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Consider a lattice in R^3. Is the some "canonical" way or ways to choose basis in it ?

I mean in R^2 we can choose a basis |h_1| < |h_2| and |(h_2, h_1)| < 1/2 |h_1|. Considering lattices with fixed determinant and up to unitary transformations we get standard picture of the PSL(2,Z) acting on the upper half plane, which has a fundamental domain Im (tau)>1 Re(tau) <1/2.

What are the similar results for other small dimensions R^3, R^4, C^4, C^8 ? What are the algorithms to find such a lattice reductions ?

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In higher dimensions, there doesn't seem to be anything as nice as in two dimensions: the fundamental domains get substantially more complicated and the algorithms become much less efficient. However, there are still some beautiful results. For example, Minkowski reduction is a natural generalization of the two-dimensional case. See, for example, Chapter 2 of Computational geometry of positive definite quadratic forms by Achill Schürmann. Minkowski reduction defines a fundamental domain, which is in fact a polyhedral cone in the space of positive-definite matrices, but the facets of this cone are known only up through seven dimensions. (It is most naturally defined using infinitely many constraints, only finitely many of which are needed in any given dimension, but figuring out exactly which ones are needed is difficult.)

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  • $\begingroup$ Thank You ver y much ! How and why positive definite matrices comes into the game ? I guess we do "polar decomposition" M = Q * P, Q - is orthogonal, P - positive definite. Is it correct ? What will be the action of sl(n,z) on P ? Does it preserve some natural structures on P ? Why not to consider "QR" decomposition: R is upper triangular ? $\endgroup$
    – Alexander Chervov
    Commented Apr 17, 2011 at 10:39
  • $\begingroup$ I actually meant the Gram matrix for a basis of the lattice, so it is both positive definite and symmetric. A change of basis matrix $U \in \textup{GL}_n(\mathbb{Z})$ acts on a Gram matrix $M$ by sending it to $UMU^t$. There are a couple of advantages of using these coordinates. One is that passing to the Gram matrix automatically mods out by the orthogonal group, and the other is that some constraints that are nonlinear in terms of a basis matrix become linear in terms of a Gram matrix. (For example, vector lengths in the lattice are linear functions of the Gram matrix entries.) $\endgroup$
    – Henry Cohn
    Commented Apr 17, 2011 at 12:46
  • $\begingroup$ Wow, indeed it is more simple that I thought. Thank You ! If have some time would You be so kind to look at : mathoverflow.net/questions/62116/… $\endgroup$
    – Alexander Chervov
    Commented Apr 18, 2011 at 12:45
  • $\begingroup$ Minkowski reduction is also described in "Rational Quadratic Forms" (By J. W. S. Cassels, Ch. 12). Some interesting detailes are given in "On the theory of construction of the Minkowski reduction domain" (by S. S. Ryshkov, M. J. Cohn). $\endgroup$
    – Alexey Ustinov
    Commented Dec 13, 2013 at 13:09
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The book by Terras "Harmonic analysis on symmetric spaces and applications" volume 2 has some stereoscopic pictures of the fundamental domains for some similar groups.

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