Affiliations: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland. kanies@mat.umk.pl; justus@mat.umk.pl; pmalicki@mat.umk.pl; lielow@mat.umk.pl
Note: [] Address for correspondence: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
Abstract: We construct a horizontal mesh algorithm for a study of a special type of mesh root systems of connected positive loop-free edge-bipartite graphs Δ, with n ≥ 2 vertices, in the sense of [SIAM J. Discrete Math. 27 (2013), 827–854] and [Fund. Inform. 124 (2013), 309-338]. Given such a loop-free edge-bipartite graph Δ, with the non-symmetric Gram matrix $\Gcaron_ \Delta \isin \mathbb{M}_n(\mathbb{Z})$ and the Coxeter transformation $\Phi_A : \mathbb{Z}^n \rarr \mathbb{Z}^n$ defined by a quasi-triangular matrix morsification $A \isin \mathbb{M}_n(\mathbb{Z})$ of Δ satisfying a non-cycle condition, our combinatorial algorithm constructs a ΦA-mesh root system structure $\Gamma(\Rscr_\Delta, \Phi_A)$ on the finite set of all ΦA-orbits of the irreducible root system $\Rscr_\Delta := {v \isin \mathbb{Z}^n; v \cdot \Gcaron_\Delta \cdot v^{tr} = 1}$. We apply the algorithm to a graphical construction of a ΦI - mesh root system structure $\Gamma(\Rscr_I, \Phi_I)$ on the finite set of ΦI -orbits of roots of any poset I with positive definite Tits quadratic form $\qIcirc : \mathbb{Z}_I \rarr \mathbb{Z}$.
