Cartesian symmetry classes associated with certain subgroups of $S_m$

Document Type : Research Paper

Authors

Department of Mathematics, Sahand University of Technology, P.O.Box 51335/1996, Tabriz, Iran

Abstract

In this paper, the problem existing $O$-basis for Cartesian symmetry classes is discussed. The dimensions of Cartesian symmetry classes associated with a cyclic subgroup of the symmetric group $S_m$ (generated by a product of disjoint cycles) and the product of cyclic subgroups of $S_m$ are explicitly expressed in terms of the Ramanajun sum. Additionally, a necessary and sufficient condition for the existence of an $O$-basis for Cartesian symmetry classes associated with the irreducible characters of dihedral group is given. The dimensions of these classes are also computed.

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References

[1] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
[2] T. W. Hungerford, Algebra, New York: Holt, Rinehart and Wilson, 1974.
[3] M. Isaacs, Character Theory of Finite Groups, Academic Press, 1976.
[4] G. James and M. Liebeck, Representations and Characters of Groups, Cambridge University press, 2001.
[5] T. G. Lei, Notes on Cartesian symmetry classes and generalized trace functions, Linear Algebra Appl., 292 (1999) 281–288.
[6] R. Merris, Multilinear Algebra, Gordon and Breach Science Publishers, 1997.
[7] G. Rafatneshan and Y. Zamani, Orthogonal bases in specific generalized symmetry classes of tensors, J. Mahani Math. Res., 13 no. 2 (2024) 209–223.
[8] M. Shahryari, On the orthogonal bases of symmetry classes, J. Algebra, 220 (1999) 327–332.
[9] Y. Zamani and M. Shahryari, On the dimensions of Cartesian symmetry classes, Asian-Eur. J. Math., 5 no. 3 (2012) 7 pp.
International Journal of Group Theory
Volume 14, Issue 3 - Serial Number 3
September 2025
Pages 125-138

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History

  • Receive Date: 01 December 2023
  • Revise Date: 29 February 2024
  • Accept Date: 14 March 2024
  • Published Online: 17 March 2024

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