Nội dung được dịch bởi AI, chỉ mang tính chất tham khảo
$${\varvec{z}}$$ -lớp trong các nhóm: một khảo sát
Tóm tắt
Bài viết khảo sát này khám phá khái niệm z-lớp trong các nhóm. Khái niệm được giới thiệu ở đây liên quan đến loại quỹ đạo trong các nhóm biến hình, và loại hoặc dạng trong lý thuyết đại diện của các nhóm hữu hạn kiểu Lie. Hai phần tử trong một nhóm được cho là z-tương đương (hoặc z-bội) nếu các trung tố (centralizer) của chúng tương đương với nhau. Đây là một khái niệm yếu hơn so với tính cộng hưởng của các phần tử. Trong bài khảo sát này, chúng tôi trình bày một số kết quả đã biết về chủ đề này và đề xuất một số câu hỏi tiếp theo.
Từ khóa
#z-lớp #nhóm #quỹ đạo #nhóm biến hình #lý thuyết đại diện #nhóm hữu hạn kiểu LieTài liệu tham khảo
Arora, Shivam; Gongopadhyay, Krishnendu, “\(z\)-classes in finite groups of conjugate type \((n,1)\)”, Proc. Indian Acad. Sci. Math. Sci. 128 (2018), no. 3, Paper No. 31, 7 pp.
Bhunia, Sushil, “Conjugacy classes of centralizers in the group of upper triangular matrices”, J. Algebra Appl. 19 (2020), no. 1, 2050008, 14 pp.
Bonnafé, Cédric, “Representations of \({SL_2}(\mathbb{F}_q)\)”, Algebra and Applications, 13. Springer-Verlag London, Ltd., London, 2011.
Bhunia, Sushil; Kaur, Dilpreet; Singh, Anupam, “\(z\)-classes and rational conjugacy classes in alternating groups”, J. Ramanujan Math. Soc. 34 (2019) no. 2, 169-183.
Bose, Anirban, “On the genus number of algebraic groups”, J. Ramanujan Math. Soc. 28 (2013), no. 4, 443-482.
Bhunia, Sushil; Singh, Anupam, “Conjugacy classes of centralizers in unitary groups”, J. Group Theory, 22 (2019), 231-151.
A. Borel and J.-P. Serre, “Théorèmes de finitude en cohomologie galoisienne”, (French) Comment. Math. Helv. 39 (1964), 111-164.
Britnell, John R.; Wildon, Mark, “On types and classes of commuting matrices over finite fields”, J. Lond. Math. Soc. (2) 83 (2011), no. 2, 470-492.
Britnell, John R.; Wildon, Mark, “On types of matrices and centralizers of matrices and permutations”, J. Group Theory 17 (2014), no. 5, 875-887.
Carter, Roger W., “Finite groups of Lie type. Conjugacy classes and complex characters”, Reprint of the 1985 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester, 1993.
Cao, Wensheng; Gongopadhyay, Krishnendu, “Commuting isometries of the complex hyperbolic space”, Proc. Amer. Math. Soc. 139 (2011), no. 9, 3317-3326.
Dolfi, Silvio; Herzog, Marcel; Jabara, Enrico, “Finite groups whose noncentral commuting elements have centralizers of equal size”, Bull. Aust. Math. Soc. 82 (2010), no. 2, 293-304.
Digne, François; Michel, Jean, “Representations of finite groups of Lie type”, London Mathematical Society Student Texts, 21. Cambridge University Press, Cambridge, 1991.
S. Eilenberg, “On the problems of topology”, Ann. of Math. (2) 50 (1949) 247-260.
E. E. Floyd, “Orbits of torus groups operating on manifolds”, Ann. of Math. (2) 65 (1957) 505-512.
Fleischmann, Peter, “Finite fields, root systems, and orbit numbers of Chevalley groups”, Finite Fields Appl. 3 (1997), no. 1, 33-47.
Fulman, Jason; Guralnick, Robert, “The number of regular semisimple conjugacy classes in the finite classical groups”, Linear Algebra Appl. 439 (2013), no. 2, 488-503.
Fong, Paul; Srinivasan, Bhama, “The blocks of finite general linear and unitary groups”, Invent. Math. 69 (1982), no. 1, 109-153.
Gongopadhyay, Krishnendu, “The z-classes of quaternionic hyperbolic isometries”, J. Group Theory 16 (2013), 941-964.
