Let $A$ be a representation-finite selfinjective (quiver) algebra, that we assume to be connected and non-semisimple. Define the Coxeter period $p_A$ of $A$ to be equal to the period of the Coxeter matrix of the Auslander algebra of $A$ in case it exists. Call $A$ Coxeter periodic in case it has finite Coxeter period.
Recall here that a matrix $M$ is periodic if $M^k$ is the identity matrix and the smallest such $k$ is called the period of $M$.
Note that two derived equivalent such algebras $A_1$ and $A_2$ have equal Coxeter period, since their Auslander algebras are derived equivalent.
Question: Is it true that all symmetric representation-finite algebras are Coxeter periodic? What is their period?
The classification of such algebras up to derived equivalence is very easy and it seems the question can be answered by force. The answer should be yes, but I wonder whether there is a nice reason avoiding ugly calculations.
The Coxeter period of the algebras $K[x]/(x^n)$ is equal to 2. I can prove that the Coxeter period of The trivial extension of type A_n quiver algebras is lcm(n,2). The Coxeter periodc for the trivial extension of type $E_6,E_7$ and $E_8$ is 22,34,58 respectively. For the trivial extension of Dynkin type $D_n$ the periods start for $n \geq 4$ with 10,14,18,22,26 and it seems to be https://oeis.org/A016825 . General symmetric Nakayama algebras also seem to have period lcm(n,2). The periods for the penny-farthing algebras starts with 6,10,14 and might be the same as for Dynkin type $D_n$ but shifted(indicating that the problem depends on the Auslander-Reiten type directly).
Question 2: When is a general representation-finite selfinjective algebra Coxeter periodic and what is the period in this case?
Not all representation-finite selfinjective algebras are Coxeter periodic, for example the Nakayama algebra with $n$ simples such that all indecomposable projectives have dimension $n$ is not Coxeter periodic. On the other hand the non-symmetric selfinjective Nakayama algebra with Kupisch series [2,2,2] has Coxeter period 6.
