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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