Gongopadhyay, Krishnendu and Kulkarni, Ravi S., “The \(z\)-classes of isometries”, J. Indian Math. Soc. (N.S.) 81 (2014), no. 3-4, 245-258.
Gongopadhyay, Krishnendu and Kulkarni, Ravi S., “z-classes of isometries of the hyperbolic space”, Conform. Geom. Dyn. 13 (2009), 91-109.
Gouraige, Rony, “z-classes in central simple algebras”, Thesis (Ph.D.)-City University of New York. 2006.
Green, J. A., “The characters of the finite general linear groups”, Trans. Amer. Math. Soc. 80 (1955), 402-447.
Garge, Shripad M.; Singh, Anupam, “Finiteness of \(z\)-classes in reductive groups”, J. Algebra 554 (2020), 41-53.
Larry C. Grove, “Classical groups and geometric algebra”, Graduate Studies in Mathematics, 39. American Mathematical Society, Providence, RI, 2002.
Hofmann, Karl H.; Morris, Sidney A., “The structure of compact groups. A primer for the student-a handbook for the expert”, Third edition, revised and augmented. De Gruyter Studies in Mathematics, 25. De Gruyter, Berlin, 2013. xxii+924 pp.
Hahn, Alexander J.; O’Meara, O. Timothy, “The classical groups and K-theory. With a foreword by J. Dieudonne”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 291. Springer-Verlag, Berlin, 1989. xvi+576 pp.
Humphreys, James E., “Linear algebraic groups”, Graduate Texts in Mathematics, No. 21. Springer-Verlag, New York-Heidelberg, 1975. xiv+247 pp.
Humphreys, James E., “Conjugacy classes in semisimple algebraic groups”, Mathematical Surveys and Monographs, 43. American Mathematical Society, Providence, RI, 1995.
Jadhav, Vikas S., Kitture, Rahul Dattatraya, “z-classes in \(p\)-groups of order \(\le p^5\)”, Bull. Allahabad Math. Soc. 29 (Part 2) (2014) 173-194.
Kulkarni, Ravindra., Kitture, Rahul Dattatraya., Jadhav, Vikas S., “\(z\)-classes in groups”, J. Algebra Appl. 15 (2016), no. 7, 1650131.
Kulkarni, Ravi S., “Dynamics of linear and affine maps”, Asian J. Math. 12 (2008), no. 3, 321-344.
Kulkarni, Ravi S., “Dynamical types and conjugacy classes of centralizers in groups”, J. Ramanujan Math. Soc. 22 (2007), no. 1, 35-56.
Khoramshahi, K., Zarrin, M., “Groups with the same number of centralizers”, to appear in the J. Algebra Appl. https://doi.org/10.1142/S0219498821500122.
Mann, L. N., “Finite orbit structure on locally compact manifolds”, Michigan Math. J. 9 (1962), 87-92.
G.D. Mostow, “On a conjecture of Montgomery”, Ann. of Math., 65(2), 1957, 513-516.
Prasad, Amritanshu; Singla, Pooja; Spallone, Steven, “Similarity of matrices over local rings of length two”, Indiana Univ. Math. J. 64 (2015), no. 2, 471-514.
Prasad, Amritanshu, “Representations of \(GL_2(\mathbb{F}_q)\) and \(SL_2(\mathbb{F}_q)\), and some remarks about \(GL_n(\mathbb{F}_q)\)”, arXiv:0712.4051.
Singla, Pooja, “On representations of general linear groups over principal ideal local rings of length two”, J. Algebra 324 (2010), no. 9, 2543-2563.
Suzuki, Michio, “Finite groups with nilpotent centralizers”, Trans. Amer. Math. Soc. 99 (1961), 425-470.
Srinivasan, Bhama, “The characters of the finite symplectic group \(Sp(4,q)\)”, Trans. Amer. Math. Soc. 131 (1968) 488-525.
Singh, Anupam, “Conjugacy Classes of Centralizers in \(G_{2}\)”, J. Ramanujan Math. Soc. 23 (2008), no. 4, 327-336.
Springer, T. A., “Linear algebraic groups”, Second edition. Progress in Mathematics, 9. Birkhäuser Boston, Inc., Boston, MA, 1998. xiv+334 pp.
Steinberg, Robert, “Conjugacy Classes in Algebraic Groups”, notes by V. Deodhar, Lecture Notes in Mathematics 366, Springer-Verlag (1974).
Zarrin, Mohammad, “Derived length and centralizers of groups”, J. Algebra Appl. 14 (2015), no. 8, 1550133, 4 pp.
