On the Dimensions of Cyclic Symmetry Classes of Tensors

M. R. Darafsheh, M. R. Pournaki

Abstract The dimensions of the symmetry classes of tensors, associated with a certain cyclic subgroup of $_{m} 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.

Keywords: Symmetry class of tensors, Ramanujan sum, Euler φ-function, Möbius function. 1995 Mathematics Subject Classification: Primary 20C30, 11L07; Secondary 15A69.

1 Introduction

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

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

a subgroup of $_{m} and let 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) : ⊗^{m}V → ⊗^{m}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 ⊗^{m}V under the T (G, χ) is called the symmetry class of tensors associated with G and χ and is denoted by V_{χ}^{m}(G). It is well known that

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

where c(σ) is the number of cycles, including cycles of length 1, in the disjoint cycle decomposition of σ (see [4]).

Several papers are devoted to calculating dim V_{χ}^{m}(G) in a more closed form than (1). Cummings [2] in the case that G is a cyclic subgroup of $_{m} generated by a cycle of length m gives a formula for dim V_{χ}^{m}(G) in terms of the Euler φ-function and considers the case that G is isomorphic to a direct product of cyclic groups as well. In [3] when G is the dihedral group of order 2m is considered and a formula is given when G is equal to the whole group $_{m} in [5] and [6]. In all cases dim V_{χ}^{m}(G) involves certain functions of n.

In [7] there is a formula for calculating dim V_{χ}^{m}(G) in the case that G = ⟨π_{1}⟩ ⋯ ⟨π_{p}⟩, where π_{i}s, 1 ≤ i ≤ p, are disjoint cycles in $_{m}. This formula involves the Euler φ-function and Möbius function and is a modification of the formula given in [2]. In this case if the order of π_{i} is m_{i}, 1 ≤ i ≤ p, then G ≅ ℤ_{m_{1}} × ⋯ × ℤ_{m_{p}}. It is mentioned in [2] and [7] that if two groups are isomorphic, then the dimensions of the symmetry classes of tensors associated with them need not be equal. For example if G = ⟨(12)⟩ and H = ⟨(12)(34)⟩ are considered as subgroups of $_{4}, then it is easy to calculate that dim V_{χ_{0}}^{4}(G) = n^{3}(n + 1)/2 and dim V_{χ_{0}}^{4}(H) = n^{2}(n^{2} + 1)/2 whereas G ≅ H ≅ ℤ_{2} and χ_{0} is the identity character of ℤ_{2}.

Now it is a natural question to consider the cyclic group G = ⟨π_{1} … π_{p}⟩ where the π_{i}s, 1 ≤ i ≤ p, are disjoint cycles and ask about the dimension of V_{χ}^{m}(G), where χ ∈ I(G). In this case if the order of π_{i}, 1 ≤ i ≤ p, is m_{i}, then G ≅ ℤ_{[m_{1},...,m_{p}]}, where [m_{1}, …, m_{p}] denotes the least common multiple of the integers m_{1}, …, m_{p}. In this paper we obtain a formula for dim V_{χ}^{m}(G) in the above case and this formula involves the generalized Ramanujan sum which itself involves the Euler φ-function and Möbius function. Therefore for the rest of this paper let G < $_{m} be of the form

G = ⟨π_{1} … π_{p}⟩,

where π_{i}s, 1 ≤ i ≤ p, are disjoint cycles in $_{m} of certain orders, say, m_{1}, …, m_{p}, respectively. Since G is cyclic, therefore the irreducible characters of G are all linear and are of the form

χ_{h} : G → ℂ^{∗}, χ_{h} ((π_{1} … π_{p})^{t}) = exp (\frac{2πiht}{[m_{1}, …, m_{p}]}) ,

0 ≤ t ≤ [m_{1}, …, m_{p}] − 1,

so

I(G) = {χ_{h} : G → ℂ^{∗}| 0 ≤ h ≤ [m_{1}, …, m_{p}] − 1},

where [m_{1}, …, m_{p}] denotes the least common multiple of the integers m_{1}, …, m_{p}. The symbol (m_{1}, …, m_{p}) denotes the greatest common divisor of m_{1}, …, m_{p}.

2 A Result About the Ramanujan Sum

The well known Ramanujan sum is

C_{m}(h) = ∑_{\begin{matrix}t = 0 \\ (t, m) = 1 \end{matrix}}^{m−1} exp (\frac{2πiht}{m}) ,

where m is a positive integer and h is a nonnegative integer. Ramanujan proved that (see [1])

C_{m}(h) = \frac{φ(m)μ(m/(m, h))}{φ(m/(m, h))},

where φ is the Euler φ-function, i.e., φ(1) = 1; for m > 1, φ(m) = the number of positive integers less than m and relatively prime to m, and μ is the Möbius function, i.e., μ(1) = 1, μ(m) = 0 if p^{2}|m for some prime number p, and μ(m) = (−1)^{r} if m = p_{1} … p_{r}, where p_{1}, …, p_{r} are distinct prime numbers.

For our main result, we need to generalize the Ramanujan sum. It seems natural to us to generalize the Ramanujan sum as follows.

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

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

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

Remark 1 It is obvious that S(h; m; 1) = C_{m}(h), and so the sum appearing in Definition 1 is a generalization of the Ramanujan sum.

In the following lemma we prove that the generalized Ramanujan sum defined in Definition 1 involves the Ramanujan sum.

Lemma 1 Let m_{1}, …, m_{p} be positive integers and let h be a nonnegative integer. Suppose d_{1}|m_{1}, …, d_{p}|m_{p} and set m′_{i} = m_{i}/d_{i}, M_{i} = m_{1} … m_{p}/m_{i}, M′_{i} = m′_{1} … m′_{p}/m′_{i}, D_{i} = d_{1} … d_{p}/d_{i} (1 ≤ i ≤ p) and

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

Then we have

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

Proof. By Definition 1 and the fact that exp (2πiht/[m_{1}, …, m_{p}]) is a periodic function of t with period [m_{1}, …, m_{p}] we have

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

Now letting t = [d_{1}, …, d_{p}]t′, we obtain

\begin{aligned}S(h; m_{1}, …, m_{p}; d_{1}, …, d_{p}) & = \frac{1}{l} ∑_{t′ = 0 ; ([d_{1}, …, d_{p}]t′, m_{1}) = d_{1} ; ⋮ ; ([d_{1}, …, d_{p}]t′, m_{p}) = d_{p}}^{[m′_{1},...,m′_{p}]−1} exp (\frac{2πihlt′}{[m′_{1}, …, m′_{p}]}) \\ & = \frac{1}{l} ∑_{t′ = 0 ; (([d_{1}, …, d_{p}]/d_{1}) t′, m′_{1}) = 1 ; ⋮ ; (([d_{1}, …, d_{p}]/d_{p}) t′, m′_{p}) = 1}^{[m′_{1},...,m′_{p}]−1} exp (\frac{2πihlt′}{[m′_{1}, …, m′_{p}]}) . \end{aligned}

If ([d_{1}, …, d_{p}]/d_{i} , m′_{i}) = 1 for all i, 1 ≤ i ≤ p, then the set of all the t′s indexing the above summation is equal to the set of all the t′s such that 0 ≤ t′ ≤ [m′_{1}, …, m′_{p}] − 1 with conditions (t′, m′_{i}) = 1, 1 ≤ i ≤ p. And if there is an i for which ([d_{1}, …, d_{p}]/d_{i} , m′_{i}) ≠ 1 then the above sum is zero and therefore we obtain:

S(h; m_{1}, …, m_{p}; d_{1}, …, d_{p}) = \{ \begin{matrix}\frac{1}{l} ∑_{t′ = 0 ; (t′, m′_{1}) = 1 ; ⋮ ; (t′, m′_{p}) = 1}^{[m′_{1},...,m′_{p}]−1} exp (\frac{2πihlt′}{[m′_{1}, …, m′_{p}]}) & , & if (\frac{[d_{1},...,d_{p}]}{d_{i}}, m′_{i}) = 1,_{1≤i≤p} \\ 0 & , & otherwise, \end{matrix}

= \{ \begin{matrix}\frac{1}{l} ∑_{\begin{matrix}t′ = 0 \\ (t′, [m′_{1}, …, m′_{p}]) = 1 \end{matrix}}^{[m′_{1},...,m′_{p}]−1} exp (\frac{2πihlt′}{[m′_{1}, …, m′_{p}]}) & , & if (\frac{[d_{1},...,d_{p}]}{d_{i}}, m′_{i}) = 1,_{1≤i≤p} \\ 0 & , & otherwise, \end{matrix}

= \{ \begin{matrix}\frac{1}{l}C_{[m′_{1},...,m′_{p}]}(hl) & , & if (\frac{[d_{1},...,d_{p}]}{d_{i}}, m′_{i}) = 1,_{1≤i≤p} \\ 0 & , & otherwise. □ \end{matrix}

In some special cases the generalized Ramanujan sum is given in the following examples:

Example 1 S(0; m; d) = C_{m/d}(0) = ^{\frac{φ(m/d)μ((m/d)/(m/d,0))}{φ((m/d)/(m/d,0))}} = φ(m/d).

Example 2 If (h, m) = 1; we have

S(h; m; d) = C_{m/d}(h) = \frac{φ(m/d)μ((m/d)/(m/d, h))}{φ((m/d)/(m/d, h))} = μ(m/d).

3 The Dimensions of Some Symmetry Classes of Tensors

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 $_{m}. Our aim is to calculate dim V_{χ}^{m}(G), where χ ∈ I(G), in terms of known functions. Our formula involves the generalized Ramanujan sum.

Theorem 1 Let G = ⟨π_{1} … π_{p}⟩, where the π_{i}s, 1 ≤ i ≤ p, are disjoint cycles in $_{m} of orders m_{1}, …, m_{p}, respectively, and let χ_{h}, 0 ≤ h ≤ [m_{1}, …, m_{p}] − 1, be an irreducible complex character of G. Then

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

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

Proof. According to (1) the dimension of V_{χ_{h}}^{m}(G) is

\frac{1}{[m_{1}, …, m_{p}]} ∑_{σ∈G} χ_{h}(σ)n^{c(σ)},  (2)

where c(σ) denotes the number of cycles, including cycles of length 1, in the disjoint cycle decomposition of σ. But every σ ∈ G is equal to σ = (π_{1} … π_{p})^{t} for some t, 0 ≤ t ≤ [m_{1}, …, m_{p}] − 1. Since π_{1}, …, π_{p} are disjoint, so are (π_{1} … π_{p})^{t} = π_{1}^{t} … π_{p}^{t}. Appealing to [7] we can obtain

c(π_{1}^{t} … π_{p}^{t}) = c(π_{1}^{t}) + ⋯ + c(π_{p}^{t}) + m − (m_{1} + ⋯ + m_{p}).

Note that if (t, m_{i}) = d, then π_{i}^{t} has d cycles of length m_{i}/d and therefore c(π_{i}^{t}) = d = (t, m_{i}). So we have

c(π_{1}^{t} … π_{p}^{t}) = (t, m_{1}) + ⋯ + (t, m_{p}) + m − (m_{1} + ⋯ + m_{p}).

Hence according to (2) we have

\begin{aligned}dim V_{χ_{h}}^{m}(G) & = \frac{1}{[m_{1}, …, m_{p}]} ∑_{σ∈G} χ_{h}(σ)n^{c(σ)} \\ & = \frac{1}{[m_{1}, …, m_{p}]} ∑_{t=0}^{[m_{1},...,m_{p}]−1} χ_{h}(π_{1}^{t} … π_{p}^{t})n^{c(π_{1}^{t}...π_{p}^{t})} \end{aligned}

\begin{matrix}= \frac{1}{[m_{1}, …, m_{p}]} ∑_{t=0}^{[m_{1},...,m_{p}]−1} exp (\frac{2πiht}{[m_{1}, …, m_{p}]}) n^{(t,m_{1})+···+(t,m_{p})+m−(m_{1}+···+m_{p})} \\ = \frac{n^{m−(m_{1}+···+m_{p})}}{[m_{1}, …, m_{p}]} ∑_{t=0}^{[m_{1},...,m_{p}]−1} exp (\frac{2πiht}{[m_{1}, …, m_{p}]}) n^{(t,m_{1})+···+(t,m_{p})}. \end{matrix}

