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Coxeter Invariants for Non-negative Unit Forms of Dynkin Type A

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Języki publikacji
EN
Abstrakty
EN
Two integral quadratic unit forms are called strongly Gram congruent if their upper triangular Gram matrices are ℤ-congruent. The paper gives a combinatorial strong Gram invariant for those unit forms that are non-negative of Dynkin type r (for r ≥ 1), within the framework introduced in [Fundamenta Informaticae 184(1):49–82, 2021], and uses it to determine all corresponding Coxeter polynomials and (reduced) Coxeter numbers.
Wydawca
Rocznik
Strony
221--246
Opis fizyczny
Bibliogr. 33 poz.
Twórcy
  • Instituto de Matemáticas, UNAM, Mexico.
Bibliografia
  • [1] Barot M, Geiss C, and Zelevinsky A. Cluster algebras of finite type and positive symmetrizable matrices. J. London Math. Soc., 2006. 73:545-564. doi:10.1112/S0024610706022769.
  • [2] Barot M, Jiménez González JA, and de la Peña JA. Quadratic Forms: Combinatorics and Numerical Results, Algebra and Applications, Vol. 25 Springer Nature Switzerland AG 2018. ISBN-13:978-3030056261, 10:3030056260.
  • [3] Barot M, Kussin D, and Lenzing H. The Lie algebra associated to a unit form. Journal of Algebra. 2006. 296:1-17. doi:10.1016/j.jalgebra.2005.11.017.
  • [4] Barot M, and de la Peña JA. The Dynkin type of a non-negative unit form, Expo. Math. 1999. 17:339-348.
  • [5] Barot M, and Rivera D. Generalized Serre relations for Lie algebras associated to positive unit forms, Journal of Pure and Applied Algebra. 2007. 211:360-373. doi:10.1016/j.jpaa.2007.01.008.
  • [6] Bocian R, Felisiak M, and Simson D. Numeric and mesh algorithms for the Coxeter spectral study of positive edge-bipartite graphs and their isotropy groups. Journal of Computational and Applied Mathematics. 2014. 259:815-827. doi:10.1016/j.cam.2013.07.013.
  • [7] Bongartz K. Algebras and quadratic forms. J. London Math. Soc. (2) 1983. 28(3):461-469. doi:10.1112/jlms/s2-28.3.461.
  • [8] Brüstle T, de la Peña JA, and Skowro´nski A. Tame algebras and Tits quadratic forms, Advances in Mathematics. 2011. 226:887-951. doi:10.1016/j.aim.2010.07.007.
  • [9] Cavaleri M, D’Angelo D, and Donno A. Characterizations of line graphs in signed and gain graphs. 2021.arXiv:2101.09677v2 [math.CO]
  • [10] Felisiak M, and Simson D. On combinatorial algorithms computing mesh root systems and matrix morsifications for the Dynkin diagram An. Discrete Mathematics. 2013. 313(12):1358-1367. doi:10.1016/j.disc.2013.02.003.
  • [11] G ˛asiorek M, Simson D, and Zaj ˛ac K. Algorithmic computation of principal posets using Maple and Python. Algebra and Discrete Math. 2014. 17:33-69. doi:10.3233/FI-2016-1345.
  • [12] Jiménez González JA. Incidence graphs and non-negative integral quadratic forms. Journal of Algebra. 2018. 513:208-245. doi:10.016/J.JALGEBRA.2018.07.020.
  • [13] Jiménez González, J.A. A graph theoretical framework for the strong Gram classification of non-negative unit forms of Dynkin type An. Fundamenta Informaticae 184(1) : 49-82 (2021)
  • [14] Jiménez González JA. A Strong Gram classification of non-negative unit forms of Dynkin type Ar . Preprint.
  • [15] Kasjan S, and Simson D. Mesh algorithms for Coxeter spectral classification of Cox-regular edge-bipartite graphs with loops, I. Mesh root systems. Fund. Inform. 2015. 139(2):153-184. doi:10.3233/FI-2015-1230.
