Graphs attached to simple Frobenius-Perron dimensions of an integral fusion category

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• 作者：Sonia Natale ; Edwin Pacheco Rodríguez
• 关键词：Fusion category ; Frobenius ; Perron dimension ; Frobenius ; Perron graph ; Equivariantization ; Braided fusion category ; Modular category ; Solvability
• 刊名：Monatshefte f¨¹r Mathematik
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：179
• 期：4
• 页码：615-649
• 全文大小：670 KB
• 参考文献：1.Alfandary, G.: On graphs related to conjugacy classes of groups. Isr. J. Math. 86, 211–220 (1994)MathSciNet CrossRef MATH
  2.Bakalov, B., Kirillov, A. Jr.: Lectures on Tensor Categories and Modular Functors. University Lecture Series, vol. 21. American Mathematical Society, Providence, RI (2001)
  3.Bertram, E., Herzog, M., Mann, A.: On a graph related to conjugacy classes of groups. Bull. Lond. Math. Soc. 22, 569–575 (1990)MathSciNet CrossRef MATH
  4.Brown, K.: Cohomology of Groups. Graduate Texts in Mathematics, vol. 87. Springer, New York (1982)
  5.Bruguières, A.: Catégories prémodulaires, modularisations et invariants des variétés de dimension 3. Math. Ann. 316, 215–236 (2000)MathSciNet CrossRef MATH
  6.Bruguières, A., Burciu, S.: On normal tensor functors and coset decompositions for fusion categories, preprint arXiv:​1210.​3922 (2013)
  7.Bruguières, A., Natale, S.: Exact sequences of tensor categories. Int. Math. Res. Not. 2011, 5644–5705 (2011)MathSciNet MATH
  8.Burciu, S., Natale, S.: Fusion rules of equivariantizations of fusion categories. J. Math. Phys. 54, 013511 (2013)MathSciNet CrossRef MATH
  9.Bruillard, P., Galindo, C., Hong, S.-M., Kashina, Y., Naidu, D., Natale, S., Plavnik, J., Rowell, E.: Classification of integral modular categories of Frobenius-Perron dimension \(pq^4\) and \(p^2q^2\) . Can. Math. Bull. 57, 721–734 (2014)MathSciNet CrossRef MATH
  10.Camina, A.R., Camina, R.D.: The influence of conjugacy class sizes on the structure of finite groups: a survey. Asian Eur. J. Math. 4, 559–588 (2011)MathSciNet CrossRef MATH
  11.Casolo, C., Dolfi, S.: The diameter of a conjugacy class graph of finite groups. Bull. Lond. Math. Soc. 28, 141–148 (1996)MathSciNet CrossRef MATH
  12.Davydov, A., Mueger, M., Nikshych, D., Ostrik, V.: The Witt group of non-degenerate braided fusion categories. J. Reine Angew. Math. 677, 135–177 (2013)MathSciNet MATH
  13.Deligne, P.: Catégories tensorielles. Mosc. Math. J. 2, 227–248 (2002)MathSciNet MATH
  14.Dijkgraaf, R., Pasquier, V., Roche, P.: Quasi-Quantum Groups Related to Orbifold Models. In: Proceedings of the International Colloquium on Modern Quantum Field Theory, Tata Institute of Fundamental Research, pp. 375–383 (1990)
  15.Dong, J., Natale, S., Vendramin, L.: Frobenius property for fusion categories of small integral dimension. J. Algebra Appl. 14, 1550011 (2015)MathSciNet CrossRef MATH
  16.Drinfeld, V., Gelaki, S., Nikshych, D., Ostrik, V.: Group-theoretical properties of nilpotent modular categories, preprint arXiv:​0704.​0195 (2007)
  17.Drinfeld, V., Gelaki, S., Nikshych, D., Ostrik, V.: On braided fusion categories I. Sel. Math. New Ser. 16, 1–119 (2010)MathSciNet CrossRef MATH
  18.Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories. Ann. Math 162, 581–642 (2005)MathSciNet CrossRef MATH
  19.Etingof, P., Nikshych, D., Ostrik, V.: Weakly group-theoretical and solvable fusion categories. Adv. Math 226, 176–205 (2011)MathSciNet CrossRef MATH
  20.Gelaki, S., Naidu, D., Nikshych, D.: Centers of graded fusion categories. Algebra Number Theory 3, 959–990 (2009)MathSciNet CrossRef MATH
  21.Gelaki, S., Nikshych, D.: Nilpotent fusion categories. Adv. Math. 217, 1053–1071 (2008)MathSciNet CrossRef MATH
  22.Isaacs, M.: Character theory of finite groups. Academic Press, New York (1976)MATH
  23.Isaacs, M.: Coprime group actions fixing all nonlinear irreducible characters. Can. J. Math. 41, 68–82 (1989)MathSciNet CrossRef MATH
  24.Isaacs, M., Praeger, C.: Permutation group subdegrees and the common divisor graph. J. Algebra 159, 158–175 (1993)MathSciNet CrossRef MATH
  25.Ito, N.: On the degrees of irreducible representations of a finite group. Nagoya Math. J. 3, 5–6 (1951)MathSciNet MATH
  26.Kazarin, L.: On groups with isolated conjugacy classes, Sov. Math. 25, 43–49 (1981) (translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1981, (230) 40–45 (1981))
  27.Lewis, M.: An overview of graphs associated with character degrees and conjugacy class sizes in finite groups. Rocky Mt. J. Math. 38, 175–211 (2008)MathSciNet CrossRef MATH
  28.Manz, O.: Degree Problems II: \(\pi \) -separable character degrees. Commun. Algebra 13, 2421–2431 (1985)MathSciNet CrossRef MATH
  29.Manz, O., Staszewski, R., Willems, W.: The number of components of a graph related to character degrees. Proc. Am. Math. Soc. 103, 31–37 (1988)MathSciNet CrossRef MATH
  30.Manz, O., Willems, W., Wolf, T.: The diameter of the character degree graph. J. Reine Angew. Math. 402, 181–198 (1989)MathSciNet MATH
  31.Manz, O., Wolf, T.: Representations of solvable groups. London Mathematics Society Lecture Notes Series 185. Cambridge University Press, Cambridge (1993)CrossRef MATH
  32.Mason, G., Ng, S.: Group cohomology and gauge equivalence of some twisted quantum doubles. Trans. Am. Math. Soc 353, 3465–3509 (2001)MathSciNet CrossRef MATH
  33.Michler, G.: A finite simple group of Lie type has \(p\) -blocks with different defects, \(p \ne 2\) . J. Algebra 104, 220–230 (1986)MathSciNet CrossRef MATH
  34.Müger, M.: Galois theory for braided tensor categories and the modular closure. Adv. Math. 150, 151–201 (2000)MathSciNet CrossRef MATH
  35.Müger, M.: Galois extensions of braided tensor categories and braided crossed \(G\) -categories. J. Algebra 277, 256–281 (2004)MathSciNet CrossRef MATH
  36.Naidu, D., Nikshych, D., Witherspoon, S.: Fusion subcategories of representation categories of twisted quantum doubles of finite groups. Int. Math. Res. Not. 22, 4183–4219 (2009)MathSciNet MATH
  37.Naidu, D., Rowell, E.: A finiteness property for braided fusion categories. Algebr. Represent. Theory 14, 837–855 (2011)MathSciNet CrossRef MATH
  38.Natale, S.: Hopf algebra extensions of group algebras and Tambara-Yamagami categories. Algebr. Represent. Theory 13, 673–691 (2010)MathSciNet CrossRef MATH
  39.Natale, S.: On weakly group-theoretical non-degenerate braided fusion categories. J. Noncommut. Geom. Preprint arXiv:​1301.​6078 (to appear)
  40.Natale, S.: Faithful simple objects, orders and gradings of fusion categories. Algebr. Geom. Topol. 13, 1489–1511 (2013)MathSciNet CrossRef MATH
  41.Tambara, D.: Invariants and semi-direct products for finite group actions on tensor categories. J. Math. Soc. Jpn. 53, 429–456 (2001)MathSciNet CrossRef MATH
  42.Turaev, V.: Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Math. 18, Berlin (1994)
  43.Turaev, V.: Homotopy quantum field theory, EMS Tracts in Mathematics 10. European Mathematical Society, Zurich (2010)CrossRef
  44.Turaev, V.: Crossed group-categories. Arab. J. Sci. Eng. 33, 484–503 (2008)MathSciNet MATH
  45.Willems, W.: Blocks of defect zero and degree problems, Proc. Sympos. Pure Math. vol. 47 (I), pp. 481–484. American Mathematics Society Providence, RI (1987)
  46.Yuster, T.: Orbit sizes under automorphism actions in finite groups. J. Algebra 82, 342–352 (1983)MathSciNet CrossRef MATH
  
