NDLI: On Coxeter Type Classification of Loop-Free Edge-Bipartite Graphs and Matrix Morsifications

Content Provider	IEEE Xplore Digital Library
Author	Bocian, R. Felisiak, M. Simson, D.
Copyright Year	2013
Description	Author affiliation: Fac. of Math. & Comput. Sci., Nicolaus Copernicus Univ., Toruń, Poland (Bocian, R.; Felisiak, M.; Simson, D.)
Abstract	We continue and complete a Coxeter spectral study (presented in our talk given in SYNASC11 and SYNASC12) of the root systems in the sense of Bourbaki, the mesh geometries $Γ(R_{Δ},$ $Φ_{A})$ of roots of Δ in the sense of [J. Pure Appl. Algebra, 215 (2010), 13-34], and matrix morsifications A ∈ $Mor_{Δ},$ for simply laced Dynkin diagrams Δ ∈ $\{A_{n},$ $D_{n},$ $E_{6},$ $E_{7},$ $E_{8}\}.$ Here we report on algorithmic and morsification technique for the Coxeter spectral analysis of connected loop-free edge-bipartite graphs Δ with n ≥ 2 vertices by means of the Coxeter matrix $Cox_{Δ}$ ∈ $M_{n}(Z),$ the Coxeter spectrum $specc_{Δ},$ and an inflation algorithm associating to any connected loop-free positive bigraph Δ a simply laced Dynkin diagram DΔ, and defining a Z-congruence of the symmetric Gram matrices $G_{Δ}$ and $G_{DΔ}.$ We also present a computer aided technique that allows us to construct a Z-congruence of the non-symmetric Gram matrices $Ğ_{Δ}$ and $Ğ_{Δ'},$ if the Coxeter spectra $specc_{Δ}$ and $specc_{Δ'}$ coincide. A complete Coxeter spectral classification of positive edge-bipartite graphs Δ of Coxeter-Dynkin types DΔ ∈ $\{A_{n},$ $D_{n},$ $E_{6},$ $E_{7}\},$ with n ≤ 7, is obtained by a reduction to computer calculation of Gl(n, $Z)_{DΔ}-orbits$ in the set $Mor_{DΔ},$ where Gl(n, $Z)_{DΔ}$ is the isotropy group of the Dynkin diagram DΔ.
Starting Page	115
Ending Page	118
File Size	267298
Page Count	4
File Format	PDF
ISBN	9781479930357
e-ISBN	9781479930364
DOI	10.1109/SYNASC.2013.22
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2013-09-23
Publisher Place	Romania
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Algorithm design and analysis Coxeter polynomial Matrix morsification Symmetric matrices Zinc Spectral analysis Geometry Coxeter spectrum Computer algorithm Algebra Mesh geometry of roots Edge-bipartite graph Dynkin diagram Polynomials
Content Type	Text
Resource Type	Article

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