Computation of the Dimensions of Symmetry Classes of

Tensors Associated with the Finite two Dimensional Projective Special Linear Group

M. R. Darafsheh, M. R. Pournaki

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 S_{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).

Keywords: Symmetry class of tensors, Action of symmetric groups, Irreducible character, Projective special linear group. 1995 Mathematics Subject Classification: Primary 20C30; Secondary 15A69.

1 Introduction

Let V be an s-dimensional vector space over the complex field ℂ. Let ⊗^{t}V be the t-th tensor power of V and write v_{1} ⊗⋯⊗v_{t} for the tensor product of the indicated vectors. To each permutation σ in the symmetric group S_{t} there corresponds a unique linear operator

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

subgroup of S_{t} 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) : ⊗^{t}V → ⊗^{t}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 ⊗^{t}V under T (G, χ) is called the symmetry class of tensors associated with G and χ and it is denoted by V_{χ}^{t}(G). It is well-known that

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

where c(σ) is the number of cycles, including cycles of length one, in the disjoint cycle decomposition of σ (see [7]).

Let Γ_{s}^{t} be the set of all sequences α = (α_{1}, …, α_{t}) with 1 ≤ α_{i} ≤ s, so α is a mapping from a set of t elements into a set of s elements. Then the group G acts on Γ_{s}^{t} by σ.α := (α_{σ^{−1}(1)}, …, α_{σ^{−1}(t)}) where σ ∈ G is a permutation on t letters and α ∈ Γ_{s}^{t} is a mapping from a set of t elements into a set of s 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_{s}^{t} and define

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

Let {e_{1}, …, e_{s}} be a basis of V. Denote by e_{α}^{∗} the tensor T (G, χ)(e_{α_{1}} ⊗ ⋯ ⊗ e_{α_{t}}). For γ ∈ \overline{Δ}, V_{γ}^{∗} = ⟨e_{σ·γ}^{∗}| σ ∈ G⟩ is called the orbital subspace of V_{χ}^{t}(G). It follows that

V_{χ}^{t}(G) = ⨁_{γ∈\overline{Δ}} V_{γ}^{∗}  (2)

is a direct sum. In [4] Freese proved that

dim V_{γ}^{∗} = \frac{χ(1)}{|G_{γ}|} ∑_{σ∈G_{γ}} χ(σ),  (3)

but the above formula can be written as

dim V_{γ}^{∗} = χ(1) (χ ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} .  (4)

If there exists γ ∈ Γ_{s}^{t} for which (χ ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} ≠ 0, then we have dim V_{γ}^{∗} > 0. Therefore by (2) we have V_{χ}^{t}(G) ≠ 0. So to show the non-triviality of the vector space V_{χ}^{t}(G), it is enough to show that there exists γ ∈ Γ_{s}^{t} for which (χ ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} ≠ 0.

Several papers are devoted to calculating dim V_{χ}^{t}(G) in a more closed form than (1). Cummings [1] in the case that G is a cyclic subgroup of S_{t} generated by a cycle of length t gives a formula for dim V_{χ}^{t}(G). In [5] the case when G is a dihedral group of order 2t is considered and a formula is given when G is equal to the whole group S_{t} in [8] and [9]. Also in [10] a formula for calculating dim V_{χ}^{t}(G) in the case that G = ⟨π_{1}⟩ ⋯ ⟨π_{p}⟩ and in [2] for G = ⟨π_{1} … π_{p}⟩ is given, where π_{i}, 1 ≤ i ≤ p, are disjoint cycles in S_{t}.

In this paper we compute dim V_{χ}^{t}(G) for the projective special linear group of degree 2 over a field with q elements, namely we let G = P SL_{2}(q), where q is a power of a prime number. We also discuss about when these vector spaces are nonzero.

2 Projective Special Linear Group

The special linear group of degree 2 is denoted by SL_{2}(q), where q = p^{n} and p is a prime number and n is a nonnegative integer. The character table of this group when p = 2 or p is an odd prime is given in [3]. We use these characters with the same name as in [3].

