Some six-dimensional rigid forms

TL;DR

This paper develops a method to enumerate all rigid positive semidefinite quadratic forms in a given dimension, confirming previous results for dimension 5 and exploring the structure of L-type domains.

Contribution

It introduces a systematic enumeration technique for rigid quadratic forms and analyzes the structure of L-type domain adjacency graphs across dimensions.

Findings

Confirmed the list of 7 rigid lattices in dimension 5

Found the adjacency graph of primitive L-type domains is an infinite tree for dimensions up to 5

Demonstrated combinatorial explosion in dimension 6

Abstract

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 adjacency 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.