Generalized Symmetry Classes of Tensors
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Czechoslovak Mathematical Journal
13 pp
Online first
GENERALIZED SYMMETRY CLASSES OF TENSORS Gholamreza Rafatneshan, Yousef Zamani, Tabriz Received January 29, 2019. Published online July 8, 2020.
Abstract. Let V be a unitary space. For an arbitrary subgroup G of the full symmetric group Sm and an arbitrary irreducible unitary representation Λ of G, we study the generalized symmetry class of tensors over V associated with G and Λ. Some important properties of this vector space are investigated. Keywords: irreducible character; generalized Schur function; orthogonal basis; symmetry class of tensors MSC 2020 : 20C30, 15A69
1. Introduction Let Sm be the full symmetric group of degree m and G a subgroup of Sm . Let U be a unitary space and End(U ) the set of all linear operators on U . Denote by Cm×m the set of all m×m complex matrices. Suppose Λ is an irreducible unitary representation of G over U . The generalized Schur function DΛ : Cm×m → End(U ) is defined by DΛ (A) =
X
σ∈G
Λ(σ)
m Y
aiσ(i)
i=1
for A = (aij )m×m ∈ Cm×m . Let V be a unitary space of dimension n and denote by V ⊗m the mth tensor power of V . Then U ⊗ V ⊗m is a unitary space with induced inner product that satisfies (u ⊗ x⊗ , v ⊗ y ⊗ ) = (u, v)
m Y
(xi , yi ),
i=1
where u, v ∈ U and x⊗ = x1 ⊗ . . . ⊗ xm , y ⊗ = y1 ⊗ . . . ⊗ ym ∈ V ⊗m . DOI: 10.21136/CMJ.2020.0044-19
1
For any σ ∈ G there is a unique permutation operator P (σ) : V ⊗m → V ⊗m satisfying P (σ −1 )(v ⊗ ) = vσ⊗ , where vσ⊗ = vσ(1) ⊗ vσ(2) ⊗ . . . ⊗ vσ(m) . The permutation operator yields a representation of G, i.e. P : G → GL(V ⊗m ). It is well known that if dim V > 2, then P is a faithful unitary reducible representation of G and Tr P (σ) = nc(σ) , where c(σ) is the number of factors in the disjoint cycle factorization of σ, see [9]. The generalized symmetrizer associated with G and Λ is defined by SΛ =
1 X Λ(σ) ⊗ P (σ) ∈ End(U ⊗ V ⊗m ). |G| σ∈G
In the following theorem we show that SΛ is an orthogonal projection on U ⊗ V ⊗m . Theorem 1.1. Suppose Λ is an irreducible unitary representation of G over unitary space U . Then SΛ is an orthogonal projection on U ⊗ V ⊗m . P r o o f . We first prove that SΛ is Hermitian. We have ∗ 1 X 1 X ∗ SΛ = Λ(σ) ⊗ P (σ) = Λ(σ)∗ ⊗ P (σ)∗ |G| |G| σ∈G σ∈G 1 X −1 −1 = Λ(σ ) ⊗ P (σ ) = SΛ . |G| σ∈G
Now we show that SΛ is idempotent. We have 1 X 1 X SΛ2 = Λ(σ) ⊗ P (σ) Λ(π) ⊗ P (π) |G| |G| σ∈G π∈G 1 XX = Λ(σ)Λ(π) ⊗ P (σ)P (π) |G|2 σ∈G π∈G 1 XX = Λ(σπ) ⊗ P (σπ) (σπ = τ ) |G|2 σ∈G π∈G 1 XX 1 X = Λ(τ ) ⊗ P (τ ) = SΛ = SΛ . 2 |G| |G| σ∈G τ ∈G
σ∈G
Definition 1.1. The range of SΛ , VΛ (G) := SΛ (U ⊗ V ⊗m ), is called the generalized symmetry class of tensors over V associated with G and Λ. 2
Online first
If dim U = 1, then VΛ (G) reduces to Vλ (G), the symmetry class of tensors associated with G and the irreducible character λ of G corresponding to the representation Λ (see [4], [5], [9], [10], [12], [13], [14]). Recently, the other types of symmetry classes have been studied by several authors (see [1], [2], [3], [7], [11], [15], [
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