It is well-known that, if N ◁ G and χ is a character of G with N ⊆ ker χ and if χ̂ is a function defined by χ̂(gN) = χ(g), then χ̂ is a character of G/N. Conversely if χ̂ is a character of G/N, then the function χ defined by χ(g) = χ̂(gN) is a character of G having N in its kernel. In both cases χ ∈ I(G), N ⊆ ker χ, if and only if χ̂ ∈ I(G/N) (see [6]). Since the projective special linear group of degree 2, G = P SL_{2}(q), where q = p^{n}, p is a prime number, n is a nonnegative integer, is a quotient of SL_{2}(q), we can compute the character table of G = P SL_{2}(q). These are given in Tables I, II and III.

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 S_{q+1}, therefore V_{χ}^{q+1}(G) is meaningful for any χ ∈ I(G) and we want to compute the dimensions of these vector spaces for all χ given in Tables I, II and III. By formula (1), to do this, we need to know the permutation structure of the elements of G.

Now we want to obtain the permutation structure of each element of G = P SL_{2}(q) as a subgroup of S_{q+1}, but since elements in the same conjugacy class have the same permutation structure, we obtain the permutation structure of a representative from each conjugacy class of elements of G.

For g ∈ G, let fix(g) = {i| 1 ≤ i ≤ q + 1, g(i) = i}. Then θ defined by θ(g) = |fix(g)|, g ∈ G, is the permutation character of G acting on the set of points. Since G acts 2-transitively on the set of points, it is well-known that ν : G → ℂ defined by ν(g) = |fix(g)| − 1, g ∈ G, is an irreducible character of G (see [6]). In the case G = P SL_{2}(q); q is odd, q ≡^{4} 1; ν is equal to one of the irreducible characters in Table I. Since ν({−I, I}1) = (q + 1) − 1 = q, ν must be equal to ψ, so |fix(g)| = 1 + ψ(g), g ∈ G, from which we obtain the |fix(g)| row in Table IV. In the other cases we obtain the |fix(g)| row in Table V and VI. Therefore using |fix(g)| row and o(g) row in Tables IV, V and VI, we obtain the permutation structure of elements of G = P SL_{2}(q). Perhaps a few words may be necessary to explain the cycle structure of elements of P SL_{2}(q). We use [3] for the matrix shapes of a, b, c and d as they appear in Tables IV, V and VI. In the case G = P SL_{2}(q); q is odd, q ≡^{4} 1; if an element x has order r and r is a prime number then all the non-trivial cycles appearing in the cycle structure of x must have length r from which the cycle structure of {−I, I}c, {−I, I}d and {−I, I}a^{(q−1)/4} that have orders p, p and 2 respectively follows. For {−I, I}a^{l} we note that {−I, I}a has two fixed points and two cycles of length (q − 1)/4. Now according to the properties of permutations we can find the cycle structures of powers of {−I, I}a. The element {−I, I}b is a Singer-cycle and consists of only one cycle and the cycle structures of its powers is clear. The discussion is similar in the other cases.

3 On the Dimensions of Symmetry Classes of Tensors Associated with G = P SL_{2}(q)

As we remarked earlier the group G = P SL_{2}(q) acts faithfully on the set of the onedimensional subspaces ⟨v⟩ of 𝕍_{2}(q). Therefore we regard G as a subgroup of the symmetric group on the q + 1 letters and find the dimensions of the symmetry classes of tensors associated with each irreducible character of G. Note that the names of the irreducible characters of G = P SL_{2}(q) are as indicated in [3]. In the following (a, b) denotes the greatest common divisor of a and b; and ρ and σ are primitive (q − 1)th and (q + 1)th roots of unity in ℂ respectively.

Theorem 1 Let G = P SL_{2}(q) as a subgroup of S_{q+1}; where q is odd, q = p^{n}, q ≡^{4} 1; and let V be an s-dimensional vector space over the complex field ℂ. Then we have the following formulae for the dimension of V_{χ}^{q+1}(G) where χ ∈ I(G).

dim V_{1_{G}}^{q+1}(G) = \frac{2}{q(q^{2} − 1)} [s^{q+1} + (q^{2} − 1)s^{1+p^{n−1}} + q(q + 1) ∑_{l=1}^{(q−5)/4} s^{2+2(l,(q−1)/2)}

+ \frac{1}{2} q(q + 1)s^{(q+3)/2} + q(q − 1) ∑_{m=1}^{(q−1)/4} s^{2(m,(q+1)/2)}],

dim V_{ψ}^{q+1}(G) = \frac{2}{q^{2} − 1} [qs^{q+1} + q(q + 1) ∑_{l=1}^{(q−5)/4} s^{2+2(l,(q−1)/2)} + \frac{1}{2} q(q + 1)s^{(q+3)/2}

