A Note on the Orthogonal Basis of a Certain Full Symmetry Class of Tensors

C. Bessenrodt, M. R. Pournaki^{*}, A. Reifegerste

Abstract It is shown that the full symmetry class of tensors associated with the irreducible character [2, 1^{n−2}] of S_{n} does not have an orthogonal basis consisting of decomposable symmetrized tensors.

Keywords: (Full) symmetry class of tensors, Orthogonal basis, Decomposable symmetrized tensor, Irreducible characters of the symmetric group. 2000 Mathematics Subject Classification: Primary 20C30; Secondary 15A69.

1 Introduction and Preliminaries

Let V be an m-unitary space. 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 S_{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 S_{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 T (G, χ) is called the symmetry class of tensors associated with G and χ and it 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 it is called a decomposable symmetrized tensor.

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}⟨u_{i}|v_{σ(i)}⟩.  (1)

*The research of the author was in part supported by a grant from IPM.

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 G-orbits on Γ_{m}^{n}.

Suppose {e_{1}, …, e_{m}} is an orthonormal basis of V. For α = (α_{1}, …, α_{n}) ∈ Γ_{m}^{n}, denote by e_{α}^{∗} the decomposable symmetrized tensor e_{α_{1}} ∗ … ∗ e_{α_{n}}. Then, by (1), one can easily obtain that for each α, β ∈ Γ_{m}^{n},

⟨e_{α}^{∗}|e_{β}^{∗}⟩ = \{ \begin{matrix}\frac{χ(1)}{|G|} ∑_{σ∈G_{β}} χ(στ^{−1}) & if α = τ · β for some τ ∈ G, \\ 0 & if O(α) ≠ O(β). \end{matrix}  (2)

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}^{∗}.  (3)

Note that it is possible for some α ∈ Δ to have V_{α}^{∗} = 0. But Freese [3] proved

dim V_{α}^{∗} = \frac{χ(1)}{|G_{α}|} ∑_{σ∈G_{α}}χ(σ),  (4)

therefore, if we set

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

then by (3) we obtain

V_{χ}^{n}(G) = ⨁_{α∈\overline{Δ}}V_{α}^{∗}.  (5)

Of course we define the right-hand side of (5) to be 0, if \overline{Δ} = ∅.

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.

Symmetry classes of tensors associated with subgroups of symmetric groups and their irreducible characters have been studied for a long time. Several papers are devoted to the investigation of the non-vanishing and existence of an O-basis for V_{χ}^{n}(G), see for example [1, 2, 5, 6, 8, 9]. In recent years the non-vanishing problem of decomposable symmetrized tensors and so the non-vanishing problem of V_{χ}^{n}(G), in the case G = S_{n}, have been studied by several authors, see for example [4, 7], and the non-vanishing problem in this case has been completely solved (see [4]). But even for G = S_{n}, no reasonable result for the structure of O-basis of an V_{χ}^{n}(G) is available. In this note we consider G = S_{n} and investigate the existence of an O-basis of V_{χ}^{n}(S_{n}), for a special irreducible character of S_{n}.

2 Main Result

Let n be a positive integer. A partition λ = (λ_{1}, …, λ_{l}) of n is a weakly decreasing sequence λ_{1} ≥ … ≥ λ_{l} > 0 of integers with ∑_{i=1}^{l} λ_{i} = n, for short we write λ ⊢ n. The number l is called the length of λ, denoted by l(λ). We often gather together equal parts of a partition and write, for example, (5^{2}, 3^{3}) for (5, 5, 3, 3, 3). If λ = (λ_{1}, …, λ_{l}) ⊢ n, then λ′ = (λ′_{1}, …, λ′_{s}), defined by

λ′_{i} = |{1 ≤ j ≤ l| λ_{j} ≥ i}|,

is a partition of n called the partition conjugate to λ.

Frobenius obtained in 1900 an explicit classification of the irreducible complex characters of S_{n}; they are naturally labelled by partitions of n. We denote the irreducible complex character labelled by the partition λ by [λ], so the set of all irreducible complex characters of S_{n} is Irr(S_{n}) = {[λ]| λ ⊢ n}.

