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P. Caldero, F. Chapoton (2004)
Cluster algebras as Hall algebras of quiver representationsCommentarii Mathematici Helvetici, 81
P. Webb (1997)
REPRESENTATION THEORY OF ARTIN ALGEBRAS (Cambridge Studies in Advanced Mathematics 36) By Maurice Auslander, Idun Reiten and Sverre O. Smalø: 423 pp., £50.00, ISBN 0 521 41134 3 (Cambridge University Press, 1995).Bulletin of The London Mathematical Society, 29
I. Assem, D. Simson, A. Skowroński (2006)
Techniques of representation theory
C. Ringel (2005)
Foundation of the Representation Theory of Artin Algebras, Using the Gabriel-Roiter Measure
C. Ringel (2005)
The Gabriel–Roiter measureBulletin Des Sciences Mathematiques, 129
M. Auslander, I. Reiten, S. Smalø (1995)
Representation Theory of Artin Algebras: Notation
C. Ringel (1985)
Tame Algebras and Integral Quadratic Forms
Steven Sam (2009)
The Caldero-Chapoton formula for cluster algebras
Christine Riedtmann (1994)
Lie Algebras Generated by IndecomposablesJournal of Algebra, 170
Bo Chen (2010)
The Gabriel-Roiter submodules of simple homogeneous modules, 138
Andrew Hubery (2007)
Hall polynomials for affine quiversRepresentation Theory of The American Mathematical Society, 14
C. Ringel (1998)
Exceptional modules are tree modulesLinear Algebra and its Applications
GAP-groups, algorithms, and programming
Bo Chen (2008)
Comparison of Auslander–Reiten Theory and Gabriel–Roiter Measure Approach to the Module Categories of Tame Hereditary AlgebrasCommunications in Algebra, 36
I. MacDonald (1979)
Symmetric functions and Hall polynomials
D. Simson, A. Skowroński (2007)
Elements of the Representation Theory of Associative Algebras
V. Dlab, C. Ringel (1976)
Indecomposable Representations of Graphs and Algebras
I. Bernstein, I. Gelfand, V. Ponomarev (1973)
COXETER FUNCTORS AND GABRIEL'S THEOREMRussian Mathematical Surveys, 28
Csaba Szántó (2006)
Hall Numbers and the Composition Algebra of the Kronecker AlgebraAlgebras and Representation Theory, 9
C. Ringel (1990)
Hall polynomials for the representation-finite hereditary algebrasAdvances in Mathematics, 84
C. Ringel (2006)
The Theorem of Bo Chen and Hall PolynomialsNagoya Mathematical Journal, 183
Let k be an arbitrary field and Q be an acyclic quiver of tame type (that is, of type A˜n,D˜n,E˜6,E˜7,E˜8). Consider the path algebra kQ, the category of finite‐dimensional right modules mod-kQ, and the minimal positive imaginary root of Q, denoted by δ. In the first part of the paper, we deduce that the Gabriel–Roiter (GR) inclusions in preprojective indecomposables and homogeneous modules of dimension δ, as well as their GR measures are field independent (a similar result due to Ringel being true in general over Dynkin quivers). Using this result, we can prove in a more general setting a theorem by Bo Chen which states that the GR submodule P of a homogeneous module R of dimension δ is preprojective of defect -1 and so the pair (R/P,P) is a Kronecker pair. The generalization consists in considering the originally missing case E˜8 and using arbitrary fields (instead of algebraically closed ones). Our proof is based on the idea of Ringel (used in the Dynkin quiver context) of comparing all possible Hall polynomials with the special form they take in case of a GR inclusion. For this purpose, we determine (with the help of a program written in GAP) a list of tame Hall polynomials which may have further interesting applications.
Bulletin of the London Mathematical Society – Wiley
Published: Apr 1, 2015
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