Representations of the general linear group over symmetry classes of polynomials

Let V be the complex vector space of homogeneous linear polynomials in the variables x 1 ,..., 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...

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Bibliographic Details
Published in:Czechoslovak mathematical journal 2018-03, Vol.68 (1), p.267-276
Main Authors: Zamani, Yousef, Ranjbari, Mahin
Format: Article
Language:English
Subjects:
Analysis
Convex and Discrete Geometry
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Ordinary Differential Equations
Polynomials
Representations
Subgroups
Symmetry
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Summary:Let V be the complex vector space of homogeneous linear polynomials in the variables x 1 ,..., 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 χ ( T ) ( f 1 ∗ f 2 ∗ ⋯ ∗ f d ) = T f 1 ∗ T f 2 ∗ ⋯ ∗ T f d . In this paper, we show that the representation T ↦ K 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) T ↦ B χ G ( T ).
ISSN:0011-4642
1572-9141
DOI:10.21136/CMJ.2017.0458-16