Let now V be an m-unitary space. The symmetry class of tensors associated with S_{n} and [λ], where λ ⊢ n, is called full symmetry class of tensors associated with λ and for short denoted by V_{λ}^{n}. Holmes [5] proved that if m, n ≥ 3, then V_{(n−1,1)}^{n} is non-zero and does not have an O-basis. In this note we consider the special case λ = (2, 1^{n−2}), that is the partition conjugate to (n − 1, 1); we prove the analogue of Holmes’ result for the space V_{λ}^{n}.

The following result was already proved in [7].

Proposition 2.1 Let V be an m-unitary space. Let λ be a partition of n. Then the full symmetry class of tensors associated with λ, i.e., V_{λ}^{n}, is non-zero if and only if m ≥ l(λ). In particular, if m ≥ n − 1, then V_{λ}^{n} ≠ 0 for all λ ≠ (1^{n}).

We are now ready to state our main result. Note that by Proposition 2.1, V_{λ}^{n} is non-zero for λ = (2, 1^{n−2}) if and only if m ≥ n − 1.

Main Theorem Let n ≥ 3 and consider λ = (2, 1^{n−2}). Let V be an m-unitary space, m ≥ n − 1. Then the full symmetry class of tensors associated with λ, i.e., V_{λ}^{n}, does not have an O-basis.

3 Proof of the Main Theorem

For the proof of the Main Theorem we need a combinatorial result on permutations. We denote by π the natural permutation character of S_{n}, i.e., for σ ∈ S_{n}, π(σ) is the number of fixed points of σ.

Lemma 3.1 Set F = {σ ∈ S_{n}| π(σ) = π(σ (1 2))}. Then we have F = {σ ∈ S_{n}| {σ(1), σ(2)} ∩ {1, 2} = ∅}.

Proof. Let σ ∈ S_{n}. There are four possibilities. If {σ(1), σ(2)} ∩ {1, 2} = ∅, then σ and σ (1 2) have exactly the same fixed points. If {σ(1), σ(2)} = {1, 2}, then the number of fixed points of σ and σ (1 2) differ by 2. Now assume that {σ(1), σ(2)} = {1, a}, with a ≠ 2. Then in one-line notation the permutations σ and σ (1 2) are (not necessarily in that order) 1 a σ(3) … σ(n) and a 1 σ(3) … σ(n), hence their fixed point numbers differ by 1. The case where {σ(1), σ(2)} = {2, a}, with a ≠ 1 is similar. This proves the claim. □

Lemma 3.2 Let F = {σ ∈ S_{n}| π(σ) = π(σ (1 2))}, and let σ_{1}, …, σ_{k} be distinct permutations of S_{n} such that σ_{i}σ_{j}^{−1} ∈ F for all i ≠ j. Then k ≤ [n/2].

Proof. Set t_{r} = σ_{1}σ_{r+1}^{−1} for r = 0, …, k −1, so t_{0} = id and t_{1}, …, t_{k−1} ∈ F. Then t_{r}^{−1}t_{s} = σ_{r+1}σ_{1}^{−1}σ_{1}σ_{s+1}^{−1} = σ_{r+1}σ_{s+1}^{−1} ∈ F for all r ≠ s, r, s ∈ {1, …, k − 1}. Hence by Lemma 3.1, {t_{r}^{−1}t_{s}(1), t_{r}^{−1}t_{s}(2)} ∩ {1, 2} = ∅, or equivalently, {t_{s}(1), t_{s}(2)} ∩ {t_{r}(1), t_{r}(2)} = ∅ for all r ≠ s, r, s ∈ {1, …, k − 1}. Thus

2k = |{t_{r}(j)| r = 0, …, k − 1, j ∈ {1, 2}}| ≤ n

and so k ≤ [n/2]. □

Remark 3.3 The bound [n/2] in Lemma 3.2 is sharp, and the proof shows how to construct such sets of permutations. In particular, we obtain such a set by setting σ_{1} = id and (in cycle notation) σ_{j} = (1 2j − 1) (2 2j) for j = 2, …, [n/2].

