Timeline for Lattice reduction in R^3 (R^4) or what is fundamental domain for SL(3,Z) , (SL(4,Z)) ?
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5 events
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| Dec 13, 2013 at 13:09 | comment | added | Alexey Ustinov | 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). | |
| Apr 18, 2011 at 12:45 | comment | added | Alexander Chervov | 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/… | |
| Apr 17, 2011 at 12:46 | comment | added | Henry Cohn | 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.) | |
| Apr 17, 2011 at 10:39 | comment | added | Alexander Chervov | 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 ? | |
| Apr 16, 2011 at 18:01 | history | answered | Henry Cohn | CC BY-SA 3.0 |