# Periodics of Coxeter matrices for truncated Nakayama algebras

For $$n \geq 3$$ and $$r \geq 3$$ let $$C_{n,r}=(c_{i,j})$$ denote the $$n \times n$$-matrix where $$c_{i,j}=1$$ for $$j=i,\dots,i+r-1$$ (we only do this until $$i+r-1>n$$). So for example for $$n=7$$ and $$r=3$$ we obtain the matrix

$$\begin{bmatrix} 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}$$

Define $$M_{n,r}:=- C_{n,r}^{-1} C_{n,r}^T$$.

Recall that a matrix $$M$$ is called periodic in case $$M^q$$ is the identity matrix for some $$q \geq 1$$. The smallest such $$q$$ is called the period in case it exists.

Question 1: Given $$r \geq 3$$. For which $$n$$ are the $$M_{n,r}$$ periodic and what is the period in case they are periodic? What is the maximal finite period for a given $$n$$ or at least a good bound?

(a more general question was asked in What are the periodic Dyck paths? )

Background: The matrices $$M_{n,r}$$ are the Coxeter matrices of linear Nakayama algebras of the form $$A_n/J^r$$, where $$A_n$$ is the hereditary Nakayama algebra with $$n$$ simples and $$J$$ its Jacobson radical. The periods are derived invariants of those algebras. It seems to be an open problem to determine the periods, see for example page 10 in https://www.math.uni-bielefeld.de/icra2012/presentations/icra2012_lenzing.pdf for a large table and section 6 for some background of this problem https://www.sciencedirect.com/science/article/pii/S0001870813000182 .

More generally one can define the Coxeter matrix of linear Nakayama algebras as in What are the periodic Dyck paths? .

Question 2: What is the maximal period of a Coxeter matrix of a linear Nakayama algebra?

for $$n=3,\dots,9$$ the sequence starts with 4,6,8,12,18,30,16.