Iterated mutations of symmetric periodic algebras

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.