On the Orthogonal Basis of the Symmetry Classes of Tensors Associated with Certain Characters

M. R. Pournaki^{*}

Abstract A necessary condition is given for the existence of an O-basis for the symmetry classes of tensors associated with a finite group and the irreducible constituents of the permutation character of the finite group. This extends a result of R. R. Holmes [see the Main theorem of Linear and Multilinear Algebra 39 (1995), 241-243].

Keywords: Symmetry class of tensors, Orthogonal basis, Permutation character, Irreducible constituent. 2000 Mathematics Subject Classification: Primary 20C30; Secondary 15A69.

1 Introduction

Let V be an m-dimensional vector space over the complex field ℂ. Let ⊗^{n}V be the nth tensor power of V and write v_{1} ⊗ ⋯ ⊗ v_{n} for the decomposable tensor product of the indicated vectors. To each permutation σ in $_{n} there corresponds a unique linear operator

P (σ): ⊗^{n}V → ⊗^{n}V determined by P (σ)(v_{1} ⊗ ⋯ ⊗ v_{n}) = v_{σ^{−1}(1)} ⊗ ⋯ ⊗ v_{σ^{−1}(n)}. Let G be a

subgroup of $_{n} and let Irr(G) be the set of all the irreducible complex characters of G. It follows from the orthogonality relations for characters that

\{T (G, χ) : ⊗^{n}V → ⊗^{n}V | T (G, χ) = \frac{χ(1)}{|G|} ∑_{σ∈G} χ(σ)P (σ), χ ∈ Irr(G)\}

is a set of annihilating idempotents which sum to the identity. The image of ⊗^{n}V under the T (G, χ) is called the symmetry class of tensors associated with G and χ and is denoted by V_{χ}^{n}(G). The image of v_{1} ⊗ ⋯ ⊗ v_{n} under T (G, χ) is denoted by v_{1} ∗ … ∗ v_{n} and is called a decomposable symmetrized tensor.

Let Γ_{m}^{n} be the set of all sequences α = (α_{1}, …, α_{n}) with 1 ≤ α_{i} ≤ m. Then the group G acts on Γ_{m}^{n} by σ · α = (α_{σ^{−1}(1)}, …, α_{σ^{−1}(n)}), where σ ∈ G and α ∈ Γ_{m}^{n}. Let O(α) = {σ · α| σ ∈ G} be the orbit of α, and G_{α} be its stabilizer subgroup, i.e., G_{α} = {σ ∈ G| σ · α = α}, and consider a system Δ of distinct representatives of the orbits of Γ_{m}^{n}.

*This research was in part supported by a grant from IPM.

Let {e_{1}, …, e_{m}} be a basis of V. Denote by e_{α}^{∗} the tensor e_{α_{1}} ∗ … ∗ e_{α_{n}}, where α = (α_{1}, …, α_{n}) ∈ Γ_{m}^{n}. For α ∈ Δ, V_{α}^{∗} = ⟨e_{σ·α}^{∗}| σ ∈ G⟩ is called the orbital subspace of V_{χ}^{n}(G), and we can easily prove that

V_{χ}^{n}(G) = ⨁_{α∈Δ}V_{α}^{∗}.  (1)

Note that it is possible for some α ∈ Δ to have V_{α}^{∗} = 0. But Freese (see [2]) proved that for α ∈ Δ

dim V_{α}^{∗} = \frac{χ(1)}{|G_{α}|} ∑_{σ∈G_{α}}χ(σ),  (2)

therefore, if we consider

\overline{Δ} = \{α ∈ Δ| ∑_{σ∈G_{α}} χ(σ) ≠ 0\},

then the sum in (1) will be reduced to the form

V_{χ}^{n}(G) = ⨁_{α∈\overline{Δ}}V_{α}^{∗}.  (3)

Of course we define the right-hand side of (3) to be 0, if \overline{Δ} = ∅.

A particular case appears when we assume that V is an m-unitary space. In this case, the inner product on V induces an inner product on ⊗^{n}V, whose restriction to V_{χ}^{n}(G) satisfies

⟨u_{1} ∗ … ∗ u_{n}|v_{1} ∗ … ∗ v_{n}⟩ = \frac{χ(1)}{|G|} ∑_{σ∈G}χ(σ) ∏_{i=1}^{n} A_{iσ(i)},

where A = [A_{ij}]_{n×n} = [⟨u_{i}|v_{j}⟩]_{n×n}.

