2004 Fiscal Year Final Research Report Summary
AN ATTEMPT OF UNIFIED INVARIANT THEORY OF ALGEBRAIC GROUPS AND RELATED TOPICS
| Project/Area Number |
14540040
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | JOSAI UNIVERSITY |
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Principal Investigator |
NAKAJIMA Haruhisa JOSAI UNIVERSITY, FACULTY OF SCIENCE, PROF., 理学部, 教授 (90145657)
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| Co-Investigator(Kenkyū-buntansha) |
ISHIBASHL Hiroyuki JOSAI UNIVERSITY, FACULTY OF SCIENCE, PROF., 理学部, 教授 (90118513)
YAMAGUCHI Hiroshi JOSAI UNIVERSITY, FACULTY OF SCIENCE, PROF., 理学部, 教授 (20137798)
KOGISO Takeyori JOSAI UNIVERSITY, FACULTY OF SCIENCE, LECT., 理学部, 講師 (20282296)
SEKIGUCHI Katsusuke KOKUSHlKAN UNIVERSITY, FACULTY OF ENGINEERING, PROF., 工学部, 教授 (20146749)
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| Project Period (FY) |
2002 – 2004
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| Keywords | INVARIANTS / ALGEBARAIC GROUPS / INVOLUTIONS |
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| Research Abstract |
(1) We give a criterion for $R_chi$ to be a free $R^G$-module of rank one for a $1$-cocycle $chi$ of $G$ in the unit group $unit(R)$ under the action of a group $G$ on a Krull domain $R$, and consequently establish a criterion for module of relative invariants of a finite central extension of algebraic tori to be free, in terms of local characters.
(2) Let $G$ be an affine algeraic group whose identity component is an algebraic torus and $X$ an affine factorial $G$-variety with trivial units, over an algebraically closed field $K$ of characteristic zero. Consider a finite dimensional generating $G$-submodule $V^^*$ of $Cal O(X)$ having a $K$-basis $Omega$ consisting of weight vectors of $G^0$ such that $Omega$ does not degenerate under the canonical morphism $Vbackslash {0} to Bbb P (V)$. Suppose that the action $(X, G)$ is stable and consider the $G$-submodule $W$ of $V$ such that $G$ is diagonal on $V/W$ and, for some $win V$, $W ni x mapsto x + w in V$ induces $Wdslash G_w cong Vdsla
… More
sh G$. We study on the Weil divisors of $Xdslash G$, $Xdslash G_w$ and $Vdslash G$. Our results are useful in determining certain representations of non-semisimple reductive algebraic groups in invariant theory.
(3) Let $varrho : G to GL(V)$ be a finite dimensional rational representation of a diagonalizable algebraic group $G$ over an algebraically closed field $K$ of characteristic zero. Using $(W, w)$ of $varrho$ defined (2), we show the existence of a cofree representation $widetilde{G_w} hookrightarrow GL(W)$ such that $varrho(G_w) subseteq widetilde{G_w}$ and $Wdslash G_w to Wdslash widetilde{G_w}$ is divisorially unramified is equivalent to the Gorensteinness of $Vdslash G$. We study on $rho$'s such that $Vdslash G$ are complete intersections and give a criterion for $Vdslash G$ to be a hypersurface in terms of characters of $G$.
(4) We shall define a linear map called a semiinvolution as a generalization of an involution, and show that any nilpotent linear endmorphism is a product of an involution and a semiinvolution. Also we shall give a new proof for Djocovi'c's theorem on a product of two involutions.
(5) We study the orthogonal group $O_n(V)$ on a quadratic module $V$ of rank $n$ with an orthogonal basis over a commutative ring $R$ satisfying some condition which holds for the ring of rational integers $Bbb Z$. The group $O_n(V)$ will be shown to be a finite group and that any element in $O_n(V)$ is a product of three elements of order 2. Less
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