  • [16] Kasjan S, and Simson D. Mesh algorithms for Coxeter spectral classification of Cox-regular edge-bipartite graphs with loops, II. Application to Coxeter spectral analysis. Fund. Inform. 2015. 139(2):185-209. doi:10.3233/FI-2015-1231.
  • [17] Kasjan S, and Simson D. Algorithms for Isotropy Groups of Cox-regular Edge-bipartite Graphs. Fund. Inform. 2015. 139(3):249-275. doi:10.3233/FI-2015-1234.
  • [18] Prasolov V. Polynomials. Algorithms and computation in Mathematics Vol. 11. Springer Verlag, Berlin 2001 second edition. ISBN-13:978-3642039799, 10:3642039790.
  • [19] Ringel CM. Tame algebras and integral quadratic forms, Springer LNM, 1984. 1099. ISBN-13:978-0387139050, 10:0387139052.
  • [20] Rotman JJ. A first course in abstract algebra with applications, 3rd Ed., Upper Saddle River, New Jersey 07458, Pearson Prentice Hall, 2006. ISBN:0-13-186267-7.
  • [21] Sato M. Periodic Coxeter matrices and their associated quadratic forms. Linear Algebra Appl. 2005. 406:99-108. doi:10.1016/j.laa.2005.03.036.
  • [22] Simson D. Mesh geometries of root orbits of integral quadratic forms, J. Pure Appl. Algebras 2011. 215(1):13-34. doi:10.1016/j.jpaa.2010.02.029.
  • [23] Simson D. A Coxeter Gram classification of positive simply laced edge-bipartite graphs. SIAM J. Discrete Math., 2013. 27(2):827-854. doi:10.1137/110843721.
  • [24] Simson D. Algorithms determining matrix morsifications, Weyl orbits, Coxeter polynomials and mesh geometries of roots for Dynkin diagrams, Fund. Inform. 2013. 123:447-490. doi:10.3233/FI-2013-820.
  • [25] Simson D. Symbolic algorithms computing Gram congruences in the Coxeter spectral classification of edge-bipartite graphs I. A Gram classification. Fund. Inform., 2016. 145(1):19-48. doi:10.3233/FI-2016-1345.
  • [26] Simson D. A Coxeter spectral classification of positive edge-bipartite graphs I. Dynkin types Bn, Cn, F4, G2, E6, E7, E8. Linear Algebra Appl. 2018. 557:105-133.
  • [27] Simson D. A computational technique in Coxeter spectral study of symmetrizable integer Cartan matrices. Linear Algebra Appl., 2020. 586:190-238. doi:10.1016/j.laa.2019.10.015.
  • [28] Simson D. A Coxeter spectral classification of positive edge-bipartite graphs II. Dynkin type Dn. Linear Algebra and its Applications Volume. 2021. 612:223-272. doi:10.1016/j.laa.2020.11.001.
  • [29] Simson, D. Weyl orbits of matrix morsifications and a Coxeter spectral classification of positive signed graphs and quasi-Cartan matrices of Dynkin type An. Advances in Mathematics Vol. 404, Part A (2022)
  • [30] Simson D, and Zając K. A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots, Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences, 2013, Article ID 743734, 22 pages. doi:10.1155/2013/743734.
  • [31] Simson D, and Zając K. Inflation algorithm for loop-free non-negative edge-bipartite graphs of corank at least two, Linear Algebra Appl. 2017. 524:109-152. doi:10.1016/j.laa.2017.02.021.
  • [32] Zaslavsky, T. Matrices in the theory of signed simple graphs, in: Advances in Discrete Mathematics and Applications: Mysore, 2008, in Ramanujan Math. Soc. Lect. Notes Ser., Vol. 13, Ramanujan Math. Soc., Mysore, 2010, pp. 207-229.
  • [33] Zhang F. Matrix Theory: Basic Results and Techniques, 2nd Ed. Springer 1999. ISBN-13:978-0387986968, 10:0387986960.
Typ dokumentu
Bibliografia
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