space of reduced positive definite quadratic forms

Question

What is the highest dimension for which the space of reduced positive definite quadratic forms (or the fundamental domain of $SL_n(\mathbb{R})/SL_n(\mathbb{Z})$) has been explicitly calculated? I know it's been done for $n \leq 7$ in 1970's. Has there been any progress since then? If not, is it because there is some sort of difficulty, or because, although the computation is very much possible in practice, people thought it's not worth the effort?

Answer 1

I very much enjoy Douglas Grenier's "Fundamental Domains for the General linear group", Pacific J. of Math., Vol 132, 2, 1988, wherein fundamental domains $F_n$ for the symmetric spaces $S_n \simeq SL(n,R) / SO(n)$ of $n \times n$ positive definite unimodular symmetric matrices are constructed inductively. That is, $F_n$ is constructed from $F_{n-1}$. The usual Minkowski reduction does not permit such an inductive description. One could therefore, in principal, see for oneself how to push the known results forward. Grenier presents an algorithm in his paper for reducing an arbitrary element in $S_n$ to his fundamental domain $F_n$, and the difficulty of pushing a fundamental domain for $S_7$ onward to $S_8$ may perhaps be phrased in terms of computing time of MAPLE or OCTAVE.

The basic element in Grenier's paper are modified Iwasawa coordinates on $S_n$. Instead of the usual Iwasawa coordinates $KAN$, where $AN$ is the identity component of an $R$-split Borel subgroup of $SL(n,R)$ and $K$ of course being $SO(n,R)$, Grenier works relative to an $R$-split maximal parabolic subgroup $P$, namely the stabilizer of a single $1$-dimensional flag. Of course one knows that $S_n$ is a homogeneous $P(R)^o$-space, and it is an interesting point to determine exactly global bijective coordinates for $S_n$ relative to a connected maximal $R$-split torus of $P(R)^o$ and the unipotent radical of $P(R)^o$.

Now my own opinion (which is negligible) is that one does not want an explicit description, even if one wants an explicit description. Any explicit description is necessarily so explicit and complicated as to be totally useless. More seriously, I don't think there is any question about the action of $SL(n,Z)$ on $S_n$ which requires a fundamental domain to be constructed. The finite presentation of $SL(n,Z)$, finite covolume, etc., can be achieved many other ways. E.g. via the Borel-Serre bordification or Siegel sets (coarse reduction theory) a la Borel/Harish-Chandra. In short, fundamental domains for $S_n$ relative to $SL(n,Z)$ are (to me) totally fictional and useless.

Answer 2

In the Lecture XV from Siegel's "Lectures on the Geometry of Numbers" is written that the volume of fundamental domain $SL_n(\mathbb{R})/SL_n(\mathbb{Z})$ is something like $$\frac{\zeta(2)\zeta(3)\ldots\zeta(n)}n.$$