MAXIMAL REPRESENTATION DIMENSION FOR GROUPS OF ORDER p^{n}

SHANE CERNELE, MASOUD KAMGARPOUR, AND ZINOVY REICHSTEIN

Abstract. The representation dimension rdim(G) of a finite group G is the smallest positive integer m for which there exists an embedding of G in GL_{m}(ℂ). In this paper we find the largest value of rdim(G), as G ranges over all groups of order p^{n}, for a fixed prime p and a fixed exponent n ≥ 1.

1. Introduction

The representation dimension of a finite group G, denoted by rdim(G), is the minimal dimension of a faithful complex linear representation of G. In this paper we determine the maximal representation dimension of a group of order p^{n}. We are motivated by a recent result of N. Karpenko and A. Merkurjev [KM07, Theorem 4.1], which states that if G is a finite p-group then the essential dimension of G is equal to rdim(G). For a detailed discussion of the notion of essential dimension for finite groups (which will not be used in this paper), see [BR97] or [JLY02, §8]. We also note that a related invariant, the minimal dimension of a faithful complex projective representation of G, has been extensively studied for finite simple groups G; for an overview, see [TZ00, §3].

Let G be a p-group of order p^{n} and r be the rank of the centre C(G). A representation of G is faithful if and only if its restriction to C(G) is faithful. Using this fact it is easy to see that a faithful representation ρ of G of minimal dimension decomposes as a direct sum

(1)  ρ = ρ_{1} ⊕ ⋯ ⊕ ρ_{r}

of exactly r irreducibles; cf. [MR09, Theorem 1.2]. Since the dimension of any irreducible representation of G is ≤ \sqrt{[G : C(G)]} (see, e.g., [W03, Corollary 3.11]) and |C(G)| ≥ p^{r}, we conclude that

(2)  rdim(G) ≤ rp^{⌊(n−r)/2⌋}.

2000 Mathematics Subject Classification. 20C15, 20D15.

Key words and phrases. p-group, representation dimension, symplectic subspace, generalized Heisenberg group.

The authors gratefully acknowledge financial support by the Natural Sciences and Engineering Research Council of Canada.

Let

f_{p}(n) := max_{r∈ℕ}(rp^{⌊(n−r)/2⌋}).

It is easy to check that f_{p}(n) is given by the following table:

n  p  f_{p}(n)
even arbitrary 2p^{(n−2)/2}
odd  odd  p^{(n−1)/2}
odd, ≥ 3  2  3p^{(n−3)/2}
1  2  1

We are now ready to state the main result of this paper.

Theorem 1. Let p be a prime and n be a positive integer. For almost all pairs (p, n), the maximal value of rdim(G), as G ranges over all groups of order p^{n}, equals f_{p}(n). The exceptional cases are

(p, n) = (2, 5), (2, 7) and (p, 4), where p is odd.

In these cases the maximal representation dimension is 5, 10, and p + 1, respectively.

The proof will show that the maximal value of rdim(G), as G ranges over all groups of order p^{n}, is always attained for a group G of nilpotency class ≤ 2. Moreover, if (p, n) is non-exceptional, n ≥ 3 and (p, n) ≠ (2, 3), (2, 4), the maximum is attained on a special class of p-groups of nilpotency class 2, which we call generalized Heisenberg groups.

The rest of this paper is structured as follows. In §2 we introduce generalized Heisenberg groups and study their irreducible representations. Theorem 1 is proved in §3.

Acknowledgement. We would like to thank Hannah Cairns, Robert Guralnick, Chris Parker, Burt Totaro, and Robert Wilson for helpful discussions.

2. Generalized Heisenberg groups

2.1. Spaces of alternating forms. Let V be a finite dimensional vector space over an arbitrary field F. Let K ⊂ Λ^{2}(V)^{∗} be a subspace. We will say that K is symplectic if every nonzero element of K is a non-degenerate alternating map. Clearly nontrivial symplectic subspaces of Λ^{2}(V)^{∗} can exist only if dim(V) is even.

Lemma 2. Suppose V is an F -vector space of dimension 2m. If F admits a field extension of degree m then there exists an m-dimensional symplectic subspace K ⊂ Λ^{2}(V)^{∗}.

Proof. Choosing a basis of V, we can identify Λ^{2}(V)^{∗} with the space of skew-symmetric 2m × 2m-matrices. Let f: M_{m}(F) → Λ^{2}(V)^{∗} be the linear map

A ↦ [\begin{matrix}0 & A \\ −A^{T} & 0 \end{matrix}] .

If W is a linear subspace of M_{m}(F) = End_{F} (F^{n}) such that W \{0} ⊂ GL_{m}(F) then K = f(W) is a symplectic subspace.

It thus remains to construct an m-dimensional linear subspace W of M_{m}(F) such that W \{0} ⊂ GL_{m}(F). Let E be a degree m field extension of F. Then E acts on itself by left multiplication. This gives an F -vector space embedding of Ψ: E ↪ End_{F} (E) such that Ψ(e) is invertible for all e ≠ 0. □

2.2. Groups associated to spaces of alternating forms. Let ω_{K} : Λ^{2}(V) → K^{∗} denote the dual of the natural injection K ↪ Λ^{2}(V)^{∗}. Note that there exists a bilinear map β : V ⊗ V → K^{∗} such that

(3)  ω_{K}(v, w) = β(v, w) − β(w, v)  ∀ v, w ∈ V.

Indeed, choose a basis {e_{1}, …, e_{n}} for V, and define β by

β(e_{i}, e_{j}) = \{ \begin{matrix}ω_{K}(e_{i}, e_{j}) & if i > j, and \\ 0 & otherwise. \end{matrix}

Definition 3. To the data (V, K, β) as above, we attach a group H = H(V, K, β). As a set H = V × K^{∗}; the group operation is given by

(4)  (v, t) · (v′, t′) = (v + v′, t + t′ + β(v, v′)).

If K is a symplectic subspace, we will refer to H as a generalized Heisenberg group.

It is easy to see that (4) is indeed a group law with the inverse given by (v, t)^{−1} = (−v, −t + β(v, v)) and the commutator given by

(5)  [(v_{1}, t_{1}), (v_{2}, t_{2})] = (0, ω_{K}(v_{1}, v_{2})) .

Remark 4. Let H = H(V, K, β), as in Definition 3. Since the inclusion K ↪ Λ^{2}(V)^{∗} is, by definition, injective, its dual ω_{K}: Λ^{2}(V) → K^{∗} is surjective. Formula (5) now tells us that [H, H] = K^{∗}.

Moreover, (5) also shows that K^{∗} ⊂ C(H), and that equality holds unless the intersection ∩_{k∈K} ker(k) is nontrivial. In particular, C(H) = K^{∗} if K contains a symplectic form.

Remark 5. The reason for the term generalized Heisenberg group is that in the special case, where F = 𝔽_{p}, p is an odd prime, K is a one-dimensional symplectic subspace and β = \frac{1}{2}ω_{K}, the group H(V, K, β) is often called the Heisenberg group.

Note that β is not uniquely determined by K; it is only unique up to adding a symmetric bilinear form V × V → K^{∗}. If β and β′ both satisfy (3) then H(V, K, β) may not be isomorphic to H(V, K, β′). For example, let V be a 2-dimensional vector space over F = 𝔽_{2}, K be the one-dimensional (symplectic) subspace generated by [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}], and β, β′ be bilinear forms on

V defined by

[\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}] and [\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}],

respectively. Then β and β′ both satisfy (3), but H(V, K, β) is isomorphic to the quaternion group while H(V, K, β′) is isomorphic to the dihedral group of order 8.