If {e_{1}, …, e_{m}} is an orthonormal basis of V, then the sum appeared in (3) is an orthogonal direct sum. In addition, we can obtain

⟨e_{g·α}^{∗}|e_{g′·α}^{∗}⟩ = \frac{χ(1)}{|G|} ∑_{σ∈G_{α}}χ(g′σg^{−1}).  (4)

Let W be a subspace of V_{χ}^{n}(G). An orthogonal basis of W of the form {e_{α}^{∗}| α ∈ S}, where S is a subset of Γ_{m}^{n}, is called an O-basis of W. Because the sum appeared in (3) is an orthogonal direct sum, V_{χ}^{n}(G) has an O-basis if and only if for all α ∈ \overline{Δ}, the orbital subspace V_{α}^{∗} has an O-basis. Note that, if χ is of degree one, since dim V_{α}^{∗} = 1 for all α ∈ \overline{Δ}, then V_{α}^{∗} has an O-basis for all α ∈ \overline{Δ}, which implies that V_{χ}^{n}(G) has such a basis.

Several papers are devoted to the investigation of the existence of an O-basis for V_{χ}^{n}(G), see for example [6]. In [4] a necessary and sufficient condition for the existing of an O-basis for V_{χ}^{n}(G) is given, where G is a cyclic or a dihedral group and in [1] when G is a dicyclic group.

The main purpose of this article is to generalize a result of Holmes (see [3, Main theorem]). In fact, he proved that if G is a 2-transitive subgroup of $_{n}, n ≥ 3, χ = θ − 1_{G}, where θ is permutation character of G, and m ≥ 2, then V_{χ}^{n}(G) does not have an O-basis.

In the following section, we will omit condition of “2-transitivity of G” and will find a necessary condition for the existence of an O-basis for the symmetry class of tensors associated with G and any constituent of its permutation character. This generalization will be independent of the permutation structure of the group G.

2 Results

Let V be an m-unitary space, G a finite group, and Ω a set of n elements. Suppose G acts faithfully on Ω, so we can assume that G is a subgroup of $_{n}. In fact, we consider {f_{σ}| σ ∈ G} as the group G, where f_{σ} : Ω ⟶ Ω defined by f_{σ}(ω) = σ · ω for all ω ∈ Ω, is a permutation on n letters. Therefore the inner product space V_{χ}^{n}(G) is meaningful for all χ ∈ Irr(G). Denote the permutation character of G by θ. For σ ∈ G, the value θ(σ) is the number of letters fixed by σ, i.e., θ(σ) = |{ω ∈ Ω| σ · ω = ω}|.

Main Theorem Let G be a finite group and let Ω be a set of n elements, n ≥ 2. Assume that G acts transitively and faithfully on Ω and let V be an m-unitary space, m ≥ 2. Let χ be an irreducible constituent of θ, where θ is the permutation character of G. If χ(1)(χ, θ)_{G} > \frac{1}{2}n, then V_{χ}^{n}(G) does not have an O-basis.

Proof. Let {e_{1}, …, e_{m}} be an orthonormal basis of V. Suppose V_{χ}^{n}(G) has an O-basis. Then, by (3), for all α ∈ \overline{Δ}, the orbital subspace V_{α}^{∗} has an O-basis. Put α = (1, 2, …, 2), then n ≥ 2 and m ≥ 2 implies that α ∈ Γ_{m}^{n}. Consider the action of G on Γ_{m}^{n} and choose Δ such that α ∈ Δ.

Without loss of generality, we can assume that Ω = {1, 2, …, n} and therefore G_{α} = G_{1}, where G_{α} is the stabilizer subgroup of α when G acts on Γ_{m}^{n} and G_{1} is the stabilizer subgroup of 1 when G acts on Ω. So by Lemma 5.14 of [5] we obtain 1_{G_{α}} ↑^{G}= 1_{G_{1}} ↑^{G}= θ.

