Algorithmic computation of principal posets using Maple and Python

Abstract

We present symbolic and numerical algorithms for a computer search in the Coxeter spectral classification problems. One of the main aims of the paper is to study finite posets \(I\) that are principal, i.e., the rational symmetric Gram matrix \(G_I : = \frac{1}{2}[C_I+ C^{tr}_I]\in\mathbb{M_I(\mathbb{Q})}\) of \(I\) is positive semi-definite of corank one, where \(C_I\in\mathbb{M}_I(\mathbb{Z})\) is the incidence matrix of \(I\). With any such a connected poset $I$, we associate a simply laced Euclidean diagram \(DI\in \{\widetilde{\mathbb{A}}_n, \widetilde{\mathbb{D}}_n, \widetilde{\mathbb{E}}_6, \widetilde{\mathbb{E}}_7, \widetilde{\mathbb{E}}_8\}\), the Coxeter matrix \(\mbox{ Cox}_I:= - C_I\cdot C^{-tr}_I\), its complex Coxeter spectrum \(\mathbf{specc}_I\), and a reduced Coxeter number \(\check{\mathbf{c}}_I\). One of our aims is to show that the spectrum \(\mathbf{specc}_I\) of any such a poset \(I\) determines the incidence matrix \(C_I\) (hence the poset \(I\)) uniquely, up to a \(\mathbb{Z}\)-congruence. By computer calculations, we find a complete list of principal one-peak posets \(I\) (i.e., \(I\) has a unique maximal element) of cardinality \(\leq 15\), together with \(\mathbf{specc}_I\), \(\check{\mathbf{c}}_I\), the incidence defect \(\partial_I:\mathbb{Z}^I \to\mathbb{Z}\), and the Coxeter-Euclidean type \(DI\). In case when  \(DI\in \{\widetilde{\mathbb{A}}_n, \widetilde{\mathbb{D}}_n, \widetilde{\mathbb{E}}_6, \widetilde{\mathbb{E}}_7, \widetilde{\mathbb{E}}_8\}\) and \(n:=|I|\) is relatively small, we show that given such a principal poset \(I\), the incidence matrix \( C_I\) is \(\mathbb{Z}\)-congruent with the non-symmetric Gram matrix \( \check G_{DI}\) of \(DI\), \(\mathbf{specc}_I = \mathbf{specc}_{DI}\) and \(\check{\mathbf{c}} _I= \check{\mathbf{c}}_{DI}\). Moreover, given a pair of principal posets \(I\) and \(J\), with \(|I|= |J| \leq 15\), the matrices \(C_I\) and \(C_J\) are \(\mathbb{Z}\)-congruent if and only if \(\mathbf{specc}_I=\mathbf{specc}_J\).