CFLat <title> </head> <body bgcolor="white" text="black" link="blue" vlink="purple" alink="red" style="font-size:12pt" <h2> CFLat - GAP package computing the canonical filtration of a lattice </h2> <p> <b>CFLat</b> is a GAP package computing the canonical filtration of an integral lattice. </p> <p> Suppose L is a lattice (a discrete subgroup of R^n). For all sublattices M of L, plot the points ( dim(M), log vol(M) ) in the xy-plane, and consider the convex hull of the plot. It is bounded below by a convex polygon. It turns out that the vertices of the polygon is represented by a unique sublattice of L, and that the sublattices representing vertices form a chain. This chain is called the <i>canonical filtration</i> of L. </p> <hr> <p> The command 'CanonicalFiltrationGram( )' accepts the Gram matrix of a lattice (positive definite square matrix) as an argument. It returns a record with the following entries.<br> < dims > - the dimensions of sublattices in the canonical filtration.<br> < dets > - the determinants of sublattices in the canonical filtration.<br> < basis > - the basis of the lattice generating the canonical filtration. <br> </p> <p> Here is a brief example. <pre> <code> <font size="-1"> gap> A := [[2,1,0,0],[1,2,0,0],[0,0,1,0],[0,0,0,3]];; Display(A); [ [ 2, 1, 0, 0 ], [ 1, 2, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 3 ] ] gap> CanonicalFiltrationGram(A); rec( dims := [ 0, 1, 3, 4 ], dets := [ 1, 1, 3, 9 ], basis := [ [ 0, 0, 1, 0 ], [ -1, 0, 1, 0 ], [ 0, -1, 0, 0 ], [ 0, 1, 0, 1 ] ] ) </font> </code> </pre> In this example, the canonical filtration of the lattice with the Gram matrix A consists of four sublattices: the zero lattice, the 1-dimensional sublattice generated by (0,0,1,0), the 3-dimensional sublattice generated by { (0,0,1,0), (-1,0,1,0), (0,-1,0,0) }, and the whole lattice. Their volumes are 1, 1, Sqrt(3), and 3. </p> <hr> <h3> Installation </h3> <p> <ul> <li>First make sure <a href="http://www.gap-system.org">GAP</a> is installed on your computer.</li> <li>Then, download <a href="CFLat-1.0.tar.gz">CFLat-1.0.tar.gz</a>. </li> <li>Extract the downloaded file in the GAP package directory. ( <code>pkg</code> directory in one of the GAP root directories. If you don't have access to the <code>pkg</code> directory of the main GAP installation, you can add private root directories by running GAP with <code> -l </code> option. ) This will create the directory <code>CFLat</code> inside the <code>pkg</code> directory.</li> <li>Now run a GAP session and type the command <code>LoadPackage("CFLat");</code> </li> </ul> </p> <hr> <h3> References </h3> <ul> <li>Youngsoo Kim, On semistability of root lattices and perfect lattices. (preprint)</li> <li>Daniel R. Grayson, Reduction theory using semistability, Comment. Math. Helv. 59 (1984), no. 4, 600–634.</li> <li>Ulrich Stuhler, Eine Bemerkung zur Reduktionstheorie quadratischer Formen, Arch. Math. (Basel) 27 (1976), no. 6, 604–610.</li> </ul> <hr> <h3> License </h3> <p> This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. <a href="http://www.gnu.org/licenses/">http://www.gnu.org/licenses/</a>. </p> <hr> </body> </html>