Brüstle, Thomas; Sergeichuk, Vladimir V. Estimate of the number of one-parameter families of modules over a tame algebra. (English) Zbl 1063.16021 Linear Algebra Appl. 365, 115-133 (2003). Summary: The problem of classifying modules over a finite dimensional algebra \(A\) reduces to a linear matrix problem whose indecomposable matrices can be parametrized by certain normal forms, called canonical matrices by V. V. Sergeichuk [Linear Algebra Appl. 317, No. 1-3, 53-102 (2000; Zbl 0967.15007)]. If the algebra \(A\) is tame, then the set of isomorphism classes of indecomposable \(A\)-modules of dimension at most \(d\) is given by a finite number \(f(d,A)\) of discrete modules and one-parameter families of modules. Accordingly, there are for every fixed size only a finite number of discrete and one-parameter families of canonical matrices. We prove that in the tame case the number of one-parameter families of canonical matrices of size \(m\times n\) and a given partition into blocks is bounded by \(4^{mn}\). Based on this estimate, we prove that \[ f(d,A)\leq{{d+r}\choose r}4^{d^2(\delta^2_1+\cdots+\delta^2_r)}, \] where \(r\) is the number of nonisomorphic indecomposable projective \(A\)-modules and \(\delta_1,\dots,\delta_r\) are their dimensions. This estimate improves significantly the double exponential estimate of T. Brüstle [C. R. Acad. Sci., Paris, Sér. I 322, No. 3, 211-215 (1996; Zbl 0846.16010)]. Cited in 3 Documents MSC: 16G60 Representation type (finite, tame, wild, etc.) of associative algebras 15A21 Canonical forms, reductions, classification 16G10 Representations of associative Artinian rings Keywords:canonical matrices; classification; tame algebras; finite-dimensional algebras; matrix problems; indecomposable modules; tame representation type Citations:Zbl 0967.15007; Zbl 0846.16010 PDFBibTeX XMLCite \textit{T. Brüstle} and \textit{V. V. Sergeichuk}, Linear Algebra Appl. 365, 115--133 (2003; Zbl 1063.16021) Full Text: DOI arXiv References: [1] Belitskiı̆, G. R., Normal forms in a space of matrices, (Marchenko, V. A., Analysis in Infinite-Dimensional Spaces and Operator Theory (1983), Naukova Dumka: Naukova Dumka Kiev), 3-15, in Russian [2] Belitskiı̆, G. R., Normal forms in matrix spaces, Integral Equations Operator Theory, 38, 3, 251-283 (2000) · Zbl 0971.65037 [3] Brüstle, Th., On the growth function of tame algebra, C. R. Acad. Sci. Paris, 322, Sèrie I, 211-215 (1996) · Zbl 0846.16010 [4] Crawley-Boevey, W. W., On tame algebras and bocses, Proc. London Math. Soc., 56, 451-483 (1988) · Zbl 0661.16026 [5] Drozd, Yu. A., Tame and wild matrix problems, (Mitropol’skiı̆, Yu. A., Representations and Quadratic Forms (1979), Inst. Mat. Akad. Nauk Ukr.: Inst. Mat. Akad. Nauk Ukr. SSR, Kiev), 39-74, in Russian · Zbl 0454.16014 [6] Drozd, Yu. A., Tame and wild matrix problems, Lect. Notes Math., 832, 242-258 (1980) [7] Drozd, Yu. A.; Kirichenko, V. V., Finite Dimensional Algebras (1994), Springer: Springer Berlin · Zbl 0469.16001 [8] Gabriel, P.; Nazarova, L. A.; Roiter, A. V.; Sergeichuk, V. V.; Vossieck, D., Tame and wild subspace problems, Ukr. Math. J., 45, 335-372 (1993) · Zbl 0869.16010 [9] Griffits, P.; Harris, J., Principles of Algebraic Geometry (1978), Wiley-Interscience: Wiley-Interscience New York [10] Sergeichuk, V. V., Canonical matrices for linear matrix problems, Linear Algebra Appl., 317, 53-102 (2000) · Zbl 0967.15007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.