# Finite fast tests for periodicity of certain matrices

MathOverflow Asked on January 12, 2021

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)

For general integer matrices $$M$$, there is no $$b(n)$$. E.g., for $$n=2$$, if $$a(a+1)+1=bc$$, and $$M=pmatrix{a&-bcr c&-a-1cr}$$ then $$M^3$$ is the identity matrix.

EDIT: for general matrices, Theorem 2.7 of James Kuzmanovich and Andrey Pavlichenkov, Finite groups of matrices whose entries are integers, The American Mathematical Monthly Vol. 109, No. 2 (Feb., 2002), pp. 173-186 shows there's a bound on $$a(n)$$ in terms of $$n$$. Let $$m=p_1^{e_1}p_2^{e_2}cdots p_t^{e_t}$$ with $$p_1. Then there is an $$ntimes n$$ integer matrix with order $$m$$ if and only if

1. $$sum_{i=1}^t(p_i-1)p_i^{e_i-1}-1le n$$ for $$p_1^{e_1}=2$$, or

2. $$sum_{i=1}^t(p_i-1)p_i^{e_i-1}le n$$ otherwise.

Answered by Gerry Myerson on January 12, 2021

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