# Connection between Coxeter-periodicity and periodicity of the trivial extension

Let $$A$$ be a finite dimensional quiver algebra with an acyclic quiver. There are two notions related to periodicity for an algebra. Call $$A$$ Coxeter-periodic in case the Coxeter matrix $$\phi_A$$ of $$A$$ is periodic, meaning that $$\phi_A^r=id$$ for some $$r \geq 1$$. On the other hand, call $$A$$ T-periodic in case the trivial extension algebra $$T(A)$$ of $$A$$ is a periodic algebra, meaning that $$T(A)$$ is a periodic module as a $$T(A)$$-bimodule.

For Nakayama algebras with a linear quiver and at most 7 simple modules the following seems to be true (tested with the computer to a good precision): $$A$$ is Coxeter-periodic iff it is $$T$$-periodic. (So it is tested for 196 Nakayama algebras, among which 18 were not Coxeter-periodic)

Question:

Is this just random or is there a deeper connection for special classes of acyclic algebras?

Note that such a connection would be very interesting, since testing Coxeter-periodicty can be done in seconds with the computer while testing T-periodicity is impossible for most algebras with the computer today (it takes too long).

Note that we have to restrict to acyclic quiver algebras (and probably restrict even more), since the Nakayama algebra with Kupisch series [2,3] is Coxeter-periodic but not T-periodic.