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 $\mathrm{\Lambda }$, and an arbitrary vertex $i$ of its Gabriel quiver, one can define mutation ${\mu }_i(\mathrm{\Lambda })$ of $\mathrm{\Lambda }$ at vertex $i$ via silting mutation of the stalk complex $\text{La}$. Then ${\mu }_i(\mathrm{\Lambda })$ 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 ${\mu }_i^{d-2}(\mathrm{\Lambda })\cong \mathrm{\Lambda }$ (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.