Frobenius reciprocity implies that

∑_{σ∈G_{α}}χ(σ) = |G_{α}|(χ ↓_{G_{α}}, 1_{G_{α}})_{G_{α}} = |G_{α}|(χ, 1_{G_{α}} ↑^{G})_{G} = |G_{α}|(χ, θ)_{G},  (5)

so by hypotheses, (χ, θ)_{G} ≠ 0, and we obtain ∑_{σ∈G_{α}} χ(σ) ≠ 0 and α ∈ \overline{Δ}. By the above discussion, V_{α}^{∗}, α = (1, 2, …, 2), has an O-basis. We now have [G : G_{α}] = [G : G_{1}] = |Ω| = n, so G = ⋃_{i=1}^{n}g_{i}G_{α}, where {g_{1}, …, g_{n}} is a system of distinct representatives of left cosets of G_{α} in G.

Since, by (2) and (5),

dim V_{α}^{∗} = \frac{χ(1)}{|G_{α}|} ∑_{σ∈G_{α}} χ(σ) = χ(1)(χ, θ)_{G} =: s,

we can assume that {e_{g_{1}·α}^{∗}, e_{g_{2}·α}^{∗}, …, e_{g_{s}·α}^{∗}} is an O-basis for V_{α}^{∗}.

Define the n × n complex matrix A = [A_{ij}] by

A_{ij} = ⟨e_{g_{i}·α}^{∗}|e_{g_{j}·α}^{∗}⟩.

Note that s < n, therefore A may be written in the form

A = [ \begin{matrix}A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix} ] ,

where A_{1}, A_{2}, A_{3}, and A_{4} are matrices of sizes s×s, s×(n−s), (n−s)×s, and (n−s)×(n−s), respectively. But, by (4), for i, j, 1 ≤ i, j ≤ s, we have

A_{ij} = ⟨e_{g_{i}·α}^{∗}|e_{g_{j}·α}^{∗}⟩

= \{ \begin{matrix}0 & if i ≠ j, \\ \frac{χ(1)}{|G|} ∑_{σ∈G_{α}} χ(σ) & if i = j, \end{matrix}

= \{ \begin{matrix}0 & if i ≠ j, \\ \frac{1}{[G:G_{α}]} dim V_{α}^{∗} & if i = j, \end{matrix}

= \{ \begin{matrix}0 & if i ≠ j, \\ \frac{s}{n} & if i = j, \end{matrix}

= (\frac{s}{n}I_{s})_{ij}, where I_{s} is the s × s identity matrix. So the block form of A has been changed to the form

A = [ \begin{matrix}\frac{s}{n}I_{s} & A_{2} \\ A_{3} & A_{4} \end{matrix} ] .

We now claim that A is idempotent. In fact,

(A^{2})_{ij} = ∑_{ℓ=1}^{n}A_{iℓ}A_{ℓj}

= ∑_{ℓ=1}^{n}⟨e_{g_{i}·α}^{∗}|e_{g_{ℓ}·α}^{∗}⟩⟨e_{g_{ℓ}·α}^{∗}|e_{g_{j}·α}^{∗}⟩

= ∑_{ℓ=1}^{n} (\frac{χ(1)}{|G|} ∑_{σ∈G_{α}}χ(g_{ℓ}σg_{i}^{−1})) (\frac{χ(1)}{|G|} ∑_{τ∈G_{α}}χ(g_{j}τg_{ℓ}^{−1}))

= \frac{χ(1)^{2}}{|G|^{2}} ∑_{ℓ=1}^{n} ∑_{σ∈G_{α}} ∑_{τ∈G_{α}}χ(g_{ℓ}σg_{i}^{−1})χ(g_{j}τg_{ℓ}^{−1})

= \frac{χ(1)^{2}}{|G|^{2}} ∑_{ℓ=1}^{n} ∑_{λ∈g_{ℓ}G_{α}} ∑_{μ∈G_{α}g_{ℓ}^{−1}}χ(λg_{i}^{−1})χ(g_{j}μ)

= \frac{χ(1)^{2}}{|G|^{2}} ∑_{ℓ=1}^{n} ∑_{λ∈g_{ℓ}G_{α}} ∑_{σ∈G_{α}}χ(λg_{i}^{−1})χ(g_{j}σλ^{−1})

