Author Details

In this paper, we compute the number of z-classes (conjugacy classes of centralizers of elements) in the symmetric group S_n, when n ≥ 3 and alternating group A_n when n ≥ 4. It turns out that the difference between the number of conjugacy classes and the number of z-classes for S_n is determined by those restricted partitions of n − 2 in which 1 and 2 do not appear as its part. In the case of alternating groups, it is determined by those restricted partitions of n −3 which has all its parts distinct, odd and in which 1 (and 2) does not appear as its part, along with an error term. The error term is given by those partitions of n which have distinct parts that are odd and perfect squares. Further, we prove that the number of rational-valued irreducible complex characters for A_n is same as the number of conjugacy classes which are rational.

Full Text

References

[AO] Armeanu, Ion and Ozturk, Didem, Alternating q-groups, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 74 no. 4, (2012) 39–46.

[BS] Bhunia, Sushil and Singh, Anupam, Conjugacy classes of centralizers in unitary groups, Journal of Group Theory, 22 (2019) 231–151.

[Ca1] R. W. Carter, Centralizers of semisimple elements in finite groups of Lie type, Proc. London Math. Soc. (3), 37 no. 3, (1978) 491–507.

[Ca2] R. W., Carter, Centralizers of semisimple elements in the finite classical groups, Proc. London Math. Soc. (3), 42 no. 1, (1981) 1–41.

[Pr] Prasad, Amritanshu, Representation theory, A combinatorial viewpoint, Cambridge Studies in Advanced Mathematics, 147. Cambridge University Press, Delhi (2015).

[Br] Brison, Owen J., Alternating groups and a conjecture about rational valued characters, Linear and Multilinear Algebra, 24 no. 3, (1989) 199–207.

[Ca] Can, Himmet, Representations of the generalized symmetric groups, Beitrge Algebra Geom., 37 no. 2, (1996) 289–307.

[FS] Feit, Walter and Seitz, Gary M., On finite rational groups and related topics, Illinois J. Math., 33 no. 1, (1989) 103–131.

[Go] Gongopadhyay, Krishnendu, The z-classes of quaternionic hyperbolic isometries, J. Group Theory, 16 no. 6, (2013) 941–964.

[GAP] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.7 (2017), (http://www.gap-system.org).

[GK] Gongopadhyay, Krishnendu and Kulkarni, Ravi S., z-classes of isometries of the hyperbolic space, Conform. Geom. Dyn., 13 (2009), 91–109.

[Gr] J. A., Green, The characters of the finite general linear groups, Trans. Amer. Math. Soc., 80 (1955) 402–447.

[JK] James, Gordon and Kerber, Adalbert, The representation theory of the symmetric group, With a foreword by P. M. Cohn. With an introduction by Gilbert de B. Robinson. Encyclopedia of Mathematics and its Applications, 16. Addison-Wesley Publishing Co., Reading, Mass. (1981).

[Kl] Kletzing, Dennis, Structure and representations of Q-groups, Lecture Notes in Mathematics, 1084. Springer-Verlag, Berlin (1984). vi+290 pp.

[Ku] Kulkarni, Ravi S., Dynamical types and conjugacy classes of centralizers in groups, J. Ramanujan Math. Soc., 22 no. 1, (2007) 35–56.

[MS] Mishra, Ashish and Srinivasan,Murali K., The Okounkov-Vershik approach to the representation theory of G ∼ S_n, J. Algebraic Combin., 44 no. 3, (2016) 519–560.

[NT] Navarro, Gabriel and Tiep, Pham Huu, Rational irreducible characters and rational conjugacy classes in finite groups, Trans. Amer. Math. Soc., 360 no. 5, (2008) 2443–2465.

[OEIS] The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org.

[Os] Osima, Masaru, On the representations of the generalized symmetric group, Math. J. Okayama Univ., 4 (1954) 39–56.

[Se] Serre, Jean-Pierre, Linear representations of finite groups, Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42. Springer-Verlag, New York-Heidelberg (1977).

[Si] Singh, Anupam, Conjugacy classes of centralizers in G₂, J. Ramanujan Math. Soc., 23 no. 4, (2008) 327–336.

Username
Password
Remember me