| Title:
|
Representations of the general linear group over symmetry classes of polynomials (English) |
| Author:
|
Zamani, Yousef |
| Author:
|
Ranjbari, Mahin |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
1 |
| Year:
|
2018 |
| Pages:
|
267-276 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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) $T\mapsto B_{\chi }^{G}(T)$. (English) |
| Keyword:
|
symmetry class of polynomials |
| Keyword:
|
general linear group |
| Keyword:
|
representation |
| Keyword:
|
irreducible character |
| Keyword:
|
induced operator |
| MSC:
|
05E05 |
| MSC:
|
15A69 |
| MSC:
|
20C15 |
| idZBL:
|
Zbl 06861580 |
| idMR:
|
MR3783598 |
| DOI:
|
10.21136/CMJ.2017.0458-16 |
| . |
| Date available:
|
2018-03-19T10:31:04Z |
| Last updated:
|
2020-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147134 |
| . |
| Reference:
|
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| Reference:
|
[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. Zbl 1330.05159, MR 3213989 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[6] Ranjbari, M., Zamani, Y.: Induced operators on symmetry classes of polynomials.Int. J. Group Theory 6 (2017), 21-35. MR 3621030 |
| Reference:
|
[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. Zbl 1297.05243, MR 3225126, 10.1142/S0219498814500595 |
| Reference:
|
[8] Shahryari, M.: Relative symmetric polynomials.Linear Algebra Appl. 433 (2010), 1410-1421. Zbl 1194.05162, MR 2680267, 10.1016/j.laa.2010.05.020 |
| Reference:
|
[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. Zbl 1277.05168, MR 3130082, 10.1142/S1793557113500332 |
| Reference:
|
[10] Zamani, Y., Babaei, E.: The dimensions of cyclic symmetry classes of polynomials.J. Algebra Appl. 13 (2014), Article ID 1350085, 10 pages. Zbl 1290.05156, MR 3119646, 10.1142/S0219498813500850 |
| Reference:
|
[11] Zamani, Y., Ranjbari, M.: Induced operators on the space of homogeneous polynomials.Asian-Eur. J. Math. 9 (2016), Article ID 1650038, 15 pages. Zbl 06580479, MR 3486726, 10.1142/S1793557116500388 |
| . |
