Tuesday, April 30, 14:00, 7.527

Csaba Szántó (Cluj)
Tame Ringel-Hall polynomials and applications

Abstract:
Classical Hall algebras associated with discrete valuation rings were introduced by Steinitz and Hall to provide an algebraic approach to the classical combinatorics of partitions. The multiplication is given by Hall polynomials which play an important role in the representation theory of the symmetric groups and the general linear groups. In 1990 Ringel defined Hall algebras for a large class of rings, namely finitary rings, including in particular path algebras of quivers over finite fields. These Ringel-Hall algebras provided a new approach to the study of quantum groups using the representation theory of finite dimensional algebras. They can also be used successfully in the theory of cluster algebras or to investigate the structure of the module category.

In case of Ringel-Hall algebras corresponding to Dynkin quivers and tame quivers we know due to Ringel and Hubery, that the structure constants of the multiplication are again polynomials in the number of elements of the base field. These are the Ringel-Hall polynomials. If we are looking at Hall polynomials associated to indecomposable modules, the classical ones are just 0 or 1, the Ringel-Hall ones in the Dynkin case are also known and have degree up to 5. However we did not have too much information about the Ringel-Hall polynomials in the tame cases.

In the first part of the talk I will present the main tools for the determination of tame Ringel-Hall polynomials associated to indecomposables and list a part of them. The second part of the talk is dedicated to the application of these polynomials in Gabriel-Roiter measure theory and in counting cardinalities of quiver Grassmannians.