What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$?
$\operatorname{GL}(n,\mathbb R)$ is the group of all real invertible matrices with matrix multiplication, $\operatorname{GL}(n,\mathbb Z)$ the group of all matrices with integer entries, whose inverses also have integer entries, with matrix multiplication. $h \in \operatorname{GL}(n, \mathbb Z) \subset \operatorname{GL}(n,\mathbb R)$ acts on $g \in \operatorname{GL}(n,\mathbb R)$ by letting $h\cdot g := hg$
Remarks:
A fundamental domain $F$ is a subset of $\operatorname{GL}(n,\mathbb R)$ such that for any $x$ in $\operatorname{GL}(n,\mathbb R)$ there is exactly one $h$ in $\operatorname{GL}(n, \mathbb Z)$ such that $hx \in F$. I'm looking for an as clean as possible description of some $F$ in terms of the matrix entries. Clearly $\operatorname{GL}(n,\mathbb R)$ can be replaced with any set of (possibly not invertible) matrices with $n$ rows (possibly with few or more than n columns). The case with one column and $n=2$ is not too different from finding a fundamental domain of the upper half plane with respect to Möbius transformations. Also, $\operatorname{GL}$ could have been replaced with $\operatorname{SL}$.
This appears as a very basic question to me, and if it turns out I'm ignorant of some useful tools or theorems I will accept pointers to such. In fact this would be even better than a direct answer to the specific question (since I have many related seemingly basic questions), as long as it helps significantly in answering the specific question.