− q(q − 1) ∑_{m=1}^{(q−1)/4} s^{2(m,(q+1)/2)}],

dim V_{χ_{i}}^{q+1}(G) = \frac{2}{q(q − 1)} [(q + 1)s^{q+1} + (q^{2} − 1)s^{1+p^{n−1}} + q(q + 1) ∑_{l=1}^{(q−5)/4}(ρ^{il} + ρ^{−il})s^{2+2(l,(q−1)/2)}

+ \frac{1}{2} q(q + 1)(ρ^{i(q−1)/4} + ρ^{−i(q−1)/4})s^{(q+3)/2}],

i = 2, 4, 6, …, (q − 5)/2

dim V_{θ_{j}}^{q+1}(G) = \frac{2}{q(q + 1)} [(q − 1)s^{q+1} − (q^{2} − 1)s^{1+p^{n−1}} − q(q − 1) ∑_{m=1}^{(q−1)/4}(σ^{jm} + σ^{−jm})s^{2(m,(q+1)/2)}],

j = 2, 4, 6, …, (q − 1)/2

dim V_{ξ_{1}}^{q+1}(G) = \frac{1}{q(q − 1)} [ \frac{1}{2} (q + 1)s^{q+1} + \frac{1}{2} (q^{2} − 1)s^{1+p^{n−1}} + q(q + 1) ∑_{l=1}^{(q−5)/4}(−1)^{l}s^{2+2(l,(q−1)/2)}

+ (−1)^{(q−1)/4} \frac{1}{2} q(q + 1)s^{(q+3)/2}],

dim V_{ξ_{2}}^{q+1}(G) = \frac{1}{q(q − 1)} [ \frac{1}{2} (q + 1)s^{q+1} + \frac{1}{2} (q^{2} − 1)s^{1+p^{n−1}} + q(q + 1) ∑_{l=1}^{(q−5)/4}(−1)^{l}s^{2+2(l,(q−1)/2)}

+ (−1)^{(q−1)/4} \frac{1}{2} q(q + 1)s^{(q+3)/2}].

Proof. It is clear that if π is a cycle of length a and (a, k) = d, then π^{k} has d cycles of length a/d, therefore c(π^{k}) = d = (a, k). Now the permutation structure of the elements of G = P SL_{2}(q) when acting on the q + 1 one-dimensional subspaces are found in Table IV and from formula (1) the theorem follows. Note that in any case P SL_{2}(q) acts 2-transitively on the q + 1 projective points and its permutation character θ is given by θ = 1 + ψ. In fact θ(g) = 1 + ψ(g) is equal to the number of points left fixed by g ∈ P SL_{2}(q) which can be computed from Table I and which is indicated by fix(g). □

Similar to the above, we obtain Theorem 2 and 3 below.

Theorem 2 Let G = P SL_{2}(q) as a subgroup of S_{q+1}; where q is odd, q = p^{n}, q ≡^{4} 3; and let V be an s-dimensional vector space over the complex field ℂ. Then we have the following formulae for the dimension of V_{χ}^{q+1}(G), where χ ∈ I(G).

dim V_{1_{G}}^{q+1}(G) = \frac{2}{q(q^{2} − 1)} [s^{q+1} + (q^{2} − 1)s^{1+p^{n−1}} + q(q + 1) ∑_{l=1}^{(q−3)/4} s^{2+2(l,(q−1)/2)}

+ q(q − 1) ∑_{m=1}^{(q−3)/4} s^{2(m,(q+1)/2)} + \frac{1}{2} q(q − 1)s^{(q+1)/2}],

dim V_{ψ}^{q+1}(G) = \frac{2}{q^{2} − 1} [qs^{q+1} + q(q + 1) ∑_{l=1}^{(q−3)/4} s^{2+2(l,(q−1)/2)} − q(q − 1) ∑_{m=1}^{(q−3)/4} s^{2(m,(q+1)/2)}

