Cartan matrices and quasi-Cartan matrices play an important role in such areas as Lie theory, representation theory, and algebraic graph theory. It is known that each (connected) positive definite quasi-Cartan matrix ...
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Cartan matrices and quasi-Cartan matrices play an important role in such areas as Lie theory, representation theory, and algebraic graph theory. It is known that each (connected) positive definite quasi-Cartan matrix A is an element of M-n (Z) is Z-equivalent with the Cartan matrix of a Dynkin diagram, called the Dynkin type of A. We present a symbolic, graph-theoretic algorithm to compute the Dynkin type of A, of the pessimistic arithmetic (word) complexity O(n(2)), significantly improving the existing algorithms. As an application we note that our algorithm can be used as a positive definiteness test for an arbitrary quasi-Cartan matrix, more efficient than standard tests. Moreover, we apply the algorithm to study a class of (symmetric and non-symmetric) quasi-Cartan matrices related to Nakayama algebras.
We continue the Coxeter spectral study of finite connected loop- free edge- bipartite graphs Delta, with m + 2 >= 3 vertices (a class of signed graphs), started in [SIAM J. Discrete Math., 27(2013), 827- 854] by me...
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ISBN:
(纸本)9781479984480
We continue the Coxeter spectral study of finite connected loop- free edge- bipartite graphs Delta, with m + 2 >= 3 vertices (a class of signed graphs), started in [SIAM J. Discrete Math., 27(2013), 827- 854] by means of the complex Coxeter spectrum specc(Delta) subset of C and presented in our talks given in SYNASC12 and SYNASC13. Here, we study non- negative edge- bipartite graphs of corank two, in the sense that the symmetric Gram matrix G Delta is an element of Mm+2(Z) of Delta is positive semi- definite of rank m >= 1. Extending each of the simply laced Euclidean diagrams (A) over tilde (m), m >= 1, D-m, m >= 4, (E) over tilde (6), (E) over tilde (7), (E) over tilde (8) by one vertex, we construct a family of loop-free corank two diagrams (A) over tilde (2)(m), (D) over tilde (2)(m), (E) over tilde (2)(6), (E) over tilde (2)(7), (E) over tilde (2)(8) (called simply extended Euclidean diagrams) such that they classify all connected corank two loop- free edge- bipartite graphs Delta, with m + 2 >= 3 vertices, up to Z-congruence Delta similar to(Z) Delta'. Here Delta similar to(Z) Delta' means that G(Delta') = B-tr.G(Delta).B, for some B is an element of Mm+2(Z) such that det B = +/- 1. We present algorithms that generate all such edge- bipartite graphs of a given size m + 2 >= 3, together with their Coxeter polynomials, and the reduced Coxeter numbers, using symbolic and numeric computer calculations in Python. Moreover, we prove that for any corank two connected loop- free edge- bipartite graph Delta, with m + 2 >= 3 vertices, there exists a simply extended Euclidean diagram D such that Delta similar to(Z) D.
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