Experiences in Computing Mesh Root Systems for Dynkin Diagrams Using Maple and C++

Felisiak, Mariusz; Simson, Daniel

doi:10.1109/synasc.2011.41

Cited by 8 publications

(18 citation statements)

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“…Here we continue and complete our Coxeter spectral study of the mesh geometries of roots and the matrix morsifications we reported in our talk given in SYNASC11, Timisoara, September 2011, see [6]. Let us start by recalling from [6] a terminology, notation and main problems of our study.…”

Section: Introductionmentioning

confidence: 72%

“…One of the main aims of the talk is to present a progress obtained during the last year in solving the problems stated in our talk [6]. We show that the reduced mesh root systems and mesh geometries of roots for each of the Dynkin diagrams Δ can be classified for n ≤ 9 by applying symbolic computer algebra computations and numeric algorithmic computations.…”

mentioning

confidence: 93%

“…We keep the terminology and notation introduced in [6]. We denote by N the set of non-negative integers, by Z the ring of integers, and by Q the rational number field.…”

Section: Introductionmentioning

confidence: 99%

“…A motivation for the study of mesh root systems, mesh geometries of roots, matrix morsifications, and their connections with some Diophantine geometry problems and the Hilbert's tenth problem (see [14]) is given in [6] and [20]- [21]. We recall that root systems and Dynkin diagrams play an important role in several branches of mathematics including Lie theory, algebraic geometry, group theory, geometry, singularity theory, and representation theory, see [1], [9], [12], [18], [25], [26].…”

Section: Introductionmentioning

confidence: 99%

“…Module categories and their derived categories D b (R) are studied by means of their AR-quivers and Φ-mesh root systems in the sense of [20], i.e., root systems R ⊆ Z n equipped with a Φ-mesh geometry structure Γ(R, Φ) defined by a non-trivial group automorphism Φ : Z n → Z n . This means that the root system R is a Φ-invariant subset of Z n , and the Φ-orbits in R form a Φ-translation quiver (oriented graph) consisting of Φ-meshes, see [6] and [20] (see also [1]- [3], [9]- [13], [15], [17], [24]- [26] for more details).…”

Section: Introductionmentioning

confidence: 99%

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On Computing Mesh Root Systems and the Isotropy Group for Simply-laced Dynkin Diagrams

Felisiak

Simson

2012

2012 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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We continue and complete a Coxeter spectral study (presented in our talk given in SYNASC11, Timisoara, September 2011 [6]) of the root systems in the sense of Bourbaki [4], the mesh geometries Γ(RΔ, ΦA) of roots of Δ in the sense of [20], and matrix morsifications A ∈ MorΔ, for simply-laced Dynkin diagrams Δ ∈ {An, Dn, E6, E7, E8}. One of the main aims of the talk is to present a progress obtained during the last year in solving the problems stated in our talk [6]. We show that the reduced mesh root systems and mesh geometries of roots for each of the Dynkin diagrams Δ can be classified for n ≤ 9 by applying symbolic computer algebra computations and numeric algorithmic computations. We also compute the isotropy group Gl(n, Z)Δ of Δ defined in [22]-[23], determine its structure and compute the Gl(n, Z)Δ-orbits of the morsifications A ∈ MorΔ, for Δ ∈ {An, Dm}, with n ≤ 9 and 4 ≤ m ≤ 8. Results of our computing experiences are presented in three tables of Section 4.

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Section: Introductionmentioning

confidence: 72%

mentioning

confidence: 93%

“…We keep the terminology and notation introduced in [6]. We denote by N the set of non-negative integers, by Z the ring of integers, and by Q the rational number field.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On Computing Mesh Root Systems and the Isotropy Group for Simply-laced Dynkin Diagrams

Felisiak

Simson

2012

2012 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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On Coxeter Spectral Study of Posets and a Digraph Isomorphism Problem

Gąsiorek

Simson

Zając

2012

2012 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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Following the spectral graph theory and algebraic techniques in graph theory, we continue a Coxeter spectral study of finite posets and edge-bipartite graphs (or signed graphs in the sense of Harary and Zaslavsky). A connection between properties of the Coxeter spectrum specc J ⊆ C and digraph isomorphism problem for Hasse digraphs of positive and non-negative posets J is also studied. In particular, we study in details a class of posets J with a non-negativity condition, in connection with the Coxeter spectral properties of the simply-laced Euclidean diagrams { Dn, n ≥ 4, E6, E7, E8}. We show that symbolic and numerical computer calculations in Python, C and Linux tools allow us to present a complete classification of these posets J, with at most 15 points, by means of their Coxeter spectra specc J . The main classification ideas and the algorithms used in the classification are presented in Sections 4 and 6. We end the paper by showing how our poset classification results apply to the isomorphism problem for a special class of digraphs. I. INTRODUCTIONFollowing the spectral graph theory problems, a graph coloring techniques, and algebraic methods in graph and poset theory studied in [1], [3], [4], [11], [15], [16], we continue a Coxeter spectral study of finite posets and edge-bipartite graphs [5], [7], [16]-[22] (or signed graphs in the sense of Harary [9] and Zaslavsky [23]). We study a connection between properties of the Coxeter spectrum specc J ⊆ C and the digraph isomorphism problem for Hasse digraphs of positive and non-negative posets J, compare with [1] and [3]. Our main inspiration for the study comes from the representation theory of posets and digraphs, Lie theory, Diophantine geometry problems, and combinatorics, see [16] and [18]. For further motivation of the study the reader is referred to Section 5 and to the articles [5]-[7], [13], [14], [17]-[20], and [22].A natural way to apply spectral graph theory to directed graphs (digraphs) is to assign to any digraph D = (V D , E D ) the adjacency matrix Ad D = [a ij ] ∈ M n (Z), where n = |V D | and we set a ij = 1 if (i, j) ∈ E D and 0 otherwise. We denote by F AdD (t) := det(t · E − Ad D ) ∈ Z[t]

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