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Published in:
2015 | OriginalPaper | Chapter
8. Combinatorial Invariants
Author : Michael Barot
Published in: Introduction to the Representation Theory of Algebras
Publisher: Springer International Publishing
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Abstract
The reduction from modules to dimension vectors in the knitting can not only be seen as a loss of information but also as a gain in simplicity. Several useful tools are presented for working with dimension vectors: The first of them, called Grothendieck group, is based on homological algebra. The second linearizes the Auslander-Reiten translation as an action on the Grothendieck group and is called Coxeter transformation. The third is a quadratic form with integer coefficients. It plays a crucial role for characterizing the dimension vector of indecomposable modules. The combinatorial description of those dimension vectors is resumed in the definition of a root.