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Keywords:
symmetry class of polynomials; general linear group; representation; irreducible character; induced operator
Summary:
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 $. \endgraf 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) $T\mapsto B_{\chi }^{G}(T)$.
References:
[1] Babaei, E., Zamani, Y.: Symmetry classes of polynomials associated with the dihedral group. Bull. Iran. Math. Soc. 40 (2014), 863-874. MR 3255403 | Zbl 1338.05271
[2] Babaei, E., Zamani, Y.: 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] Babaei, E., Zamani, Y., Shahryari, M.: Symmetry classes of polynomials. Commun. Algebra 44 (2016), 1514-1530. DOI 10.1080/00927872.2015.1027357 | MR 3473866 | Zbl 1338.05272
[4] Isaacs, I. M.: Character Theory of Finite Groups. Pure and Applied Mathematics 69, Academic Press, New York (1976). MR 0460423 | Zbl 0337.20005
[5] Merris, R.: Multilinear Algebra. Algebra, Logic and Applications 8, Gordon and Breach Science Publishers, Amsterdam (1997). MR 1475219 | Zbl 0892.15020
[6] Ranjbari, M., Zamani, Y.: Induced operators on symmetry classes of polynomials. Int. J. Group Theory 6 (2017), 21-35. MR 3621030
[7] Rodtes, K.: 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] Shahryari, M.: Relative symmetric polynomials. Linear Algebra Appl. 433 (2010), 1410-1421. DOI 10.1016/j.laa.2010.05.020 | MR 2680267 | Zbl 1194.05162
[9] Zamani, Y., Babaei, E.: 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] Zamani, Y., Babaei, E.: 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] Zamani, Y., Ranjbari, M.: 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
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