Some Non-trivial Symmetry Classes of Tensors Associated with Certain Characters

M. R. Pournaki^{*}

Abstract Let G be a finite group and Ω a set of n elements. Assume that G acts faithfully on Ω and let V be a vector space over the complex field ℂ, with dimV = m ≥ 2. It is shown that for each irreducible constituent χ of permutation character of G, the symmetry class of tensors associated with G and χ is non-trivial. This extends a result of Merris and Rashid (see [6, Theorem 2]).

Keywords: Symmetry class of tensors, Permutation character, Irreducible constituent. 1995 Mathematics Subject Classification: Primary 20C30; Secondary 15A69.

1 Introduction

Let V be an m-dimensional vector space over the complex field ℂ, and let G be a subgroup of the symmetric group on n letters, $_{n}, and let χ be an irreducible complex character of G. Suppose ϕ : ×^{n}V → U be an n-linear function, where U is a finite dimensional vector space over ℂ. We say ϕ is symmetric with respect to G and χ if for all v_{1}, …, v_{n} ∈ V,

\frac{χ(1)}{|G|} ∑_{σ∈G} χ(σ)ϕ(v_{σ^{−1}(1)}, …, v_{σ^{−1}(n)}) = ϕ(v_{1}, …, v_{n}).

A finite dimensional vector space S over ℂ is called a symmetry class of tensors associated with G and χ if there is an n-linear function ϕ : ×^{n}V → S which is symmetric with respect to G and χ such that

(i) ⟨Im ϕ⟩ = S, (ii) for each finite dimensional vector space U over ℂ and for each n-linear function ψ : ×^{n}V → U, symmetric with respect to G and χ, there exists a unique linear transformation f : S → U such that the following diagram commutes.

\begin{matrix}×^{n}V & ⟶^{ϕ} S \\ ψ ↓ & ↙ f \\ U \end{matrix}

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

One easily obtains that the symmetry class of tensors associated with G and χ exists and it is unique up to isomorphisms of vector spaces. We denote the symmetry class of tensors associated with G and χ by V_{χ}^{n}(G). Let ⊗^{n}V be the n-th 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)}.

Now if we define the linear operator T (G, χ) : ⊗^{n}V ⟶ ⊗^{n}V by

T (G, χ) = \frac{χ(1)}{|G|} ∑_{σ∈G} χ(σ)P (σ),

then one easily obtains that the symmetry class of tensors associated with G and χ is isomorphic with the image of T (G, χ) and so we can assume that the symmetry class of tensors associated with G and χ is equal to this image: V_{χ}^{n}(G) = T (G, χ)(⊗^{n}V). The image of v_{1} ⊗ … ⊗ v_{n} under T (G, χ) is denoted by v_{1} ∗ … ∗ v_{n} and it is called a decomposable symmetrized tensor. 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 | χ ∈ Irr(G)},

is a set of annihilating idempotents which sum to the identity and therefore by a classical theorem in linear algebra we have

⨂^{n}V = ⨁_{χ∈Irr(G)}V_{χ}^{n}(G).

It is well-known that

dim V_{χ}^{n}(G) = \frac{χ(1)}{|G|} ∑_{σ∈G} χ(σ)m^{c(σ)},

where c(σ) is the number of cycles, including cycles of length one, in the disjoint cycle decomposition of σ (see [5]).

Let Γ_{m}^{n} be the set of all sequences α = (α_{1}, …, α_{n}) with 1 ≤ α_{i} ≤ m, so α is a mapping from a set of n elements into a set of m elements. Then the group G acts on Γ_{m}^{n} by σ · α = (α_{σ^{−1}(1)}, …, α_{σ^{−1}(n)}) where σ ∈ G is a permutation on n letters and α ∈ Γ_{m}^{n} is a mapping from a set of n elements into a set of m elements. Therefore the action may be written as σ · α = ασ^{−1} which is a composition of two functions.

Let O(α) = {σ · α| σ ∈ G} be the orbit with representative α, also let G_{α} be the stabilizer subgroup of α, i.e., G_{α} = {σ ∈ G| σ · α = α}, and consider Δ be a system of distinct representatives of the orbits of Γ_{m}^{n}, when G acts on Γ_{m}^{n}.

Consider {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 we have V_{α}^{∗} = 0. But Freese (see [3]) proved that for α ∈ Δ

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

and therefore, if we consider

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

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

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

of course we define that the right hand of (3) equals 0, if \overline{Δ} = ∅.

Non-triviality of V_{χ}^{n}(G) were studied by several authors. In [1], the authors discuss the non-triviality of the symmetry class of tensors associated with the projective special linear group, P SL_{2}(q), as a subgroup of the symmetric group $_{q+1}. Also in [2], the authors considered G as a subgroup of $_{n} by Cayley representation and proved that if m ≥ 2, then for all χ, χ ∈ Irr(G), V_{χ}^{n}(G) ≠ 0. The main purpose of this article is to generalize a result of Merris and Rashid (see [6, Theorem 2]). In fact, they proved that if G is a doubly transitive subgroup of $_{n} and χ = θ − 1_{G}, where θ is permutation character of G, then for all m, m ≥ 2, V_{χ}^{n}(G) ≠ 0.