− \frac{1}{2} q(q − 1)s^{(q+1)/2}],

dim V_{χ_{i}}^{q+1}(G) = \frac{2}{q(q − 1)} [(q + 1)s^{q+1} + (q^{2} − 1)s^{1+p^{n−1}} + q(q + 1) ∑_{l=1}^{(q−3)/4}(ρ^{il} + ρ^{−il})s^{2+2(l,(q−1)/2)}],

i = 2, 4, 6, …, (q − 3)/2

dim V_{θ_{j}}^{q+1}(G) = \frac{2}{q(q + 1)} [(q − 1)s^{q+1} − (q^{2} − 1)s^{1+p^{n−1}} − q(q − 1) ∑_{m=1}^{(q−3)/4}(σ^{jm} + σ^{−jm})s^{2(m,(q+1)/2)}

− \frac{1}{2} q(q − 1)(σ^{j(q+1)/4} + σ^{−j(q+1)/4})s^{(q+1)/2}],

j = 2, 4, 6, …, (q − 3)/2

dim V_{η_{1}}^{q+1}(G) = \frac{1}{q(q + 1)} [ \frac{1}{2} (q − 1)s^{q+1} − \frac{1}{2} (q^{2} − 1)s^{1+p^{n−1}} − q(q − 1) ∑_{m=1}^{(q−3)/4}(−1)^{m}s^{2(m,(q+1)/2)}

+ (−1)^{(q+5)/4} \frac{1}{2} q(q − 1)s^{(q+1)/2}],

dim V_{η_{2}}^{q+1}(G) = \frac{1}{q(q + 1)} [ \frac{1}{2} (q − 1)s^{q+1} − \frac{1}{2} (q^{2} − 1)s^{1+p^{n−1}} − q(q − 1) ∑_{m=1}^{(q−3)/4}(−1)^{m}s^{2(m,(q+1)/2)}

+ (−1)^{(q+5)/4} \frac{1}{2} q(q − 1)s^{(q+1)/2}].

Theorem 3 Let G = P SL_{2}(q) as a subgroup of S_{q+1}; where q is even, q = 2^{n}; and let V be an s-dimensional vector space over the complex field ℂ. Then we have the following formulae for the dimension of V_{χ}^{q+1}(G), where χ ∈ I(G).

dim V_{1_{G}}^{q+1}(G) = \frac{1}{q(q^{2} − 1)} [s^{q+1} + (q^{2} − 1)s^{(q+2)/2} + q(q + 1) ∑_{l=1}^{(q−2)/2} s^{2+(l,q−1)}

+ q(q − 1) ∑_{m=1}^{q/2} s^{(m,q+1)}],

dim V_{ψ}^{q+1}(G) = \frac{1}{q^{2} − 1} [qs^{q+1} + q(q + 1) ∑_{l=1}^{(q−2)/2} s^{2+(l,q−1)} − q(q − 1) ∑_{m=1}^{q/2} s^{(m,q+1)}] ,

dim V_{χ_{i}}^{q+1}(G) = \frac{1}{q(q − 1)} [(q + 1)s^{q+1} + (q^{2} − 1)s^{(q+2)/2} + q(q + 1) ∑_{l=1}^{(q−2)/2}(ρ^{il} + ρ^{−il})s^{2+(l,q−1)}] ,

i = 1, 2, …, (q − 2)/2

dim V_{θ_{j}}^{q+1}(G) = \frac{1}{q(q + 1)} [(q − 1)s^{q+1} − (q^{2} − 1)s^{(q+2)/2} − q(q − 1) ∑_{m=1}^{q/2}(σ^{jm} + σ^{−jm})s^{(m,q+1)}] ,

j = 1, 2, …, q/2

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

In 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 Consider G = P SL_{2}(q) as a subgroup of S_{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. Proof. Let γ = (1, 1, 2, 2, …, 2) ∈ Γ_{2}^{q+1} and consider the action of G on the set of

2-subsets of Ω consisting of the q + 1 points; Ω^{{2}}. This action is transitive and let G_{{Ω̃}}, Ω̃ ⊆ Ω, |Ω̃| = 2, denote the setwise stabilizer of Ω̃. It is easy to see that G_{γ} = G_{{Ω̃}}. By Frobenius reciprocity we have (χ ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} = (χ, 1_{G_{γ}} ↑^{G})_{G}. But 1_{G_{γ}} ↑^{G}= 1_{G_{{Ω̃}}} ↑^{G}= ξ is the permutation character of G acting on Ω^{{2}}. So

