Cartesian symmetry classes associated with certain subgroups of $S_m$

Gholami, Seyyed Sadegh; Zamani, Yousef

doi:10.22108/ijgt.2024.139990.1882

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	Cartesian symmetry classes associated with certain subgroups of $S_m$
International Journal of Group Theory
مقاله 3، دوره 14، شماره 3، آذر 2025، صفحه 125-138 اصل مقاله (455.66 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2024.139990.1882
نویسندگان
Seyyed Sadegh Gholami؛ Yousef Zamani^*
Department of Mathematics, Sahand University of Technology, P.O.Box 51335/1996, Tabriz, Iran
چکیده
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.
کلیدواژه‌ها
Irreducible characters؛ Cartesian symmetry classes؛ orthogonal basis؛ cyclic group؛ Dihedral group؛ Ramanujan sum

مراجع
[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.
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Cartesian symmetry classes associated with certain subgroups of $S_m$