We are now ready to prove the Main Theorem. Let {e_{1}, …, e_{m}} be an orthonormal basis of V. Assume that V_{λ}^{n}, λ = (2, 1^{n−2}), has an O-basis. Put γ = (1, 1, 2, 3, …, n − 1), then n ≥ 3 and m ≥ n − 1 implies that γ ∈ Γ_{m}^{n}. Consider the action of S_{n} on Γ_{m}^{n} and choose Δ such that γ ∈ Δ. It is easy to see that the stabilizer subgroup of γ is equal to (S_{n})_{γ} = {(1), (12)}. Therefore

∑_{σ∈(S_{n})_{γ}}[λ](σ) = (n − 1) + (−1)(n − 3) = 2,

so it is non-zero and we obtain γ ∈ \overline{Δ}. (In fact, this sum is non-zero for any partition λ ≠ (1^{n}), so γ ∈ \overline{Δ} for any λ ≠ (1^{n}).) Now, by (5), we can decompose V_{λ}^{n} into the orthogonal direct sum of orbital subspaces indexed by \overline{Δ}. Since V_{λ}^{n} has an O-basis, and γ ∈ \overline{Δ}, the orthogonality of this decomposition implies that V_{γ}^{∗} has an O-basis. But, by (4), we have

dim V_{γ}^{∗} = \frac{n − 1}{2} · 2 = n − 1,

so we can assume {e_{g_{1}·γ}^{∗}, …, e_{g_{n−1}·γ}^{∗}} is an O-basis for V_{γ}^{∗}. Therefore for each i ≠ j, 1 ≤ i, j ≤ n − 1, we have ⟨e_{g_{i}·γ}^{∗}|e_{g_{j}·γ}^{∗}⟩ = 0. On the other hand if α = g_{i} · γ and β = g_{j} · γ, then g_{i}g_{j}^{−1} · β = α, so if we set τ = g_{i}g_{j}^{−1} and use (2), then we obtain

⟨e_{g_{i}·γ}^{∗}|e_{g_{j}·γ}^{∗}⟩ = \frac{[λ](1)}{|S_{n}|} ∑_{σ∈(S_{n})_{γ}}[λ](g_{i}^{−1}g_{j}σ)
= \frac{n − 1}{n!} (ε(g_{i}g_{j}^{−1})(π(g_{i}g_{j}^{−1}) − 1) + ε(g_{i}g_{j}^{−1}(12))(π(g_{i}g_{j}^{−1}(12)) − 1))
= \frac{n − 1}{n!} ε(g_{i}g_{j}^{−1})(π(g_{i}g_{j}^{−1}) − π(g_{i}g_{j}^{−1}(12))),

where ε is the sign character. Now the condition ⟨e_{g_{i}·γ}^{∗}|e_{g_{j}·γ}^{∗}⟩ = 0, for each i ≠ j, 1 ≤ i, j ≤ n − 1, implies that for such i, j we have π(g_{i}g_{j}^{−1}) = π (g_{i}g_{j}^{−1}(12)). Hence we deduce from Lemma 3.2 that n − 1 ≤ [n/2]. Since n ≥ 3, the last inequality is a contradiction and thus V_{λ}^{n}, λ = (2, 1^{n−2}), does not have an O-basis. □

Acknowledgment: This work was done while the second author was a Postdoctoral Research Associate at the School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran, and he was visiting the Otto-von-Guericke University, Magdeburg, Germany. He would like to express his thanks to Professor Christine Bessenrodt for the hospitality enjoyed at this university. Also he would like to thank the IPM for the financial support.

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The Authors’ Addresses

C. Bessenrodt, Institut für Mathematik, Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany.

E-mail address: bessen@math.uni-hannover.de

M. R. Pournaki, School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran.

E-mail address: pournaki@ipm.ir

A. Reifegerste, Institut für Mathematik, Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany.

E-mail address: reifegerste@math.uni-hannover.de