4
$\begingroup$

$\DeclareMathOperator\GL{GL}$Consider the unimodular group $\GL_n(\mathbb{Z})$, consisting of integral matrices $A \in \mathbb{Z}^{n \times n}$ such that that $\det(A) =\pm 1$.

It is well known that any $A \in \GL_n(\mathbb{Z})$ can be written as a product of signed permutation matrices and 'Gauss moves', the latter referring to matrices that have 1 on the diagonal and at most one non-zero entry away from the diagonal. This is proven, for example, in M. Newman's book 'Integral Matrices' (Theorem II.7).

My question is: is there an upper bound (depending on $n$, but independent of $A$) on the number of moves required? If not, can you provide a sequence of matrices that require an increasing number of factors in the above decomposition?

Thank you in advance for your help.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
11
$\begingroup$

They’re usually called “elementary matrices”, not Gauss moves. Your question is equivalent to asking whether the integer special linear group is boundedly generated by elementary matrices. The answer is no for $n=2$ (an easy exercise using the free product with amalgamation description of the group in that case), but amazingly it is “yes” for larger $n$. This is a theorem of Carter, Keller, and Paige. See here.

Improve this answer
$\endgroup$
2
  • $\begingroup$ Thank you very much, this addresses it. Do you know whether explicit bounds were derived for the number of elementary matrices required? In the paper you mention, it says that no explicit bounds are given unfortunately since some of the arguments used are implicit. Do you know if they are derived somewhere else? $\endgroup$
    – gm01
    Commented Aug 30, 2023 at 12:55
  • 3
    $\begingroup$ @gm01: I didn’t know the history very well, so I went and looked at the mathscinet references and discovered that the case you care about was originally done by Carter and Keller with an explicit bound (though a much uglier and more computational proof). See the paper Carter, David (1-VA); Keller, Gordon (1-VA) Elementary expressions for unimodular matrices. Comm. Algebra 12 (1984), no. 3-4, 379–389. $\endgroup$
    – Andy Putman
    Commented Aug 30, 2023 at 13:37

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .