Symmetry Class of Tensors

Associated with Certain Groups

A Thesis Submitted to Faculty of Science University of Tehran

By M. R. Pournaki

In Partial

Fulfillment of the Requirements for the Degree of Doctor of Philosophy (Ph.D.) in

Pure Mathematics

Supervisor Professor M. R. Darafsheh

Preface

In the last two decades, the theory of symmetry class of tensors has been one of the attractive subjects in the multilinear algebra. So many mathematicians have been working on the variety of problems concerning symmetry class of tensors. Below we have a glimpse on the main ideas of the thesis.

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}

We can prove that the symmetry class of tensors associated with G and χ exists and it is unique up to isomorphisms of vector spaces. Also we can prove that the symmetry class of tensors associated with G and χ is (isomorphic with) the image of the linear operator T (G, χ) : ⊗^{n}V → ⊗^{n}V, where

T (G, χ)(v_{1} ⊗ ⋯ ⊗ v_{n}) = \frac{χ(1)}{|G|} ∑_{σ∈G} χ(σ)v_{σ^{−1}(1)} ⊗ ⋯ ⊗ v_{σ^{−1}(n)},

where ⊗^{n}V is the n-th tensor power of V and v_{1} ⊗ ⋯ ⊗ v_{n} is a typical element of it. We denote the symmetry class of tensors associated with G and χ by V_{χ}^{n}(G). In addition, if we assume that V is an m-unitary space, then V_{χ}^{n}(G) will become a unitary space.

Finding an explicit formula for the dimension of V_{χ}^{n}(G), investigation of the existence of an orthogonal basis of decomposable symmetrized tensors for V_{χ}^{n}(G) and non-vanishing of V_{χ}^{n}(G), for a general or certain group G, are open problems in this branch of mathematics. In this thesis, we consider certain groups G and an irreducible character χ of G and answer the above questions. The certain groups which we considered are G = ⟨π_{1} … π_{p}⟩, where the π_{i}’s, 1 ≤ i ≤ p, are disjoint cycles in $_{n}, G = T_{4n}, G = P SL_{2}(q) and a group G of order n as a subgroup of $_{n} with Cayley representation. The results of this thesis appeared in some international journals (see [2], [3], [4] and [5]).

M. R. Pournaki March 2000

Extended Abstract

1 Introduction

Let V be an m-dimensional vector space over the complex field ℂ. 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)}.

Let G be a subgroup of $_{n} and I(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 ∈ I(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. It is well-known that

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

where c(σ) is the number of cycles, including cycles of length one, in the disjoint cycle decomposition of σ (see [9]). Also we have that

⨂^{n}V = ⨁_{χ∈I(G)}V_{χ}^{n}(G)  (2)

is a direct sum.

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 of α, i.e., G_{α} = {σ ∈ G| σ.α = α}. Let Δ be a system of distinct representatives of the orbits of G acting on Γ_{m}^{n} and define

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

and let Ω be the union of those equivalence classes represented by elements of \overline{Δ}.

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 ∈ \overline{Δ}, V_{α}^{∗} = ⟨e_{σ.α}^{∗}| σ ∈ G⟩ is called the orbital subspace of V_{χ}^{n}(G). It follows that

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

is a direct sum. In [6] Freese proved that

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

in particular, if χ is of degree one, then dim V_{α}^{∗} = 1 for all α ∈ \overline{Δ}.

A particular case appear, 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|}d_{χ}^{G}(A)

where A = [a_{ij}]_{n×n} = [⟨u_{i}|v_{j}⟩]_{n×n} and d_{χ}^{G}(A) = ∑_{σ∈G} χ(σ)a_{1σ(1)} … a_{nσ(n)} is the gen-

eralized matrix function.

With respect to this inner product the sums appearing in (2) and (3) are orthogonal direct sums. Also if we suppose {e_{1}, …, e_{m}} is an orthonormal basis of V, then we obtain

⟨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}

In particular, taking the norm of e_{α}^{∗}, with respect to the induced inner product, one easily obtains the condition e_{α}^{∗} ≠ 0 if and only if α ∈ Ω.

If α = g.γ and β = g′.γ, then gg^{′−1}.β = α, so if we set τ = gg^{′−1} and use the above formula for ⟨e_{α}^{∗} | e_{β}^{∗}⟩, then we obtain

⟨e_{g.γ}^{∗} | e_{g′.γ}^{∗}⟩ = \frac{χ(1)}{|G|} ∑_{σ∈G_{γ}} χ(g′σg^{−1}).  (5)

An orthogonal basis of the form {e_{α}^{∗}| α ∈ S}, where S is a subset of Γ_{m}^{n}, is called an O-basis for V_{χ}^{n}(G). By (3), 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. In particular, 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 in investigation of the non-vanishing of V_{χ}^{n}(G) and finding a formula for dim V_{χ}^{n}(G) in a more closed form than (1), also discuss about the existence of an O-basis for these vector spaces, see for example [1], [7] and [14]. In [10] and [12] a formula for dim V_{χ}^{n}(G) is given when G is equal to the whole group of $_{n}, also in [13] a formula for calculating dim V_{χ}^{n}(G) in the case that G = ⟨π_{1}⟩ ⋯ ⟨π_{p}⟩ is given, where the π_{i}’s, 1 ≤ i ≤ p, are disjoint cycles in $_{n}. Also, in [7] a necessary and sufficient condition for the existence of O-basis for V_{χ}^{n}(G) is given, when G is a cyclic or a dihedral group. The books [8] and [11] are good sources of information about characters of finite groups and the symmetry class of tensors respectively.

2 On the Dimensions of Symmetry Classes of Tensors Associated with the Group G = ⟨π_{1} … π_{p}⟩

Abstract The dimensions of the symmetry classes of tensors, associated with a certain cyclic subgroup of $_{n} which is generated by a product of disjoint cycles is explicitly given in terms of the generalized Ramanujan sum. These dimensions can also be expressed as the Euler φ-function and the Möbius function. In the following we show some results appeared in [2].

Definition 2.1 Let n_{1}, …, n_{p} be positive integers and let h be a nonnegative integer. Suppose d_{1}|n_{1}, …, d_{p}|n_{p}. The generalized Ramanujan sum denoted by S(h; n_{1}, …, n_{p}; d_{1}, …, d_{p}) is defined by

S(h; n_{1}, …, n_{p}; d_{1}, …, d_{p}) = ∑_{t = 0 ; (t, n_{1}) = d_{1} ; ⋮ ; (t, n_{p}) = d_{p}}^{[n_{1},...,n_{p}]−1} exp (\frac{2πiht}{[n_{1}, …, n_{p}]}) .

If the set {0 ≤ t ≤ [n_{1}, …, n_{p}] − 1| (t, n_{i}) = d_{i}; 1 ≤ i ≤ p} is empty, then we define S(h; n_{1}, …, n_{p}; d_{1}, …, d_{p}) = 0.

Lemma 2.2 Let n_{1}, …, n_{p} be positive integers and let h be a nonnegative integer. Suppose d_{1}|n_{1}, …, d_{p}|n_{p} and set n′_{i} = n_{i}/d_{i}, N_{i} = n_{1} … n_{p}/n_{i}, N′_{i} = n′_{1} … n′_{p}/n′_{i}, D_{i} = d_{1} … d_{p}/d_{i} (1 ≤ i ≤ p) and

l = \frac{(N_{1}, …, N_{p})}{(N′_{1}, …, N′_{p})(D_{1}, …, D_{p})}.

Then we have

S(h; n_{1}, …, n_{p}; d_{1}, …, d_{p}) = \{ \begin{matrix}\frac{1}{l}C_{[n′_{1},...,n′_{p}]}(hl) & , & if (\frac{[d_{1},...,d_{p}]}{d_{i}}, n′_{i}) = 1,_{1≤i≤p} \\ 0 & , & otherwise. \end{matrix}

In this section, as we mentioned earlier, the group G = ⟨π_{1} … π_{p}⟩ is considered, where the π_{i}’s, 1 ≤ i ≤ p, are disjoint cycles in $_{n}. Our main result, which appears in the following theorem, is to calculate dim V_{χ}^{n}(G), where χ ∈ I(G), in terms of known number theoretical functions. Our formula involves the generalized Ramanujan sum.

Theorem 2.3 Let G = ⟨π_{1} … π_{p}⟩, where the π_{i}’s, 1 ≤ i ≤ p, are disjoint cycles in $_{n} of orders n_{1}, …, n_{p}, respectively, and let χ_{h}, 0 ≤ h ≤ [n_{1}, …, n_{p}] − 1, be an irreducible complex character of G. If V be an m-dimensional vector space over ℂ, then

dim V_{χ_{h}}^{n}(G) = \frac{m^{n−(n_{1}+···+n_{p})}}{[n_{1}, …, n_{p}]} ∑_{d_{1}|n_{1} ; ⋮ ; d_{p}|n_{p}} S(h; n_{1}, …, n_{p}; d_{1}, …, d_{p})m^{d_{1}+···+d_{p}}

where S(h; n_{1}, …, n_{p}; d_{1}, …, d_{p}) denotes the generalized Ramanujan sum.

3 On the Orthogonal Basis of the Symmetry Classes of Tensors Associated with the Dicyclic Group

Abstract A necessary and sufficient condition for the existence of O-basis for the symmetry classes of tensors associated with the dicyclic group is given. In particular we apply these conditions to the generalized quaternion group, for which the dimensions of the symmetry classes of tensors are computed. In the following we show some results appeared in [3].

Let G = T_{4n} = ⟨r, s| r^{2n} = 1, r^{n} = s^{2}, s^{−1}rs = r^{−1}⟩, n ≥ 1, be the dicyclic group as a subgroup of $_{4n} with Cayley representation, and let V be an m-unitary space, with orthonormal basis {e_{1}, …, e_{m}}. For n = 1, the dicyclic group T_{4} is cyclic, T_{4} ≃ ℤ_{4}, therefore all of its irreducible characters are of degree 1 and so V_{χ}^{4}(T_{4}) has an O-basis for all χ ∈ I(T_{4}). Therefore we assume that n ≥ 2. If m = 1, then dim ⊗^{4n}V = 1, so dim V_{χ}^{4n}(G) = 0 or 1, therefore we do not have any problem about the existence of O-basis for V_{χ}^{4n}(G) for all χ ∈ I(G), therefore we assume that m ≥ 2.

For irreducible characters of T_{4n} of degree 1, ψ_{i}, ϕ_{i}, 1 ≤ i ≤ 4, V_{ψ_{i}}^{4n}(T_{4n}) and V_{ϕ_{i}}^{4n}(T_{4n}) have O-basis and so we do not deal with the ψ_{i}’s and ϕ_{i}’s.

Therefore we investigate the problem for irreducible characters of degree 2 of T_{4n}, i.e., χ_{h}, 1 ≤ h ≤ n − 1, which are given by

χ_{h}(r^{k}) = 2 cos \frac{khπ}{n} ,  χ_{h}(r^{k}s) = 0,  0 ≤ k < 2n.

The following results are the main results in this section.

Theorem 3.1 Let G = T_{4n}, n ≥ 2, and χ = χ_{h}, 1 ≤ h ≤ n − 1, dim V = m ≥ 2. Then V_{χ}^{4n}(G) has an O-basis if and only if ν_{2}(\frac{h}{n}) < 0, where ν_{2} is 2-adic valuation.

Corollary 3.2 Let G = T_{4n}, n ≥ 2 is odd, and χ = χ_{h}, 1 ≤ h ≤ n − 1, dim V = m ≥ 2. Then V_{χ}^{4n}(G) does not have an O-basis.

Theorem 3.3 Let G = Q_{2^{n+1}} = T_{4(2^{n−1})}, n ≥ 2, the generalized quaternion group, and χ = χ_{h}, 1 ≤ h ≤ 2^{n−1} − 1, dim V = m ≥ 2. Then V_{χ}^{2^{n+1}}(G) has an O-basis.

4 On the Non-Vanishing of the Symmetry Classes of tensors Associated with G = P SL_{2}(q)

Abstract The dimensions of the symmetry classes of tensors associated with the projective special linear group of degree 2 over a field with q elements, P SL_{2}(q), are found. Of course we will assume P SL_{2}(q) as a subgroup of the symmetric group $_{q+1} because this group has a faithful action on the points of the underlying projective space. We also discuss the non-triviality of the symmetry classes of tensors associated with each irreducible character of P SL_{2}(q). In the following we show some results appeared in [4].