In the next section, we will prove that the condition of “doubly transitivity of G” can be omitted and so, when m ≥ 2, the symmetry class of tensors associated with G and each constituent of its permutation character is non-trivial. This generalization will be independent of the permutation structure of the group G and we will obtain the result appeared in [2] that we mentioned earlier.

2 Results

Let V be an m-dimensional vector space over the complex field ℂ and let G be 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 vector 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 Ω a set of n elements. Assume that G acts faithfully on Ω and let V be an m-dimensional vector space over the complex field ℂ. If m ≥ 2, then for all irreducible constituent χ of permutation character of G, we have V_{χ}^{n}(G) ≠ 0.

Proof. If n = 1, the theorem is obvious, so we assume that n ≥ 2. Denote the permutation character of G by θ and suppose χ be any irreducible constituent of θ, therefore (θ, χ)_{G} ≠ 0. Without losing generality, we can assume that Ω = {1, 2, …, n}. Consider {i_{1}, …, i_{s}} ⊆ Ω be a system of distinct representatives of the orbits of Ω, when G acts on Ω. It is well-know that

θ = 1_{G_{i_{1}}} ↑^{G} + … + 1_{G_{is}} ↑^{G},

where G_{i_{j}}, 1 ≤ j ≤ s, is the stabilizer subgroup of i_{j} (see [4, Lemma 5.14]). Therefore (θ, χ)_{G} ≠ 0 implies that

(1_{G_{i_{1}}} ↑^{G}, χ)_{G} + … + (1_{G_{is}} ↑^{G}, χ)_{G} ≠ 0,

and so for some j, 1 ≤ j ≤ s,

(1_{G_{i_{j}}} ↑^{G}, χ)_{G} ≠ 0.  (4)

Now we consider α = (1, …, 1, 2, 1, …, 1) where 2 appeared in i_{j}-th place. Since n ≥ 2 and m ≥ 2, we obtain that α ∈ Γ_{m}^{n}. Consider the action of G on Γ_{m}^{n} and choose Δ such that α ∈ Δ. Note that G_{α} = G_{i_{j}}, where G_{α} is the stabilizer subgroup of α when G acts on Γ_{m}^{n} and G_{i_{j}} is the stabilizer subgroup of i_{j} when G acts on Ω. So by Frobenius reciprocity we have

(1_{G_{i_{j}}} ↑^{G}, χ)_{G} = (1_{G_{α}} ↑^{G}, χ)_{G} = (χ ↓_{G_{α}}, 1_{G_{α}})_{G_{α}} = \frac{1}{|G_{α}|} ∑_{σ∈G_{α}}χ(σ),

and by (4) we obtain that

∑_{σ∈G_{α}} χ(σ) ≠ 0.

Therefore α ∈ \overline{Δ} and so we have \overline{Δ} ≠ ∅. Now by (2) and (3) we obtain that V_{χ}^{n}(G) ≠ 0. □

Corollary 1 ([Theorem 2 of 6]) Let G be a doubly transitive subgroup of $_{n}. Let χ = θ − 1_{G}, where θ is permutation character of G. If dimV = m ≥ 2, then V_{χ}^{n}(G) ≠ 0.

Proof. G has a natural faithful action on set Ω = {1, …, n} by σ · i = σ(i). Doubly transitivity of this action implies that χ ∈ Irr(G). So χ is an irreducible constituent of θ and therefore by the main theorem we have V_{χ}^{n}(G) ≠ 0. □

Now we consider a special case. Suppose G is a group of order n and G acts on G by right multiplication, i.e., for all g, g′ ∈ G, g.g′ = gg′. This action is faithful, therefore G is a subgroup of $_{n} and in this case we say G is a subgroup of $_{n} by Cayley representation.

Corollary 2 ([Theorem 5 of 2]) Let G be a group of order n, that is a subgroup of $_{n} by Cayley representation. If dimV = m ≥ 2, then for all χ, χ ∈ Irr(G), V_{χ}^{n}(G) ≠ 0.

Proof. When G is a subgroup of $_{n} by Cayley representation, each elements of G are fixed point free, except identity, therefore permutation character of G, in this case, is regular character and so any irreducible character of G is a constituent for θ. So by main theorem for all χ, χ ∈ Irr(G), V_{χ}^{n}(G) ≠ 0. □

References

[1] M. R. Darafsheh, M. R. Pournaki, Computation of the Dimensions of Symmetry Classes of Tensors Associated with the Finite two Dimensional Projective Special Linear Group, Appl. Algebra Engrg. Comm. Comput. 10 (2000), no. 3, 237-250.

[2] M. R. Darafsheh, M. R. Pournaki, Non-Vanishing and Orthogonal Basis of Symmetry Classes of Tensors, Southeast Asian Bull. Math. 24 (2000), no. 4, 525-531.

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

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

[5] M. Marcus, “Finite Dimensional Multilinear Algebra”, Part 1, Marcel Dekker, New York, 1973.

[6] R. Merris, M. A. Rashid, The Dimension of Certain Symmetry Classes of Tensors, Linear and Multilinear Algebra 2 (1974), 245-248.

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