Iterated mutations of symmetric periodic algebras

TL;DR

This paper investigates the properties of iterated mutations of symmetric periodic algebras, establishing that for a vertex with period d, the mutation operation has order d-2 under certain conditions, revealing deep algebraic symmetries.

Contribution

It introduces a new understanding of the order of mutation operations on symmetric periodic algebras, extending previous methods to a specific algebraic context.

Findings

Mutation at a d-periodic vertex has order d-2 under certain conditions.

Mutations preserve symmetry and can be iterated multiple times.

Examples suggest the property may hold more generally without restrictions.

Abstract

Following methods used by A. Dugas for investigating derived equivalent pairs of (weakly) symmetric algebras, we apply them in a specific situation, obtaining new deep results concerning iterated mutations of symmetric periodic algebras. More specifically, for any symmetric algebra $\Lambda$, and an arbitrary vertex $i$ of its Gabriel quiver, one can define mutation $\mu_i(\Lambda)$ of $\Lambda$ at vertex $i$ via silting mutation of the stalk complex $\La$. Then $\mu_i(\Lambda)$ is again symmetric, and we can iterate this process. We want to understand the order of $\mu_i$, in case the vertex $i$ is $d$-periodic, i.e. the simple module $S_i$ associated to $i$ is periodic of period $d$ (with respect to the syzygy). The main result of this paper shows that then $\mu_i$ has order $d-2$, that is $\mu_i^{d-2}(\Lambda)\cong\Lambda$ (modulo socle), under some additional assumption on the (periodic) projective-injective resolution of $S_i$. Besides, we present briefly some consequences concerning arbitrary periodic vertex and give few sugestive examples showing that this property should hold in general, i.e. without restrictions on the periodic projective resolution.