It is well-known that G = P SL_{2}(q) acts faithfully and 2-transitively on the q + 1 points of the projective line Ω and so we can assume that G = P SL_{2}(q) is a subgroup of $_{q+1}, therefore V_{χ}^{q+1}(G) is meaningful for any χ ∈ I(G).

In the following we write the main results of this section. We discuss the question of when the symmetry classes of tensors associated with G = P SL_{2}(q) are nonzero vector spaces. If dim V = s = 1, then it is clear that for all χ, χ ∈ I(G) − {1_{G}}, V_{χ}^{q+1}(G) = 0 and V_{1_{G}}^{q+1}(G) ≠ 0. Therefore we deal with the case dim V = s = 2 in the following theorem.

Theorem 4.1 Consider G = P SL_{2}(q) as a subgroup of $_{q+1} and let V be a vector space over the complex field ℂ, such that dim V = s = 2.

a) If q is odd, q = p^{n}, q ≡^{4} 1; then for all χ, χ ∈ I(G) − {χ_{i}, ξ_{1}, ξ_{2}| i = 2, 4, …, (q −

5)/2; i ≡^{4} 2}, we have V_{χ}^{q+1}(G) ≠ 0. Additionally if q ≡^{8} 1, then V_{ξ_{1}}^{q+1}(G) ≠ 0 and V_{ξ_{2}}^{q+1}(G) ≠ 0,

b) If q is odd, q = p^{n}, q ≡^{4} 3; then for all χ, χ ∈ I(G) − {θ_{j}, η_{1}, η_{2}| j = 2, 4, …, (q −

3)/2; j ≡^{4} 0}, we have V_{χ}^{q+1}(G) ≠ 0. Additionally if q ≡^{8} 3, then V_{η_{1}}^{q+1}(G) ≠ 0 and V_{η_{2}}^{q+1}(G) ≠ 0,

c) If q is even, q = 2^{n}; then for all χ, χ ∈ I(G) − {θ_{j}| 1 ≤ j ≤ q/2}, we have V_{χ}^{q+1}(G) ≠ 0.

Theorem 4.2 Consider G = P SL_{2}(q) as a subgroup of $_{q+1} and let V be a vector space over the complex field ℂ, such that dim V = s ≥ 3. Then for all χ, χ ∈ I(G), we have V_{χ}^{q+1}(G) ≠ 0.

5 On the Orthogonal Basis and Non-Vanishing of the Symmetry Classes of tensors Associated with a Group G with Cayley Representation

Abstract By Cayley’s theorem, any finite group G of order n can be regarded as a subgroup of the symmetric group $_{n}. Let χ be any irreducible complex character of G and let V_{χ}^{n}(G) denote the symmetry classes of tensors associated with G and χ. In this section assuming the Cayley representation of G, we obtain a formula for the dimension of V_{χ}^{n}(G) and discuss its non-vanishing in general. A necessary condition for the existence of the O-basis for V_{χ}^{n}(G) is also obtained. In the following we show some results appeared in [5].

Let V be an m-dimensional vector space over the complex field ℂ and let G be a finite group and Ω be a set of n elements. Suppose G acts faithfully on Ω, so we can assume that G is a subgroup of $_{n}, i.e., G = {g| g ∈ G} = {σ_{g}| g ∈ G} where σ_{g} : Ω → Ω is defined by σ_{g}(ω) = g.ω for all ω ∈ Ω, is a permutation on n letters. Therefore the vector space V_{χ}^{n}(G) is meaningful for all χ ∈ I(G). The following results are main results of this section.

Theorem 5.1 Let G be a group of order n, that is, a subgroup of $_{n} by Cayley representation. If V is an m-dimensional vector space over the complex field ℂ, then for all χ ∈ I(G), we have

dim V_{χ}^{n}(G) = \frac{χ(1)}{n} ∑_{g∈G} χ(g)m^{n/o(g)},

in particular, for all m ≥ 2, V_{χ}^{n}(G) ≠ 0.

Theorem 5.2 Let G be a non-trivial group of order n, that is, a subgroup of $_{n} by Cayley representation. If V is an m-unitary space, m ≥ 2, and χ ∈ I(G) such that χ(1)^{2} > |G|/2, then V_{χ}^{n}(G) dose not have an O-basis.

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