Journal of the Ramanujan Mathematical Society

Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

𝓏-Classes and Rational Conjugacy Classes in Alternating Groups


Affiliations
1 IISER Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, India
     

   Subscribe/Renew Journal


In this paper, we compute the number of z-classes (conjugacy classes of centralizers of elements) in the symmetric group Sn, when n ≥ 3 and alternating group An when n ≥ 4. It turns out that the difference between the number of conjugacy classes and the number of z-classes for Sn 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 An is same as the number of conjugacy classes which are rational.
User
Subscription Login to verify subscription
Notifications
Font Size

  • [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 ∼ Sn, 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.


Abstract Views: 517

PDF Views: 0




  • 𝓏-Classes and Rational Conjugacy Classes in Alternating Groups

Abstract Views: 517  |  PDF Views: 0

Authors

Sushil Bhunia
IISER Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, India
Dilpreet Kaur
IISER Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, India
Anupam Singh
IISER Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, India

Abstract


In this paper, we compute the number of z-classes (conjugacy classes of centralizers of elements) in the symmetric group Sn, when n ≥ 3 and alternating group An when n ≥ 4. It turns out that the difference between the number of conjugacy classes and the number of z-classes for Sn 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 An is same as the number of conjugacy classes which are rational.

References