Algorithms Determining Matrix Morsifications, Weyl orbits, Coxeter Polynomials and Mesh Geometries of Roots for Dynkin Diagrams

Abstract

By computer algebra technique and computer computations, we solve the mesh morsification problems 1.10 and present a classification of irreducible mesh roots systems, for some of the simply-laced Dynkin diagrams $\rmDelta \in \{\mathbb{A}_n,\mathbb{D}_n, \mathbb{E}_6, \mathbb{E}_7, \mathbb{E}_8\}$. The methods we use show an importance of computer algebra tools in solving difficult modern algebra problems of enough high complexity that had no solution by means of standard theoretical tools only. Inspired by results of Sato [Linear Algebra Appl. 4062005, 99-108] and a mesh quiver description of indecomposable representations of finite-dimensional algebras and their derived categories explained in [London Math. Soc. Lecture Notes Series, Vol. 119, 1988] and [Fund. Inform. 1092011, 425-462] see also 5.11, given a Dynkin diagram Δ, with n vertices and the Euler quadratic form q$_\rmDelta : \mathbb{Z}^n \rightarrow \mathbb{Z}$, we study the set Mor$_\rmDelta \subseteq \mathbb{M}_{n} \mathbb{Z}$ of all morsifications of q$_\rmDelta$ [37], i.e., the non-singular matrices A $in \mathbb{M}_{n}\mathbb{Z}$ such that its Coxeter matrix Cox$_A$ := -A · A$^{-tr}$ lies in Gln, \mathbb{Z} and q$_{\rmDelta}$ v = v · A · v$^{tr}$, for all v $\in \mathbb{Z}n$. The matrix Weyl group \mathbb{W}$_\rmDelta$ 2.13 acts on Mor$_\rmDelta$ and the determinant detA $\in$ \mathbb{Z}, the order cA $\ge2$ of CoxA i.e. the Coxeter number, and the Coxeter polynomial cox$_A$ t := dett ·E-Cox$_A$ $\in$ \mathbb{Z}[t] are $\mathbb{W}_\rmDelta$-invariant. Moreover, the finite set $R_{q\rmDelta} = \{v \in \mthbb{Z}^n; q_\rmDelta v = 1\}$ of roots of q$_\rmDelta$ is Cox$_A$-invariant. The following problems are studied in the paper: a determine the $\mathbb{W}_\rmDelta$-orbits \cal{Orb}A of Mor$_\rmDelta$ and the set $\cal{CPol}_\rmDelta = \{cox_{A}t; A \in Mor_\rmDelta\}$, b construct a finite minimal Cox$_A$-mesh quiver in $\mathbb{Z}^n$ containing all Cox$_A$-orbits of the finite set $R_{q\rmDelta}$ of roots of q$_\rmDelta$;. We prove that \cal{CPol}$_\rmDelta$ is a finite set and we construct algorithms allowing us to solve the problems for the morsifications $A = [a_{ij}] \in Mor_\rmDelta$, with $|a_{ij}| \le 2$. In this case, by computer algebra technique and computer computations, we prove that, for $n \le 8$, the number of the $\mathbb{W}_\rmDelta$-orbits \cal{Orb}A is at most 6, $s_\rmDelta := |\cal{CPol}_\rmDelta| \le 9$ and, given A,A' $\in$ Mor$_\rmDelta$ and $n \le 7$, the following three conditions are equivalent: i A' = $B^{tr}$ · A · B, for some B $\in$ Gln, \mathbb{Z}, ii cox$_{A}$t = cox$_{A'}$ t, and iii cA · det A = c$_{A'}$ · det A'. We also show that s$_{\rmDelta}$ equals 6, 5, and 9, if $\rmDelta$ is the diagram $\mathbb{E}_6$, $\mathbb{E}_7$, and $\mathbb{E}_8$, respectively.