Remark 6. Two groups S and T are isoclinic if there are isomorphisms f : S/C(S) → T/C(T) and g : [S, S] → [T, T] such that if a, b ∈ S and a′, b′ ∈ T with f(aC(S)) = a′C(T) and f(bC(S)) = b′C(T), then we have g([a, b]) = [a′, b′]; see [PH40]. Let K be a subspace of Λ^{2}(V)^{∗}, and suppose β and β′ are bilinear forms on V satisfying (3). Then H = H(V, K, β) and H′ = H(V, K, β′) are isoclinic (in this case f and g are identity maps).

2.3. Representations. Let p be an arbitrary prime, F = 𝔽_{p} be the finite field of p elements, V be a vector space over F and K be a subspace of Λ^{2}(V)^{∗}. Let ω = ω_{K} : Λ^{2}(V) → K^{∗} denote the dual of K ↪ Λ^{2}(V)^{∗}.

Let ψ be a morphism K^{∗} → ℂ^{×}. Identify (non-canonically) 𝔽_{p} with the group of p^{th} roots of unity in ℂ^{×}, so that we can view ψ as being in (K^{∗})^{∗}. Using the canonical isomorphism between (K^{∗})^{∗} and K we associate to ψ an element k ∈ K such that k = ψ ∘ ω. In particular, k is non-degenerate if and only if ψ ∘ ω is non-degenerate; this condition does not depend on the way we identify 𝔽_{p} with the group of p^{th} roots of unity in ℂ^{×}. Conversely, to each k ∈ K we can associate a character ψ of K^{∗} such that if we view ψ ∈ (K^{∗})^{∗}, we have k = ψ ∘ ω.

Lemma 7. Let G = V (V, K, β) = V × K^{∗} be as in Definition 3. Let ρ be an irreducible representation of G such that K^{∗} acts by ψ. Assume ψ ∘ ω : V ⊗ V → ℂ^{×} is non-degenerate.

(a) If g ∈ G, g ∉ K^{∗}, then Tr(ρ(g)) = 0.

(b) dim(ρ) = \sqrt{|V |}.

(c) ρ is uniquely determined (up to isomorphism) by ψ.

Proof. (a) Let g ∈ G\K^{∗}. Since ψ ∘ ω is non-degenerate there exists h ∈ G such that ψ ∘ ω(gK^{∗}, hK^{∗}) ≠ 1. Observe that ρ([g, h]) = ψ([g, h]) Id, and that ρ(h^{−1}gh) = ρ(g)ρ([g, h]). Taking the trace of both sides, we have Tr(ρ(g)) = ψ([g, h]) Tr(ρ(g)). Since ψ([g, h]) ≠ 1 we must have Tr(ρ(g)) = 0.

(b) Since ρ is irreducible, and the trace of ρ vanishes outside of K^{∗}, we have:

1 = \frac{1}{|G|} ∑_{g∈G} Tr(ρ(g))\overline{Tr(ρ(g))}

= \frac{1}{|G|} ∑_{g∈K^{∗}} Tr(ρ(g))\overline{Tr(ρ(g))}

= \frac{1}{|G|} dim(ρ)^{2} ∑_{g∈K^{∗}} Tr(ψ(g))\overline{Tr(ψ(g))}

= dim(ρ)^{2} \frac{|K^{∗}|}{|G|}

Thus dim ρ = \sqrt{|G|/|K^{∗}|} = \sqrt{|V |}.

(c) We have completely described the character of ρ, and it follows that ρ is uniquely determined by ψ. Indeed,

Tr(ρ(g)) = \begin{cases}\sqrt{|V |} · ψ(g), & if g ∈ K^{∗} and \\ 0 & otherwise. \end{cases}
□

Henceforth, let K be a symplectic subspace of Λ^{2}(V)^{∗}, H = H(V, K, β) = V × K^{∗} be a generalized Heisenberg group, for some β as in (3). The proposition below is a direct consequence of Lemma 7.