(χ ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} = (χ, ξ)_{G}  (5)

where ξ(g) is the number of 2-subsets of Ω fixed by g. Consider g as a permutation on Ω and recall that the permutation character of G on Ω is denoted by θ. Therefore there are \binom{θ(g)}{2} subsets of Ω of size 2 which are fixed by g setwise. A simple calculation shows that there are \frac{1}{2}(θ(g^{2}) − θ(g)) cycles of length 2 in the cycle structure of g and therefore the total number of 2-subsets of Ω fixed by g is

ξ(g) = \binom{θ(g)}{2} + \frac{θ(g^{2}) − θ(g)}{2} = \frac{1}{2} (θ(g)^{2} + θ(g^{2})) − θ(g).

Using Tables IV, V and VI we computed the values of ξ which are given in Tables VII, VIII and IX.

If q ≡^{4} 1, then the following decomposition of ξ is computed using Tables I and VII,

ξ = \{ \begin{matrix}1_{G} + 2ψ + 2 ∑_{i≡^{4}0} χ_{i} + ∑_{i odd} χ_{i} + ∑_{j} θ_{j} + ξ_{1} + ξ_{2} & if & q ≡^{8} 1, \\ 1_{G} + 2ψ + 2 ∑_{i≡^{4}0} χ_{i} + ∑_{i odd} χ_{i} + ∑_{j} θ_{j} & & otherwise. \end{matrix}

Therefore by (5) if χ ∈ I(G) − {χ_{i}, ξ_{1}, ξ_{2}| i = 2, 4, …, (q − 5)/2; i ≡^{4} 2}, then (χ ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} ≠ 0 and so V_{χ}^{q+1}(G) ≠ 0, additionally if q ≡^{8} 1, then (ξ_{1} ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} ≠ 0 and (ξ_{2} ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} ≠ 0 and so V_{ξ_{1}}^{q+1}(G) ≠ 0 and V_{ξ_{2}}^{q+1}(G) ≠ 0.

Furthermore, using Tables II and VIII when q ≡^{4} 3 we are able to decompose ξ as follows,

ξ = \{ \begin{matrix}1_{G} + ψ + ∑_{i} χ_{i} + 2 ∑_{j≡^{4}2} θ_{j} + ∑_{j odd} θ_{j} + η_{1} + η_{2} & if & q ≡^{8} 3, \\ 1_{G} + ψ + ∑_{i} χ_{i} + 2 ∑_{j≡^{4}2} θ_{j} + ∑_{j odd} θ_{j} & & otherwise. \end{matrix}

Therefore by (5) if χ ∈ I(G) − {θ_{j}, η_{1}, η_{2}| j = 2, 4, …, (q − 3)/2; j ≡^{4} 0}, then (χ ↓_{G_{γ}} , 1_{G_{γ}})_{G_{γ}} ≠ 0 and so V_{χ}^{q+1}(G) ≠ 0, additionally if q ≡^{8} 3, then (η_{1} ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} ≠ 0 and (η_{2} ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} ≠ 0 and so V_{η_{1}}^{q+1}(G) ≠ 0 and V_{η_{2}}^{q+1}(G) ≠ 0.

Also if q is even, then by Tables III and IX we have

ξ = 1_{G} + ψ + ∑_{i} χ_{i}.

Therefore by (5) if χ ∈ I(G) − {θ_{j}| 1 ≤ j ≤ q/2}, then (χ ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} ≠ 0 and so V_{χ}^{q+1}(G) ≠ 0. □

Theorem 5 Consider G = P SL_{2}(q) as a subgroup of S_{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.

Proof. First we assume that dim V = s = 3. Let γ = (1, 2, 3, 3, …, 3) ∈ Γ_{3}^{q+1} and consider the action of G on the set of ordered pairs of points of Ω namely Ω^{(2)}. This action is transitive and let G_{(Ω̂)}, Ω̂ ⊆ Ω, |Ω̂| = 2, denote the pointwise stabilizer of Ω̂. Therefore G_{γ} = G_{(Ω̂)}. By Frobenius reciprocity we have (χ ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} = (χ, 1_{G_{γ}} ↑^{G})_{G}. But 1_{G_{γ}} ↑^{G}= 1_{G_{(Ω̂)}} ↑^{G}= ξ′ is the permutation character of G acting on Ω^{(2)}. So

(χ ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} = (χ, ξ′)_{G}  (6)

where ξ′(g) is the number of ordered pairs fixed by g and so using similar techniques as in the previous theorem we get

ξ′(g) = 2\binom{θ(g)}{2} = θ(g)^{2} − θ(g).

