### Some six-dimensional rigid forms

Mathieu Dutour, Frank Vallentin
<span title="2005-01-04">2005</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
One can always decompose Dirichlet-Voronoi polytopes of lattices non-trivially into a Minkowski sum of Dirichlet-Voronoi polytopes of rigid lattices. In this report we show how one can enumerate all rigid positive semidefinite quadratic forms (and thereby rigid lattices) of a given dimension d. By this method we found all rigid positive semidefinite quadratic forms for d = 5 confirming the list of 7 rigid lattices by Baranovskii and Grishukhin. Furthermore, we found out that for d <= 5 the
more &raquo; ... ency graph of primitive L-type domains is an infinite tree on which GL_d(Z) acts. On the other hand, we demonstrate that in d = 6 we face a combinatorial explosion.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0401191v3">arXiv:math/0401191v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/borzmbhklfgclpqh7mnadzgvhi">fatcat:borzmbhklfgclpqh7mnadzgvhi</a> </span>
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