Proposition 8. The irreducible representations of H are exhausted by the following list:

(i) |V | one-dimensional representations, one for every character of V. (ii) |K| − 1 representations of dimension \sqrt{|V |}, one for every nontrivial

character ψ : K^{∗} → ℂ^{×}.

The next corollary is also immediate upon observing that C(H) = K^{∗}; see Remark 4.

Corollary 9. The representation dimension of H equals dim(K)\sqrt{|V |}.

If G is a finite Heisenberg group in the usual sense (as in Remark 5) then for each nontrivial character χ of C(G) there is a unique irreducible representation ψ of G whose central character is χ; cf. [GH07, §1.1]. This is a finite group variant of the celebrated Stone-von Neumann Theorem. For a detailed discussion of the history and the various forms of the Stonevon Neumann theorem we refer the reader to [R04]. Proposition 8 tells us that, in fact, every generalized Heisenberg group over 𝔽_{p} has the Stone-von Neumann property. This observation, stated as Corollary 10 below, will not be used in the sequel.

Corollary 10. Two irreducible representations of H with the same nontrivial central character are isomorphic.

3. Proof of Theorem 1

The case where n ≤ 2 is trivial; clearly rdim(G) = rank (G) if G is abelian. We will thus assume that n ≥ 3.

In the non-exceptional cases of the theorem, in view of the inequality (2), it suffices to construct a group G of order p^{n} with rdim(G) = f_{p}(n). Here f_{p}(n) is the function defined just before the statement of Theorem 1.

If (p, n) = (2, 3) or (2, 4), we take G to be the elementary abelian group 𝔽_{2}^{3} and 𝔽_{2}^{4}, yielding the desired representation dimension of 3 and 4, respectively. For all other non-exceptional pairs (p, n), we take G to be a generalized Heisenberg group as described in the table below. Here H(V, K) stands for H(V, K, β), for some β as in (3). In each instance, the existence of a symplectic subspace K of suitable dimension is guaranteed by Lemma 2 and the value of rdim(H(V, K)) is given by Corollary 9.

n  p  dim(V) dim(K) rdim(H(V, K))
even, ≥ 6 arbitrary n − 2  2  2p^{(n−2)/2}
odd, ≥ 3  odd  n − 1  1  p^{(n−1)/2}
odd, ≥ 9  2  n − 3  3  3p^{(n−3)/2}

This settles the generic case of Theorem 1. We now turn our attention to the exceptional cases. We will need the following upper bound on rdim(G), strengthening (2).

Let C(G)_{p} be the subgroup of central elements g ∈ C(G) such that g^{p} = 1. If ρ: G → GL(V) is an irreducible representation then C(G) (and hence, its subgroup C(G)_{p}) acts on V by scalar multiplication, g · v ↦ χ(g)v, where χ is a multiplicative character of C(G). Following [MR09, Lemma 2.2], we will call χ: C(G)_{p} → ℂ^{×} the associated character (to ρ).

Lemma 11. Let G be a p-group and r = rank (C(G)) = rank (C(G)_{p}).

(a) Suppose there exists an irreducible representation ρ_{1} such that Ker(ρ_{1}) does not contain C(G)_{p}. Then there are irreducible representations ρ_{2}, …, ρ_{r} of G such that ρ_{1} ⊕ ⋯ ⊕ ρ_{r} is faithful. In particular,

rdim(G) ≤ dim(ρ_{1}) + (r − 1)\sqrt{[G : C(G)]} .

(b) If C(G)_{p} is not contained in [G, G], then

rdim(G) ≤ 1 + (r − 1)\sqrt{[G : C(G)]} .

The lemma can be deduced from [KM07, Remark 4.7] or [MR09, Theorem 1.2]; for the sake of completeness we give a self-contained proof.

