English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

A classification of 2-simple prehomogeneous vector spaces is completed by using some P.V.-equivalences together with [3]. Some part is very different from the previous classification of the irreducible or simple cases [1, 2], and some new method is necessary. This result shows the difficult point of a classification problem of reductive prehomogeneous vector spaces. Introduction. In M. Sato and T. Kimura [1], all irreducible prehomogeneous vector spaces (abbreviated irreducible P.V.'s) are classified up to castling-equivalence, and it is proved that any irreducible P.V. is castling-equivalent to a 2-simple P.V. (or to (SL(m) X SL(m) X GL(2), A1 X A1 X A1) with m = 2,3). Therefore, as a step to a classification of the general case, it is natural to classify all nonirreducible 2-simple P.V.'s. In this paper, we shall classify all 2-simple P.V.'s of type II and give the complete list of them up to strong equivalence in the sense of Definition 4, p. 36 in [1]. However we need various P.V.-equivalences (cf. Propositions 1.7; 1.12; 1.34; 1.36; 1.37, Theorem 1.16 and its Corollary; Corollary 1.26, Definition 1.31, Remark 1.32) to carry out the classification, and hence, first we shall prove them. We also investigate the regularity in some cases. Since all 2-simple P.V.'s of type I are already classified in [3], this completes the classification of all 2-simple P.V.'s. Here we say that (G, p, V) (or simply (G, p)) is a 2-simple P.V. if it is a prehomogeneous vector space of the following form: G = GL(1)k?s?t XG, X G2 where G1 and G2 are simple algebraic groups, p = the composition of the scalar multiplications GL (l)k+s + on each irreducible component and the representation p' = p1 ? p'1 + *** +Pk ? P'k+ (a1 + *** +s) ?1 + 1 ? (T1, + *** +T,) ofthegroupG1 x G2 where pi, aj (resp. p', Tj) are nontrivial irreducible representations of G1 (resp. G2). We write (G, p', V) instead of (G, p, V) for simplicity. To avoid confusion, we write + instead of (. Clearly each (GL(1) X G1 x G2, pi ? p) (i = 1,..., k) is an irreducible 2-simple P.V. If at least one of them is a nontrivial P.V. (see Definition 1.1), we call it a 2-simple P.V. of type I. On the other hand, if all of them are trivial P.V.'s, we call it a 2-simple P.V. of type II. In this case, we have G2= SL(n), Received by the editors March 3, 1986 and, in revised form, July 30, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 11R20, 11R29; Secondary 11N25, 11R18. 01988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

Some P.V.-equivalences and a classification of 2-simple prehomogeneous vector spaces of type 𝐼𝐼