Geodesic
Representations of the general linear group over symmetry classes of polynomials
Czechoslovak Mathematical Journal, Tome 68 (2018) no. 1, pp. 267-276 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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) $T\mapsto B_{\chi }^{G}(T)$.
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) $T\mapsto B_{\chi }^{G}(T)$.
DOI : 10.21136/CMJ.2017.0458-16
Classification : 05E05, 15A69, 20C15
Keywords: symmetry class of polynomials; general linear group; representation; irreducible character; induced operator 
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Zamani, Yousef; Ranjbari, Mahin. Representations of the general linear group over symmetry classes of polynomials. Czechoslovak Mathematical Journal, Tome 68 (2018) no. 1, pp. 267-276. doi: 10.21136/CMJ.2017.0458-16

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