Czechoslovak Mathematical Journal, online first, 10 pp.

# Representations of the general linear group over symmetry classes of polynomials

## Yousef Zamani, Mahin Ranjbari

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

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

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 2010): 20C15, 15A69, 05E05

DOI: 10.21136/CMJ.2017.0458-16

Full text available as PDF.

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