We computed the values of ξ′ in Tables VII, VIII and IX. Using these tables and the character table of G = P SL_{2}(q) we obtain

ξ′ = 1_{G} + 3ψ + 2 ∑_{i} χ_{i} + 2 ∑_{j} θ_{j} + ξ_{1} + ξ_{2},

when q ≡^{4} 1 and therefore by (6) if χ ∈ I(G), then (χ ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} ≠ 0 and so V_{χ}^{q+1}(G) ≠ 0.

If q ≡^{4} 3, then

ξ′ = 1_{G} + 3ψ + 2 ∑_{i} χ_{i} + 2 ∑_{j} θ_{j} + η_{1} + η_{2}.

Therefore by (6) if χ ∈ I(G), then (χ ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} ≠ 0 and so V_{χ}^{q+1}(G) ≠ 0. If q is even, then

ξ′ = 1_{G} + 2ψ + ∑_{i} χ_{i} + ∑_{j} θ_{j}.

Therefore by (6) if χ ∈ I(G), then (χ ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} ≠ 0 and so V_{χ}^{q+1}(G) ≠ 0. So we proved that if dim V = s = 3, then V_{χ}^{q+1}(G) ≠ 0, for all χ, χ ∈ I(G).

Now we assume that dim V = s ≥ 4. In this case if we consider γ = (1, 2, 3, 4, 4, …, 4) ∈ Γ_{s}^{q+1}, then by Tables IV, V and VI the elements of G; except the identity, fix at most 2 points of Ω and so G_{γ} is the trivial subgroup of G, i.e., G_{γ} = {1}; therefore for all χ, χ ∈ I(G), (χ ↓_{G_{γ}}, 1_{G_{γ}})_{G_{γ}} = χ(1) ≠ 0 and so V_{χ}^{q+1}(G) ≠ 0. □

Table I The character table of G = P SL_{2}(q); q is odd, q = p^{n}, q ≡^{4} 1; |G| = \frac{1}{2} q(q^{2} − 1)

g {−I, I}1  {−I, I}c  {−I, I}d  {−I, I}a^{l}  {−I, I}a^{(q−1)/4}  {−I, I}b^{m}
|(g)| 1  \frac{1}{2} (q^{2} − 1)  \frac{1}{2} (q^{2} − 1)  q(q + 1)  \frac{1}{2} q(q + 1)  q(q − 1)
o(g) 1  p  p  \frac{(q−1)/2}{(l,(q−1)/2)}  2  \frac{(q+1)/2}{(m,(q+1)/2)}
1_{G} 1  1  1  1  1  1
ψ q  0  0  1  1  −1
χ_{i} q + 1  1  1  ρ^{il} + ρ^{−il} ρ^{i(q−1)/4} + ρ^{−i(q−1)/4}  0
θ_{j} q − 1  −1  −1  0  0  −(σ^{jm} + σ^{−jm})
ξ_{1}  \frac{1}{2} (q + 1)  \frac{1}{2} (1 + \sqrt{q})  \frac{1}{2} (1 − \sqrt{q})  (−1)^{l}  (−1)^{(q−1)/4}  0
ξ_{2}  \frac{1}{2} (q + 1)  \frac{1}{2} (1 − \sqrt{q})  \frac{1}{2} (1 + \sqrt{q})  (−1)^{l}  (−1)^{(q−1)/4}  0
i = 2, 4, 6, …, (q − 5)/2  ,  l = 1, 2, …, (q − 5)/4  ,  ρ = e^{2π\sqrt{−1}/q−1}
j = 2, 4, 6, …, (q − 1)/2  ,  m = 1, 2, …, (q − 1)/4  ,  σ = e^{2π\sqrt{−1}/q+1}

|I(G)| = 4 + \frac{q−5}{4} + \frac{q−1}{4} = \frac{q+5}{2}

Table II The character table of G = P SL_{2}(q); q is odd, q = p^{n}, q ≡^{4} 3; |G| = \frac{1}{2} q(q^{2} − 1)

