One-parameter equipped posets and their representations
Representations of equipped posets can be interpreted in terms of matrix problems over the field \(\mathbb{R}\) of real numbers corresponding to a special type of vector space categories over \(\mathbb{R}\). In particular the considered vector space categories have indecomposable objects with endomorphism ring \(\mathbb{R}\) or \(\mathbb{C}\) (the field of complex numbers) only. The possible bimodules of morphisms between indecomposable objects are \(\mathbb{R}\), \(\mathbb{C}\) and \(\left(\begin{smallmatrix}\mathbb{R} &\mathbb{R}\\ \mathbb{R} &\mathbb{R}\end{smallmatrix}\right)\). An equipped poset is a partially ordered set with an additional structure encoding the corresponding matrix problem. The paper announces the description of equipped posets which are of infinite representation type and (domestic) one-parameter in the sense analogous to the usual one [see Section 15.10 in \textit{D. Simson}, ``Linear representations of partially ordered sets and vector space categories'', Algebra, Logic and Applications, Vol. 4, Gordon \& Breach Science Publishers, New York (1992; Zbl 0818.16009)]. The combinatorial criterion for an equipped poset to be one-parameter generalizes the result of \textit{V. V. Otrashevskaya} [in: Matrix problems, Kiev 1977, 144-149 (1977; Zbl 0461.16025)]. The possible coordinate vectors of indecomposable representations of one-parameter equipped posets are described in terms of roots and imaginary roots of the associated Tits quadratic form and so-called special vectors. Also a classification of sincere nontrivially equipped posets is given, cf. \textit{A. G. Zavadskij} [Inst. Mat. Akad. Nauk Ukr. SSR, Kiev 1981, Preprint 81.27, 30-42 (1981)].
one-parameter families of representations
scientific article; zbMATH DE number 1505731