Proof. (a) Let χ_{1} be the character of C(G)_{p} associated to ρ_{1}. By our assumption χ_{1} is nontrivial. Complete χ_{1} to a basis χ_{1}, χ_{2}, …, χ_{r} of the r-dimensional 𝔽_{p}-vector space C(G)_{p}^{∗} and choose an irreducible representation ρ_{i} with associated character χ_{i}. (The representation ρ_{i} can be taken to be any irreducible component of the induced representation Ind_{C(G)_{p}}^{G}(χ).) The restriction of ρ: = ρ_{1} ⊕ ⋯ ⊕ ρ_{r} to C(G)_{p} is faithful. Hence, ρ is a faithful representation of G. As we mentioned in the introduction dim(ρ_{i}) ≤ \sqrt{[G : C(G)]} for every i ≥ 2, and part (a) follows.

(b) By our assumption there exists one-dimensional representation ρ_{1} of G whose restriction to C(G)_{p} is nontrivial. Now apply part (a). □

We are now ready to prove Theorem 1 in the three exceptional cases.

3.1. Exceptional case 1: p is odd and n = 4.

Lemma 12. Let p be an odd prime and G be a group of order p^{4}. (a) Then rdim(G) ≤ p + 1.

(b) Suppose C(G) ≃ 𝔽_{p}^{2} and G/C(G) ≃ 𝔽_{p}^{2}. Then rdim(G) = p + 1.

Proof. (a) We argue by contradiction. Assume there exists a group of order p^{4} such that rdim(G) ≥ p + 2. If |C(G)| ≥ p^{3} or G/C(G) is cyclic then G is abelian and rdim(G) = rank (G) ≤ 4 ≤ p + 1, a contradiction. If C(G) is cyclic then rdim(G) ≤ p by (2), again a contradiction.

Thus C(G) ≃ G/C(G) ≃ 𝔽_{p}^{2}. This reduces part (a) to part (b).

(b) Here C(G)_{p} = C(G) has rank 2. Hence, a faithful representation ρ of G of minimal dimension is the sum of two irreducibles ρ_{1} ⊕ ρ_{2}, as in (1), each of dimension 1 or p.

Clearly dim(ρ_{1}) = dim(ρ_{2}) = 1 is not possible, since in this case G would be abelian, contradicting [G : C(G)] = p^{2}. It thus remains to show that rdim(G) ≤ p + 1. Since G/C(G) is abelian, [G, G] ⊂ C(G). Hence, by Lemma 11(b) we only need to establish that [G, G] ⊊ C(G).

To show that [G, G] ⊊ C(G), note that the commutator map

Ψ : G/C(G) × G/C(G) → [G, G]
(gC(G), g′C(G)) ↦ [g, g′]

can be thought of as an alternating bilinear map from 𝔽_{p}^{2} to itself. Viewed in this way, Ψ can be written as Ψ(v, v′) = (w_{1}(v, v′), w_{2}(v, v′)) for alternating maps w_{1} and w_{2} from 𝔽_{p}^{2} to 𝔽_{p}. Since Λ^{2}(𝔽_{p}^{2})^{∗} is a one-dimensional vector space over 𝔽_{p}, w_{1} and w_{2} are scalar multiples of each other. Hence, the image of Ψ is a cyclic group of order p, and [G, G] ⊊ C(G), as claimed. □

To finish the proof in this case, note that G = 𝔽_{p} × G_{0}, where G_{0} is a non-abelian group of order p^{3}, satisfies the conditions of Lemma 12(b). Thus the maximal representation dimension of a group of order p^{4} is p + 1, for any odd prime p.

3.2. Exceptional case 2: p = 2 and n = 5.

Lemma 13. Let G be a group of order 32. Then rdim(G) ≤ 5.

Proof. We argue by contradiction. Assume there exists a group of order 32 and representation dimension ≥ 6. Let r = rank (C(G)). Then 1 ≤ r ≤ 5 and (2) shows that rdim(G) ≤ 5 for every r ≠ 3.

