# Symmetry classes of polynomials associated with the ‎direct ‎product of permutation groups

Document Type : Research Paper

Authors

1 Sahand University of technology

2 Sahand University of Technology

Abstract

‎Let $G_{i}$ be a subgroup of $S_{m_{i}}‎ ,‎\ 1 \leq i \leq k$‎. ‎Suppose $\chi_{i}$ is an irreducible complex character of $G_{i}$‎. ‎We consider $G_{1}\times \cdots \times G_{k}$ as subgroup of $S_{m}$‎, ‎where $m=m_{1}+\cdots‎ +‎m_{k}$‎. ‎In this paper‎, ‎we give a formula for the dimension of $H_{d}(G_{1}\times \cdots \times G_{k}‎, ‎\chi_{1}\times\cdots \times \chi_{k})$ and investigate the existence of an o-basis of this type of classes‎.

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#### References

E. Babaei and Y. Zamani Symmetry classes of polynomials associated with the dihedral group. to appear in Bull. Iranian Math. Soc.. M. R. Darafsheh and M. R. Pournaki (2000). On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group. Linear and Multilinear Algebra. 47, 137-149 M. R. Darafsheh and N. S. Poursalavati (2001). On the existence of the orthogonal basis of the symmetry classes of tensors associated with certain groups. SUT J. Math.. 37, 1-17 R. R. Holmes (1995). Orthogonal bases of symmetrized tensor spaces. Linear and Multilinear Algebra. 39, 241-243 I. M. Isaacs (1976). Character Theory of Finite Groups. Academic Press, New York. R. Merris (1997). Multilinear Algebra. Gordon and Breach Science Publisher, Amsterdam. M. Shahryari (2010). Relative symmetric polynomials. Linear Algebra Appl.. 433, 1410-1421 M. Shahryari (1999). On the orthogonal bases of symmetry classes. J. Algebra. 220, 327-332 M. Shahryari and Y. Zamani (2011). Symmetry classes of tensors associated with Young subgroups. Asian-Eur. J. Math.. 4, 179-185 Y. Zamani and E. Babaei (2014). The dimensions of cyclic symmetry classes of polynomials. J. Algebra Appl., (10 pages). 13 Y. Zamani (2007). On the special basis of a certain full symmetry class of tensors. Pure Math. Appl.. 18, 357-363 Y. Zamani and E. Babaei (2013). Symmetry classes of polynomials associated with the dicyclic group. Asian-Eur. J. Math., (10 pages). 6