Czechoslovak Mathematical Journal, first online, pp. 10, 2017


Representations of the general linear group over symmetry classes of polynomials

Yousef Zamani, Mahin Ranjbari

Received August 28, 2016.  First published May 4, 2017.

Abstract:  Let $V$ be the complex vector space of homogeneous linear polynomials in the variables $x_1, \ldots, x_m$. Suppose $G$ is a subgroup of $S_m$, and $\chi$ is an irreducible character of $G$. Let $H_d(G,\chi)$ be the symmetry class of polynomials of degree $d$ with respect to $G$ and $\chi$. For any linear operator $T$ acting on $V$, there is a (unique) induced operator $K_{\chi} (T)\in{\rm End}(H_d(G,\chi))$ acting on symmetrized decomposable polynomials by $K_{\chi}(T)(f_1\ast f_2\ast\ldots\ast f_d)=Tf_1\ast Tf_2\ast\ldots\ast Tf_d.$ In this paper, we show that the representation $T\mapsto K_{\chi} (T)$ of the general linear group $GL(V)$ is equivalent to the direct sum of $\chi(1)$ copies of a representation (not necessarily irreducible) $T\mapsto B_{\chi}^G(T)$.
Keywords:  symmetry class of polynomials; general linear group; representation; irreducible character; induced operator
Classification MSC:  20C15, 15A69, 05E05
DOI:  10.21136/CMJ.2017.0458-16

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References:
[1] E. Babaei, Y. Zamani: Symmetry classes of polynomials associated with the dihedral group. Bull. Iran. Math. Soc. 40 (2014), 863-874. MR 3255403 | Zbl 1338.05271
[2] E. Babaei, Y. Zamani: Symmetry classes of polynomials associated with the direct product of permutation groups. Int. J. Group Theory 3 (2014), 63-69. MR 3213989 | Zbl 1330.05159
[3] E. Babaei, Y. Zamani, M. Shahryari: Symmetry classes of polynomials. Commun. Algebra 44 (2016), 1514-1530. DOI 10.1080/00927872.2015.1027357 | MR 3473866 | Zbl 1338.05272
[4] I. M. Isaacs: Character Theory of Finite Groups. Pure and Applied Mathematics 69, Academic Press, New York (1976). MR 0460423 | Zbl 0337.20005
[5] R. Merris: Multilinear Algebra. Algebra, Logic and Applications 8, Gordon and Breach Science Publishers, Amsterdam (1997). MR 1475219 | Zbl 0892.15020
[6] M. Ranjbari, Y. Zamani: Induced operators on symmetry classes of polynomials. Int. J. Group Theory 6 (2017), 21-35.
[7] K. Rodtes: Symmetry classes of polynomials associated to the semidihedral group and o-bases. J. Algebra Appl. 13 (2014), Article ID 1450059, 7 pages. DOI 10.1142/S0219498814500595 | MR 3225126 | Zbl 1297.05243
[8] M. Shahryari: Relative symmetric polynomials. Linear Algebra Appl. 433 (2010), 1410-1421. DOI 10.1016/j.laa.2010.05.020 | MR 2680267 | Zbl 1194.05162
[9] Y. Zamani, E. Babaei: Symmetry classes of polynomials associated with the dicyclic group. Asian-Eur. J. Math. 6 (2013), Article ID 1350033, 10 pages. DOI 10.1142/S1793557113500332 | MR 3130082 | Zbl 1277.05168
[10] Y. Zamani, E. Babaei: The dimensions of cyclic symmetry classes of polynomials. J. Algebra Appl. 13 (2014), Article ID 1350085, 10 pages. DOI 10.1142/S0219498813500850 | MR 3119646 | Zbl 1290.05156
[11] Y. Zamani, M. Ranjbari: Induced operators on the space of homogeneous polynomials. Asian-Eur. J. Math. 9 (2016), Article ID 1650038, 15 pages. DOI 10.1142/S1793557116500388 | MR 3486726 | Zbl 06580479

Affiliations:   Yousef Zamani (corresponding author), Mahin Ranjbari, Department of Mathematics, Faculty of Sciences, Sahand University of Technology, P.O. Box 51335/1996, Tabriz, East Azerbaijan, Iran, e-mail: zamani@sut.ac.ir, m_ranjbari@sut.ac.ir


 
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