On the number of absolutely indecomposable representations of a quiver
type: Item
instance of: scholarly article
title: On the number of absolutely indecomposable representations of a quiver
zbMATH Open document ID: 0965.16007
author: Bert Sevenhant
author: Michel Van den Bergh
published in: Journal of Algebra
publication date: 2000-05-02T00:00:00Z
review text: A quiver \(Q\) consists of a set \(Q_0\) of vertices, a set \(Q_1\) of arrows and two maps \(t,h : Q_1 \rightarrow Q_0\) assigning to an arrow \(\phi\) its tail and its head. A representation of a quiver is a family of finite dimensional vector spaces over \(k\), one for each vertex, together with a linear map for each arrow. If \(k\) is a finite field with \(q\) elements then \textit{V. G. Kac} [ Lect. Notes Math. 996, 74-108 (1993; Zbl 0534.14004)] has shown that the number of absolutely indecomposable representations of a quiver over \(k\) is a polynomial in \(q\) with integral coefficients. He also conjectured that the constant term of this polynomial is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. The present paper is devoted to reformulate such a conjecture into a property of Gaussian multinomial coefficients.
reviewed by: Stefano Capparelli
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 1384436
zbMATH Keywords: quivers
zbMATH Keywords: indecomposable representations
zbMATH Keywords: Gaussian multinomial coefficients
MaRDI profile type: Publication
full work available at URL: jabr.1999.7937
OpenAlex ID: W2024515503
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DOI: 10.1006/JABR.1999.7937
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description: scientific article; zbMATH DE number 1384436