Representations of the general linear group over symmetry classes of polynomials

Zamani, Yousef, and Ranjbari, Mahin. "Representations of the general linear group over symmetry classes of polynomials." Czechoslovak Mathematical Journal 68.1 (2018): 267-276. <http://eudml.org/doc/294362>.

@article{Zamani2018,
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) $T\mapsto B_\{\chi \}^\{G\}(T)$.},
author = {Zamani, Yousef, Ranjbari, Mahin},
journal = {Czechoslovak Mathematical Journal},
keywords = {symmetry class of polynomials; general linear group; representation; irreducible character; induced operator},
language = {eng},
number = {1},
pages = {267-276},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Representations of the general linear group over symmetry classes of polynomials},
url = {http://eudml.org/doc/294362},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Zamani, Yousef
AU - Ranjbari, Mahin
TI - Representations of the general linear group over symmetry classes of polynomials
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 1
SP - 267
EP - 276
AB - 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) $T\mapsto B_{\chi }^{G}(T)$.
LA - eng
KW - symmetry class of polynomials; general linear group; representation; irreducible character; induced operator
UR - http://eudml.org/doc/294362
ER -