Let $M=- U^{-1} U^T$ be an $n \times n$-integer matrix, where $U$ is an upper triangular 0-1-matrix where all diagonal entries are equal to one. $M$ is called periodic if $M^r=id$ for some $r \geq 1$. The question is about whether there is a fast finite test to check whether a matrix is periodic, so that one just has to look for "small" $r$. See Distributive lattices with periodic Coxeter matrix for a motivation.

Question 1: Given $n$, is there a minimal number $a(n)$ such that if $M^r \neq id$ for all $r=1,...,a(n)$, then $M$ is not periodic?

Question 2: Given $n$, is there a minimal number $b(n)$ such that if $M$ has an entry of absolute value at least $b(n)$, then $M$ is not periodic?

If yes, what are good bounds for $a(n)$ and $b(n)$?

Of course one might ask those questions also for integer matrices $M$ with more general properties.

Let $M=- U^{-1} U^T$ be an $n \times n$-integer matrix, where $U$ is an upper triangular 0-1-matrix where all diagonal entries are equal to one. $M$ is called periodic if $M^r=id$ for some $r \geq 1$. The question is about whether there is a fast finite test to check whether a matrix is periodic, so that one just has to look for "small" $r$. See Distributive lattices with periodic Coxeter matrix for a motivation.

Question 1: Given $n$, is there a good bound for the minimal number $a(n)$ such that if $M^r \neq id$ for all $r=1,...,a(n)$, then $M$ is not periodic?

Question 2: Given $n$, is there a good bound for the minimal number $b(n)$ such that if $M$ has an entry of absolute value at least $b(n)$, then $M$ is not periodic?

Of course one might ask those questions also for integer matrices $M$ with more general properties. (there it does not work for question 2 as Gerry Myerson showed)