Posets of finite prinjective type and a class of orders
Let \(K\) be a field and \(J\) a finite partially ordered set with \(n\) maximal elements, called peaks. The author defines the category of socle projective representations of \(J\) over \(K\) (or peak \(J\)-spaces) generalizing the notion of \(I\)-spaces in the sense of \textit{P. Gabriel} [Manuscr. Math. 6, 71-103 (1972; Zbl 0232.08001)] and the definition of the author [in Proc. Tsukuba Int. Conf. Representations of finite dimensional algebras, CMS Conf. Proc. 11, 287-298 (1991; Zbl 0797.16015)], where the case \(n = 2\) is considered. The main result is a characterization of those posets \(J\) having only finitely many isomorphism classes of indecomposable socle projective representations. The characterization is given in terms of the Tits quadratic form associated to \(J\), in terms of algebraic varieties with an algebraic group action and by presenting a critical list of 110 minimal posets having infinitely many isomorphism classes of indecomposable socle projective representations. The result generalizes the characterization of representation-finite one peak posets of \textit{M. M. Kleiner} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 28, 32-41 (1972; Zbl 0345.06001)] and the author's characterization dealing with the case of two peaks given in the paper mentioned above. One of the methods used in the proof is the study of prinjective \(KJ\)-modules investigated by \textit{J. A. de la Peña} and the author [in Trans. Am. Math. Soc. 329, 733-753 (1992; Zbl 0789.16010)]. An application to the study of lattices over orders is given.
category of socle projective representations
indecomposable socle projective representations
scientific article; zbMATH DE number 500601
scientific article