with this, we have studied relative equidimensionalities and relative stabilities of actions of nonsemisimple reductive groups and obtain some results which are useful in classifying coregular or equidimensional rep
… More
resentaions. We generalize a part of the classical ramification theory of finite Galois groups to one of quotient morphisms under affine group actions and give a criterion the result similar to in finite covering cases to hold in affine groups case, which is related to an extension of some results on semiinvariants of finite groups to in the case of centric diconnected tori. In order to study on invariant theory of classical groups over local rings, we give a nice criterion for a set of symplectic trasvections to be a genrating system of the sympectic group Sp(V) defined over local rings (due to Ishibashi). On representaion theory of finite groups: We determine essential ideals and primary decompositions of mod 2cohomology ring of finite abelian 2groups (due to Ogawa) and obtain partial results on extensions of some 2groups which preserve the irresucibilities of induced characters (due to Sekiguchi). These results seem to be useful in studying functor properties in invariant theory of finite groups. Less

Research Products
(14 results)