Orthogonal bases in specific generalized symmetry classes of tensors

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran

Abstract

Let V be a unitary vector space. Suppose G is a permutation group of degree m and Λ is an irreducible unitary representation of G. We denote by VΛ(G) the generalized symmetry class of tensors associated with G and Λ. 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 U6n, as well as dihedral and dicyclic groups.

Keywords

Main 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