Combinatorial methods in the representation theory of finite dimensional tame algebras - SZÁNTÓ, Csaba; SZÖLLŐSI, István (2023)
Starting from classical problems in linear algebra and experiencing a spectacular development in thelast forty years, the theory of representations of finite dimensional algebras is today one of the mostpopular and active fields in algebra. New theoretical approaches (like the use of quiver-theoreticaltechniques, the introduction of almost split sequences and Auslander-Reiten theory, the developmentof tilting theory, the notion of Gabriel-Roiter measure, the theory of Ringel-Hall algebras, the results incluster theory) and the development of new computational algebra packages and tools (like GAP andMagma) have contributed not only to the deep understanding of the structure of module categoriesover some finite dimensional algebras, but have also connected representation theory with many otherfields even outside algebra (for example quantum groups, algebraic geometry, control theory of lineardynamical systems).The present monograph records the progress made by the authors in the last fifteen years regardingtame Ringel-Hall polynomials and their various applications, using a large spectrum of combinatorial,computational and representation theoretical tools.