= \frac{χ(1)^{2}}{|G|^{2}} ∑_{λ∈G} ∑_{σ∈G_{α}}χ(λg_{i}^{−1})χ(g_{j}σλ^{−1})

= \frac{χ(1)^{2}}{|G|^{2}} ∑_{λ∈G} ∑_{σ∈G_{α}}χ(λ)χ(g_{j}σg_{i}^{−1}λ^{−1})

= \frac{χ(1)^{2}}{|G|^{2}} ∑_{σ∈G_{α}}∑_{λ∈G}χ(λ)χ(g_{j}σg_{i}^{−1}λ^{−1})

= \frac{χ(1)^{2}}{|G|^{2}} ∑_{σ∈G_{α}} (\frac{|G|}{χ(1)}χ(g_{j}σg_{i}^{−1}))

= \frac{χ(1)}{|G|} ∑_{σ∈G_{α}}χ(g_{j}σg_{i}^{−1})

\begin{matrix}= & ⟨e_{g_{i}·α}^{∗}|e_{g_{j}·α}^{∗}⟩ \\ = & A_{ij}. \end{matrix}

Therefore, the claim holds. Now, using A^{2} = A, we obtain

A_{2}A_{3} = (\frac{s}{n} − \frac{s^{2}}{n^{2}} )I_{s}.

Since 0 < s < n, A_{2}A_{3} is an invertible matrix and hence s ≤ n−s, s ≤ \frac{1}{2}n, or χ(1)(χ, θ)_{G} ≤ \frac{1}{2}n, which is a contradiction. Thus V_{χ}^{n}(G) does not have an O-basis. □

Corollary 1 (Main theorem [3]) Let G be a 2-transitive subgroup of $_{n}, n ≥ 3. Let χ = θ − 1_{G}, where θ is the permutation character of G. If dimV = m ≥ 2, then V_{χ}^{n}(G) does not have an O-basis.

Proof. G has a natural transitive and faithful action on the set Ω = {1, 2, …, n}, given by σ · i = σ(i). 2-Transitivity of this action implies that χ is an irreducible constituent of θ . On the other hand, n ≥ 3 and so χ(1)(χ, θ)_{G} = (n − 1) > \frac{1}{2}n. Hence, by our main theorem, V_{χ}^{n}(G) does not have an O-basis. □

Example Let G=Ω=𝔸_{4} be the alternating group of degree 4. We know that G acts transitively and faithfully on Ω by right multiplication, so G is a subgroup of $_{12}. G has an irreducible complex character, say χ, of degree 3. As θ, the permutation character of G, is regular, χ is an irreducible constituent of θ of multiplicity 3. Therefore, χ(1)(χ, θ)_{G} = 9 > 6 = \frac{1}{2}|Ω|, so by the main theorem V_{χ}^{12}(G) does not have an O-basis. Note that in this example the action of G on Ω is not 2-transitive, so our main result really extends the result of [3].

References

[1] M. R. Darafsheh, M. R. Pournaki, On the Orthogonal Basis of the Symmetry Classes of Tensors Associated with the Dicyclic Group, Linear and Multilinear Algebra 47 (2000), no. 2, 137-149.

[2] R. Freese, Inequalities for Generalized Matrix Functions Based on Arbitrary Characters, Linear Algebra Appl. 7 (1973), 337-345.

[3] R. R. Holmes, Orthogonal Bases of Symmetrized Tensor Space, Linear and Multilinear Algebra 39 (1995), 241-243.

[4] R. R. Holmes, T. Y. Tam, Symmetry Classes of Tensors Associated with Certain Groups, Linear and Multilinear Algebra 32 (1992), 21-31.

[5] I. M. Isaacs, “Character Theory of Finite Groups”, Academic Press, New York, 1976.

[6] B. Y. Wang, M. P. Gong, The Subspace and Orthonormal Bases of Symmetry Classes of Tensors, Linear and Multilinear Algebra 30 (1991), 195-204.

The Author’s Address

M. R. Pournaki, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11365-9415, Tehran, Iran, and School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran.

E-mail address: pournaki@karun.ipm.ac.ir