Periodicity shadows I: A new approach to combinatorics of periodic algebras

TL;DR

This paper introduces the concept of periodicity shadows as a new combinatorial tool for analyzing the structure of Gabriel quivers in generalized quaternion type algebras, revealing their global shape and cycle placement rules.

Contribution

It defines the notion of periodicity shadow and demonstrates its application in describing Gabriel quivers of algebras of generalized quaternion type.

Findings

Gabriel quivers are obtained from basic shadows by attaching 2-cycles.

Positions of 2-cycles are restricted by specific rules.

The main theorem describes the global shape of these quivers.

Abstract

This article is devoted to introduce a new notion of periodicity shadow, which appeared naturally in the study of combinatorics of tame symmetric algebras of period four, or more generally, algebras of generalized quaternion type. For any such an algebra $\La$, we consider its shadow $\bS_\La$, which is the (signed) adjacency matrix of the Gabriel quiver of $\La$. Studying properties of shadows $\bS_\La$ leads us to the definition of the periodicity shadow, which is basically, a skew-symmetric integer matrix satisfying certain set of conditions motivated by the properties of shadows $\bS_\La$. This turned out to be a very useful tool in describing the combinatorics of Gabriel quivers of algebras of generalized quaternion type, not only for algebras with small Gabriel quivers (i.e. up to $6$ vertices), which it was originally desined for. In this paper, we introduce and briefly discuss this notion and present one of its theoretical applications, which shows how significant it is. Namely, the main result of this paper describes the global shape of the Gabriel quivers of algebras of generalized quaternion type, as quivers obtained from some basic shadows by attaching $2$-cycles, and moreover, postion of the $2$-cycles is restricted by precise rules (see the Main Theorem). Computational aspects are reported in the second part.