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By applying computer algebra tools (mainly, Maple and C++), given the Dynkin diagram ∆ = An, with n > 2 vertices and the Euler quadratic form q∆ : Zn → Z, we study the problem of classifying mesh root systems and the mesh geometries of roots of A (see Section 1 for details). The problem reduces to the computation of the Weyl orbits in the set Mor∆ ⊆ Mn(Z) of all matrix modifications A of q∆, i.e., the non-singular matrices A ∈ Mn(Z) such that (i) q∆ (v) = v A vtr, for all v ∈ Zn, and (ii) the Coxeter matrix CoxA := —A A-tr lies in Gl(n, Z). The Weyl group W∆ ⊆ Gl(n, Z) acts on MorA and the determinant det A ∈ Z, the order cA > 2 of CoxA (i.e. the Coxeter number), and the Coxeter polynomial coxA(t) := det(t E — CoxA) ∈ Z[t] are W∆ -invariant. The problem of determining the W∆ -orbits Orb(A) of MorA and the Coxeter polynomials coxA(t), with A e MorA, is studied in the paper and we get its solution for n < 8, and A = [aij] ∈ MorAn, with [aij] | < 1. In this case, we prove that the number of the W∆ - orbits Orb(A) and the number of the Coxeter polynomials coxA(t) equals two or three, and the following three conditions are equivalent: (i) Orb(A) = Orb(A'), (ii) coxA(t) = coxA> (t), (iii) cA det A = cA det A!. We also construct: (a) three pairwise different W∆ -orbits in Mor∆, with pairwise different Coxeter polynomials, if ∆ = A2m-i and m > 3; and (b) two pairwise different W∆ -orbits in Mor∆, with pairwise different Coxeter polynomials, if ∆ = A2m and m > 1.
Wydawca
Czasopismo
Rocznik
Tom
Strony
21--49
Opis fizyczny
Bibliogr. 45 poz.
Twórcy
autor
- Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Bibliografia
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- [7] V. M. Bondarenko and M. V. Stepochkina, On finite posets of inj-finite type and their Tits form, Algebra and Discrete Math. 2(2006), 17-21.
- [8] V. M. Bondarenko and M. V. Stepochkina, (Min, max)-equivalency of posets and nonnegative Tits forms, Ukrain. Mat. Zh. 60 (2008), pp. 1157-1167.
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- [20] J. Kosakowska, Inflation algorithms for positive and principal edge-bipartite graphs and unit quadratic forms, Fund. Inform. 119(2012), 149-162, doi: 10.3233/FI-2012-731.
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- [39] D. Simson, A framework for Coxeter spectral analysis of loop-free edge-bipartite graphs, their rational mor- sifications and mesh geometries of root orbits, Fund. Inform. 124 (2013), 59-88, doi: 10.3233/FI-2013-835.
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Bibliografia
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