g {−I, I}1  {−I, I}c  {−I, I}d  {−I, I}a^{l}  {−I, I}b^{m}  {−I, I}b^{(q+1)/4}
|(g)| 1  \frac{1}{2} (q^{2} − 1)  \frac{1}{2} (q^{2} − 1)  q(q + 1)  q(q − 1)  \frac{1}{2} q(q − 1)
o(g) 1  p  p  \frac{(q−1)/2}{(l,(q−1)/2)}  \frac{(q+1)/2}{(m,(q+1)/2)}  2
1_{G} 1  1  1  1  1  1
ψ q  0  0  1  −1  −1
χ_{i} q + 1  1  1  ρ^{il} + ρ^{−il}  0  0
θ_{j} q − 1  −1  −1  0  −(σ^{jm} + σ^{−jm}) −(σ^{j(q+1)/4} + σ^{−j(q+1)/4})
η_{1}  \frac{1}{2} (q − 1) − \frac{1}{2} (1 − \sqrt{−q}) − \frac{1}{2} (1 + \sqrt{−q})  0  (−1)^{m+1}  (−1)^{(q+5)/4}
η_{2}  \frac{1}{2} (q − 1) − \frac{1}{2} (1 + \sqrt{−q}) − \frac{1}{2} (1 − \sqrt{−q})  0  (−1)^{m+1}  (−1)^{(q+5)/4}

i = 2, 4, 6, …, (q − 3)/2 , l = 1, 2, 3, …, (q − 3)/4 , ρ = e^{2π\sqrt{−1}/q−1} j = 2, 4, 6, …, (q − 3)/2 , m = 1, 2, 3, …, (q − 3)/4 , σ = e^{2π\sqrt{−1}/q+1} |I(G)| = 4 + \frac{q−3}{4} + \frac{q−3}{4} = \frac{q+5}{2}

Table III

The character table of G = P SL_{2}(q); q is even, q = 2^{n}; |G| = q(q^{2} − 1)

g {I}1 {I}c  {I}a^{l}  {I}b^{m}
|(g)| 1  q^{2} − 1  q(q + 1)  q(q − 1)
o(g) 1  2  \frac{q−1}{(l,q−1)}  \frac{q+1}{(m,q+1)}
1_{G} 1  1  1  1
ψ q  0  1  −1
χ_{i} q + 1  1  ρ^{il} + ρ^{−il}  0
θ_{j} q − 1 −1  0  −(σ^{jm} + σ^{−jm})
1 ≤ i ≤ (q − 2)/2 , 1 ≤ l ≤ (q − 2)/2 , ρ = e^{2π\sqrt{−1}/q−1}
1 ≤ j ≤ q/2 , 1 ≤ m ≤ q/2 , σ = e^{2π\sqrt{−1}/q+1}

|I(G)| = 2 + \frac{q−2}{2} + \frac{q}{2} = q + 1

Table IV

The permutation structure of elements of the group G = P SL_{2}(q) ≤ S_{q+1}; q is odd, q = p^{n}, q ≡^{4} 1; |G| = \frac{1}{2} q(q^{2} − 1)

g  {−I, I}1 {−I, I}c {−I, I}d  {−I, I}a^{l}  {−I, I}a^{(q−1)/4}  {−I, I}b^{m}
|(g)|  1  \frac{1}{2} (q^{2} − 1) \frac{1}{2} (q^{2} − 1)  q(q + 1)  \frac{1}{2} q(q + 1)  q(q − 1)
o(g)  1  p  p  \frac{(q−1)/2}{(l,(q−1)/2)}  2  \frac{(q+1)/2}{(m,(q+1)/2)}
|fix(g)|  q + 1  1  1  2  2  0
per. stru. of g 1^{q+1}  1^{1}p^{p^{n−1}} 1^{1}p^{p^{n−1}} 1^{2} \frac{(q−1)/2}{(l,(q−1)/2)}^{2(l,(q−1)/2)}  1^{2}2^{(q−1)/2}  \frac{(q+1)/2}{(m,(q+1)/2)}^{2(m,(q+1)/2)}