• 作者单位：Sonia Natale (1)
  Edwin Pacheco Rodríguez (1)
  
  1. Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, CIEM-CONICET, Ciudad Universitaria, 5000, Córdoba, Argentina
  
• 刊物主题：Mathematics, general;
• 出版者：Springer Vienna
• ISSN：1436-5081

文摘

Let \({\mathcal C}\) be an integral fusion category. We study some graphs, called the prime graph and the common divisor graph, related to the Frobenius-Perron dimensions of simple objects in the category \({\mathcal C}\), that extend the corresponding graphs associated to the irreducible character degrees and the conjugacy class sizes of a finite group. We describe these graphs in several cases, among others, when \({\mathcal C}\) is an equivariantization under the action of a finite group, a \(2\)-step nilpotent fusion category, and the representation category of a twisted quantum double. We prove generalizations of known results on the number of connected components of the corresponding graphs for finite groups in the context of braided fusion categories. In particular, we show that if \({\mathcal C}\) is any integral non-degenerate braided fusion category, then the prime graph of \({\mathcal C}\) has at most \(3\) connected components, and it has at most \(2\) connected components if \({\mathcal C}\) is in addition solvable. As an application we prove a classification result for weakly integral braided fusion categories all of whose simple objects have prime power Frobenius-Perron dimension.