Periodicity Shadows II: Computational Aspects

This article is the second part of the research project initiated last year, in which we introduced and investigated so-called periodicity shadows, i.e. special skew-symmetric integer matrices related to symmetric algebras with periodic simple modules. These matrices provide a new tool for describing the structure of quivers arising in the problem classification of all tame symmetric algebras of period four, an important class of algebras with various links to different branches of algebra. In the first part of the project, we focused on the theoretical aspects of this notion setting a general framework, and we obtained a few nice properties of these quivers. The part of the project described in this paper is devoted to complementary considerations concerning computational issues. We present here an algorithm for computing all tame periodicity shadows of a given size, briefly discuss the output for small sizes, i.e., up to 7×7, and provide some graphical visualizations. We also mention that by applying the computations for sizes up to 5×5, we obtained a classification of tame symmetric algebras of period four defined by quivers with at most 5 vertices.