1 ≤ l ≤ (q − 5)/4 1 ≤ m ≤ (q − 1)/4

Table V

The permutation structure of elements of the group G = P SL_{2}(q) ≤ S_{q+1}; q is odd, q = p^{n}, q ≡^{4} 3; |G| = \frac{1}{2} q(q^{2} − 1)

g  {−I, I}1 {−I, I}c {−I, I}d  {−I, I}a^{l}  {−I, I}b^{m}  {−I, I}b^{(q+1)/4}
|(g)|  1  \frac{1}{2} (q^{2} − 1) \frac{1}{2} (q^{2} − 1)  q(q + 1)  q(q − 1)  \frac{1}{2} q(q − 1)
o(g)  1  p  p  \frac{(q−1)/2}{(l,(q−1)/2)}  \frac{(q+1)/2}{(m,(q+1)/2)}  2
|fix(g)|  q + 1  1  1  2  0  0
per. stru. of g 1^{q+1}  1^{1}p^{p^{n−1}} 1^{1}p^{p^{n−1}} 1^{2} \frac{(q−1)/2}{(l,(q−1)/2)}^{2(l,(q−1)/2)} \frac{(q+1)/2}{(m,(q+1)/2)}^{2(m,(q+1)/2)}  2^{(q+1)/2}

1 ≤ l ≤ (q − 3)/4 1 ≤ m ≤ (q − 3)/4

Table VI The permutation structure of elements of the group G = P SL_{2}(q) ≤ S_{q+1}; q is even, q = 2^{n}; |G| = q(q^{2} − 1)

g  {I}1 {I}c  {I}a^{l}  {I}b^{m}
|(g)|  1  q^{2} − 1  q(q + 1)  q(q − 1)
o(g)  1  2  \frac{q−1}{(l,q−1)}  \frac{q+1}{(m,q+1)}
|fix(g)|  q + 1  1  2  0
per. stru. of g 1^{q+1} 1^{1}2^{q/2} 1^{2} \frac{q−1}{(l,q−1)}^{(l,q−1)} \frac{q+1}{(m,q+1)}^{(m,q+1)}
1 ≤ l ≤ (q − 2)/2
1 ≤ m ≤ q/2

Table VII

G = P SL_{2}(q); q is odd, q = p^{n}, q ≡^{4} 1

g  {−I, I}1 {−I, I}c {−I, I}d {−I, I}a^{l} {−I, I}a^{(q−1)/4} {−I, I}b^{m}
|C_{G}(g)| \frac{1}{2} q(q^{2} − 1)  q  q  \frac{1}{2} (q − 1)  q − 1  \frac{1}{2} (q + 1)
θ(g) q + 1  1  1  2  2  0
ξ(g) \frac{1}{2} q(q + 1)  0  0  1  \frac{1}{2} (q + 1)  0
ξ′(g) q(q + 1)  0  0  2  2  0

l = 1, 2, …, (q − 5)/4 m = 1, 2, …, (q − 1)/4

Table VIII

G = P SL_{2}(q); q is odd, q = p^{n}, q ≡^{4} 3

g  {−I, I}1 {−I, I}c {−I, I}d {−I, I}a^{l} {−I, I}b^{m} {−I, I}b^{(q+1)/4}
|C_{G}(g)| \frac{1}{2} q(q^{2} − 1)  q  q  \frac{1}{2} (q − 1) \frac{1}{2} (q + 1)  q + 1
θ(g) q + 1  1  1  2  0  0
ξ(g) \frac{1}{2} q(q + 1)  0  0  1  0  \frac{1}{2} (q + 1)
ξ′(g) q(q + 1)  0  0  2  0  0

l = 1, 2, …, (q − 3)/4 m = 1, 2, …, (q − 3)/4

Table IX G = P SL_{2}(q); q is even, q = 2^{n}

g  {I}1  {I}c {I}a^{l} {I}b^{m}
|C_{G}(g)| q(q^{2} − 1)  q q − 1 q + 1
θ(g) q + 1  1  2  0
ξ(g) \frac{1}{2} q(q + 1) \frac{1}{2} q  1  0
ξ′(g) q(q + 1)  0  2  0
1 ≤ l ≤ (q − 2)/2
1 ≤ m ≤ q/2

References

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

[2] M. R. Darafsheh, M. R. Pournaki, On the Dimensions of Cyclic Symmetry Classes of Tensors, J. Algebra 205 (1998), no. 1, 317-325.

[3] L. Dornhoff, “Group Representation Theory”, Part I, Marcel Dekker, Inc., 1972.

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

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

[6] I. M. Isaacs, “Character Theory of Finite Groups”, Dover: Academic Press, 1994.

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

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

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

[10] 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