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Czechoslovak Mathematical Journal

, Volume 68, Issue 1, pp 267–276 | Cite as

Representations of the general linear group over symmetry classes of polynomials

  • Yousef Zamani
  • Mahin Ranjbari
Article

Abstract

Let V be the complex vector space of homogeneous linear polynomials in the variables x1,..., x m . Suppose G is a subgroup of S m , and χ is an irreducible character of G. Let H d (G, χ) be the symmetry class of polynomials of degree d with respect to G and χ.

For any linear operator T acting on V, there is a (unique) induced operator K χ (T) ∈ End(H d (G, χ)) acting on symmetrized decomposable polynomials by
$${K_\chi }\left( T \right)\left( {{f_1} * {f_2} * \cdots * {f_d}} \right) = T{f_1} * T{f_2} * \cdots * T{f_d}.$$
In this paper, we show that the representation TK χ (T) of the general linear group GL(V) is equivalent to the direct sum of χ(1) copies of a representation (not necessarily irreducible) TB χ G (T).

Keywords

symmetry class of polynomials general linear group representation irreducible character induced operator 

MSC 2010

20C15 15A69 05E05 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesSahand University of TechnologyTabriz, East AzerbaijanIran

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