Journal of Mahani Mathematical Research

Journal of Mahani Mathematical Research

Orthogonal bases in specific generalized symmetry classes of tensors

Document Type : Research Paper

Authors
Gholamreza Rafatneshan
Yousef Zamani
Department of Mathematics, Faculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran
10.22103/jmmr.2024.22282.1519
Abstract
Let $V$ be a unitary vector space. Suppose $G$ is a permutation group of degree $m$ and $\Lambda$ is an irreducible unitary representation of $G$. We denote by $V_{\Lambda}(G)$ the generalized symmetry class of tensors associated with $G$ and $\Lambda$. In this paper, we prove the existence of orthogonal bases consisting of generalized decomposable symmetrized tensors for the generalized symmetry classes of tensors associated with unitary irreducible representations of group $U_{6n}$, as well as dihedral and dicyclic groups.
Keywords
Generalized symmetry classes of tensors
orthogonal basis
the group $U_{6n}$
dihedral group
dicyclic group
Subjects
[1] Babaei, E., & Zamani, Y. (2014). Symmetry classes of polynomials associated with the dihedral group. Bull. Iranian Math. Soc., 40(4), 863{874.
[2] Babaei, E., & Zamani, Y. (2014). Symmetry classes of polynomials associated with the direct product of permutation groups. Int. J. Group Theory, 3(4), 63{69. https://doi.org/10.22108/ijgt.2014.5479
[3] Babaei, E., Zamani, Y., & Shahryari. M. (2016). Symmetry classes of polynomials. Commun. Algebra, 44(4), 1514{1530.  https://doi.org/10.1080/00927872.2015.1027357
[4] Darafsheh, M. R., & Pournaki, M. R. (2000). On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group. Linear Multilinear Algebra, 47(2), 137{149. https://doi.org/10.1080/03081080008818639
[5] Darafsheh, M. R., & Poursalavati, N. S. (2001). On the existence of the orthogonal basis of the symmetry classes of tensors associated with certain groups. SUT J. Math., 37(1), 1{17. 
[6] Dias da Silva, J. A., & Torres, M. M. (2005). On the orthogonal dimensions of orbital sets. Linear Algebra Appl., 401 (15 May), 77{107. https://doi.org/10.1016/j.laa.2003.11.005
[7] Gao, R., Liu, H., & Zhou, F. (2020). Symmetry classes of tensors associated with certain groups. Linear Algebra Appl., 602(1 October), 240{251. https://doi.org/10.1016/j.laa.2020.05.016
[8] Holmes, R. R., & Kodithuwakku, A. (2013). Orthogonal bases of Brauer symmetry classes of tensors for the dihedral group. Linear Multilinear Algebra, 61(8), 1136{1147. https://doi.org/10.1080/03081087.2012.729583
[9] Holmes, R. R., & Kodithuwakku, A. (2016). Symmetry classes of tensors associated with principal indecomposable characters and Osima idempotents. Linear Multilinear Algebra, 64(4), 574{586. https://doi.org/10.1080/03081087.2015.1040220
[10] Holmes, R. R., & Tam, T. Y. (1992). Symmetry classes of tensors associated with certain groups. Linear Multilinear Algebra, 32(1), 21{31. https://doi.org/10.1080/03081087.2012.729583
[11] Hormozi, M., & Rodtes, K. (2013). Symmetry classes of tensors associated with the semi-dihedral groups SD8n. Colloq. Math., 131(1), 59{67. DOI: 10.4064/cm131-1-6
[12] James, G., & Liebeck, M. (1993). Representations and Characters of Groups. Cambridge University press.
[13] Lei, T. G. (1997). Generalized Schur functions and generalized decompossble symmetric tensors. Linear Algebra Appl., (263), 311{332. https://doi.org/10.1016/S0024-3795(96)00542-3
[14] Marcus, M. (1973). Finite Dimensional Multilinear Algebra (Part I). Marcel Dekker, Inc., New York.
[15] Merris, R. (1997). Multilinear Algebra. Gordon and Breach Science Publisher, Amsterdam.
[16] Rafatneshan, G., & Zamani, Y. (2020). Generalized symmetry classes of tensors. Czechoslovak Math. J., 70(145), 921{933. https://doi.org/10.21136/CMJ.2020.0044-19
[17] Rafatneshan, G., & Zamani, Y. (2021). Induced operators on the generalized symmetry classes of tensors, Int. J. Group Theory, 10(4), 197{211. http://dx.doi.org/10.22108/ijgt.2020.122990.1622
[18] Shahabi, M. A., Azizi, K., & Jafari, M. H. (2001). On the orthogonal basis of symmetry classes of tensors. J. Algebra, 237(2), 637{646. https://doi.org/10.1006/jabr.2000.8332
[19] Shahryari, M. (1999). On the orthogonal bases of symmetry classes. J. Algebra, 220(1), 327{32. https://doi.org/10.1006/jabr.1999.7932
[20] Shelash, H. B., & Ashra , A. R. (2019). Computing maximal and minimal subgroups with respect to a given property in certain  nite groups. Quasigroups and Related Systems, 27, 133{146.
[21] Zamani, Y. (2007). On the special basis of a certain full symmetry class of tensors. Pure Math. Appl., 18(3), 357{363.
[22] Zamani, Y., & Babaei, B. (2013). Symmetry classes of polynomials associated with the dicyclic group. Asian-Eur. J. Math., 6(3), Article ID:1350033 (10 pages). https://doi.org/10.1142/S1793557113500332
[23] Zamani, Y., & Shahryari, M. (2011). Symmetry classes of tensors associated with Young subgroups. Asian-Eur. J. Math., 4(1), 179{185. https://doi.org/10.1142/S1793557111000150

  • Receive Date 03 October 2023
  • Revise Date 12 January 2024
  • Accept Date 28 January 2024