On Coxeter type study of non-negative posets using matrix morsifications and isotropy groups of Dynkin and Euclidean diagrams.

DOI10.1016/j.ejc.2015.02.015zbMath1318.06004OpenAlexW2014289472MaRDI QIDQ2346584

Katarzyna Zając, Daniel Simson, Marcin Gąsiorek

Publication date: 2 June 2015

Published in: European Journal of Combinatorics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.ejc.2015.02.015

zbMATH Keywords

incidence matricesfinite posetssymmetric matricesCoxeter polynomialsCoxeter spectraCoxeter-Dynkin type classificationnon-negative posetssimply laced Dynkin diagramsEuclidean diagrams

Mathematics Subject Classification ID

Symbolic computation and algebraic computation (68W30) Combinatorics of partially ordered sets (06A07) Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50) Representations of quivers and partially ordered sets (16G20) Quadratic and bilinear forms, inner products (15A63) Algebraic aspects of posets (06A11)

Related Items (15)

A Gram classification of non-negative corank-two loop-free edge-bipartite graphs ⋮ A class of highly symmetric graphs, symmetric cylindrical constructions and their spectra ⋮ A Gram classification of principal Cox-regular edge-bipartite graphs via inflation algorithm ⋮ On algorithmic Coxeter spectral analysis of positive posets ⋮ Symbolic computation of strong Gram congruences for Cox-regular positive edge-bipartite graphs with loops ⋮ On mesh geometries of root Coxeter orbits and mesh algorithms for corank two edge-bipartite signed graphs ⋮ A Coxeter spectral classification of positive edge-bipartite graphs. II: Dynkin type \(\mathbb{D}_n\) ⋮ Structure and a Coxeter-Dynkin type classification of corank two non-negative posets. ⋮ Applications of mesh algorithms and self-dual mesh geometries of root Coxeter orbits to a Horn-Sergeichuk type problem ⋮ Inflation algorithm for loop-free non-negative edge-bipartite graphs of corank at least two ⋮ On the structure of loop-free non-negative edge-bipartite graphs ⋮ A computational technique in Coxeter spectral study of symmetrizable integer Cartan matrices ⋮ A Coxeter spectral classification of positive edge-bipartite graphs. I: Dynkin types \(\mathcal{B}_n\), \(\mathcal{C}_n\), \(\mathcal{F}_4\), \(\mathcal{G}_2\), \(\mathbb{E}_6\), \(\mathbb{E}_7\), \(\mathbb{E}_8\) ⋮ A Coxeter type classification of one-peak principal posets ⋮ Congruence of rational matrices defined by an integer matrix

Uses Software

• Maple
• Python

Cites Work

• Kleiner's theorem for unitary representations of posets
• Structure and a Coxeter-Dynkin type classification of corank two non-negative posets.
• Applications of matrix morsifications to Coxeter spectral study of loop-free edge-bipartite graphs
• The problems of classifying pairs of forms and local algebras with zero cube radical are wild.
• Mesh geometries of root orbits of integral quadratic forms
• Schurian sp-representation-finite right peak PI-rings and their indecomposable socle projective modules
• Integral bilinear forms, Coxeter transformations and Coxeter polynomials of finite posets
• Socle reductions and socle projective modules
• Towards the classification of sincere weakly positive unit forms
• Congruences of a square matrix and its transpose
• Systems of subspaces of a unitary space
• The Dynkin type of a non-negative unit form.
• Numeric and mesh algorithms for the Coxeter spectral study of positive edge-bipartite graphs and their isotropy groups
• One-peak posets with positive quadratic Tits form, their mesh translation quivers of roots, and programming in Maple and Python
• Coxeter spectral classification of almost \(TP\)-critical one-peak posets using symbolic and numeric computations.
• On Algorithmic Study of Non-negative Posets of Corank at Most Two and their Coxeter-Dynkin Types
• A Framework for Coxeter Spectral Analysis of Edge-bipartite Graphs, their Rational Morsifications and Mesh Geometries of Root Orbits
• Computer Algebra Technique for Coxeter Spectral Study of Edge-bipartite Graphs and Matrix Morsifications of Dynkin Type $\mathbb{A}_n$
• A Coxeter--Gram Classification of Positive Simply Laced Edge-Bipartite Graphs
• A computation of positive one-peak posets that are Tits-sincere
• (Min, max)-equivalence of posets and nonnegative Tits forms
• Mesh Algorithms for Solving Principal Diophantine Equations, Sand-glass Tubes and Tori of Roots
• Torsionless modules over 1-gorensteinl-hereditary artinian rings
• Eigenvalues of coxeter transformations and the structure of regular componentsof an auslander-reiten quiver
• Algebras whose Euler form is non-negative
• Algorithms Determining Matrix Morsifications, Weyl orbits, Coxeter Polynomials and Mesh Geometries of Roots for Dynkin Diagrams
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