Now letting (t, m_{i}) = d_{i}, 1 ≤ i ≤ p, we obtain

\begin{aligned}dim V_{χ_{h}}^{m}(G) & = \frac{n^{m−(m_{1}+···+m_{p})}}{[m_{1}, …, m_{p}]} ∑_{d_{1}|m_{1} ; ⋮ ; d_{p}|m_{p}} ( ∑_{t = 0 ; (t, m_{1}) = d_{1} ; ⋮ ; (t, m_{p}) = d_{p}}^{[m_{1},...,m_{p}]−1} exp (\frac{2πiht}{[m_{1}, …, m_{p}]})) n^{d_{1}+···+d_{p}} \\ & = \frac{n^{m−(m_{1}+···+m_{p})}}{[m_{1}, …, m_{p}]} ∑_{d_{1}|m_{1} ; ⋮ ; d_{p}|m_{p}} S(h; m_{1}, …, m_{p}; d_{1}, …, d_{p})n^{d_{1}+···+d_{p}}. □ \end{aligned}

Using Theorem 1 we obtain Theorems 1 and 2 of [2] in the following corollaries.

Corollary 1 If G is a cyclic subgroup of $_{m} generated by an m-cycle and χ is the identity character 1, then dim V_{χ}^{m}(G) = (1/m) ∑_{d|m} φ(m/d)n^{d}.

Proof. Since χ = χ_{0}, by Example 1 and using Theorem 1 we obtain

dim V_{χ}^{m}(G) = \frac{n^{m−m}}{m} ∑_{d|m} S(0; m; d)n^{d} = \frac{1}{m} ∑_{d|m} φ(m/d)n^{d}. □

Remark 2 In Corollary 1, if dim V = n = 1, then dim ⊗^{m}V = 1, and so dim V_{χ}^{m}(G) = 0 or 1. So (1/m) ∑_{d|m} φ(m/d) = 0 or 1. But (1/m) ∑_{d|m} φ(m/d) = 0 is impossible, therefore (1/m) ∑_{d|m} φ(m/d) = 1 or ∑_{d|m} φ(d) = m, which is well known identity in number theory.

Corollary 2 If G is a cyclic subgroup of $_{m} generated by an m-cycle and χ is a primitive linear character, then dim V_{χ}^{m}(G) = (1/m) ∑_{d|m} μ(m/d)n^{d}.

Proof. We know that a linear character of a cyclic subgroup of $_{m} is primitive if its value on a generator of the subgroup is a primitive mth root of unity, so χ = χ_{h} where (h, m) = 1 and by Example 2 and using Theorem 1, we have

dim V_{χ}^{m}(G) = \frac{n^{m−m}}{m} ∑_{d|m} S(h; m; d)n^{d} = \frac{1}{m} ∑_{d|m} μ(m/d)n^{d}. □

Example 3 Let G = ⟨(12)(34)(5678)⟩ be a subgroup of $_{9}. Suppose χ is the identity character 1, i.e., χ = χ_{0}. Then by Theorem 1 we have

dim V_{χ}^{9}(G) = \frac{n^{9−(2+2+4)}}{4} ∑_{\begin{matrix}d_{1}|2 \\ d_{2}|2 \\ d_{3}|4 \end{matrix}} S(0; 2, 2, 4; d_{1}, d_{2}, d_{3})n^{d_{1}+d_{2}+d_{3}}

\begin{matrix}= \frac{n}{4} [S(0; 2, 2, 4; 2, 2, 4)n^{8} + S(0; 2, 2, 4; 2, 2, 2)n^{6} + S(0; 2, 2, 4; 1, 1, 1)n^{3}] \\ = \frac{n}{4}[n^{8} + n^{6} + 2n^{3}]. \end{matrix}

References

[1] T. M. Apostol, “Introduction to Analytic Number Theory”, Springer-Verlag, New York, 1976.

[2] L. J. Cummings, Cyclic Symmetry Classes, J. Algebra 40 (1976), 401-405.

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

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

[5] R. Merris, The Dimension of Certain Symmetry Classes of Tensors II, Linear and Multilinear Algebra 4 (1976), 205-207.

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

[7] T. Y. Tam, On the Cyclic Symmetry Classes, J. Algebra 182 (1996), 557-560.

The Authors’ Addresses

M. R. Darafsheh, Department of Mathematics and Computer Science, University of Tehran, and Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran.

E-mail address: darafshe@vax.ipm.ac.ir

M. R. Pournaki, Department of Mathematics and Computer Science, University of Tehran, and Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran.

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