Thus we may assume r = 3. If |C(G)| ≥ 16 or G/C(G) is cyclic then G is abelian, and rdim(G) = rank (G) ≤ 5. We conclude that C(G) ≃ 𝔽_{2}^{3} and G/C(G) ≃ 𝔽_{2}^{2}. Applying the same argument as in the proof of Lemma 12(b), we see that [G, G] ⊊ C(G), and hence rdim(G) ≤ 5 by Lemma 11(b), a contradiction. □

To finish the proof in this case, note that the elementary abelian group of order 2^{5} has representation dimension 5. Thus the maximal representation dimension of a group of order 2^{5} is 5.

3.3. Exceptional case 3: p = 2 and n = 7.

Lemma 14. If |G| = 128 then rdim(G) ≤ 10.

Proof. Again, we argue by contradiction. Assume there exists a group G of order 128 and representation dimension ≥ 11. Let r be the rank of C(G). By (2), r = 3; otherwise we would have rdim(G) ≤ 10.

As we explained in the introduction, this implies that a faithful representation ρ of G of minimal dimension is the direct sum of three irreducibles ρ_{1}, ρ_{2} and ρ_{3}, each of dimension ≤ \sqrt{2^{7}/|C(G)|}. If |C(G)| > 8, then dim(ρ_{i}) ≤ 2 and rdim(G) = dim(ρ_{1}) + dim(ρ_{2}) + dim(ρ_{3}) ≤ 6, a contradiction.

Therefore, C(G) ≅ (𝔽_{2})^{3} and dim(ρ_{1}) = dim(ρ_{2}) = dim(ρ_{3}) = 4. By Lemma 11(a) this implies that the kernel of every irreducible representation of G of dimension 1 or 2 must contain C(G). In other words, any such representation factors through the group G/C(G) of order 16. Consequently, if m_{i} is the number of irreducible representations of G of dimension i then m_{1}+4m_{2} = 16. We can now appeal to [JNO90, Tables I and II], to show that no group of order 2^{7} has these properties. From Table I we can determine which groups G (up to isoclinism, cf. Remark 6) have |C(G)| = 8 and using Table II we can determine m_{1} and m_{2} for these groups. There is no group G with |C(G)| = 8 and m_{1} + 4m_{2} = 16. □

We will now construct an example of a group G of order 2^{7} with rdim(G) = 10. Let V = 𝔽_{2}^{4} and let K be the 3-dimensional subspace of Λ^{2}(V)^{∗} generated by the following three elements:

[\begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}] , [\begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}] , [\begin{matrix}0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 \end{matrix}] .

Let G := H(V, K, β) = V × K^{∗} for some β as in (3). Note that K contains only one non-zero degenerate element (the sum of the three generators). In other words, there is only one character χ of K^{∗} such that χ∘ω : V ×V → ℂ^{×} is degenerate. By Remark 4

[G, G] = C(G) = K^{∗} .

Let ρ be a faithful representation of G of minimal dimension. As we explained in the Introduction, ρ is the sum of rank (C(G)) = 3 irreducibles.

Denote them by ρ_{1}, ρ_{2}, and ρ_{3}, and their associated characters by χ_{1}, χ_{2} and χ_{3}, respectively. Since ρ is faithful, χ_{1}, χ_{2} and χ_{3} form an 𝔽_{2}-basis of C(G)_{p}^{∗} ≃ 𝔽_{2}^{3}. By Lemma 7, for each nontrivial character χ of K^{∗} except one, there is a unique irreducible representation ψ of G such that χ is the associated character to ψ, and dim ψ = 4. Thus at least 2 of the irreducible components of ρ, say, ρ_{1} and ρ_{2} must have dimension 4. By Lemma 14, dim(ρ) ≤ 10, i.e., dim(ρ_{3}) ≤ 2. But every one-dimensional representation of G has trivial associated character. We conclude that dim(ρ_{3}) = 2 and consequently rdim(G) = dim(ρ) = 4 + 4 + 2 = 10.

Thus the maximal representation dimension of a group of order 2^{7} is 10.

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Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

scernele@math.ubc.ca, masoud@math.ubc.ca, reichst@math.ubc.ca