On Generalized Characters of Nilpotent Groups Geoffrey R. Robinson, Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, Scotland, United Kingdom g.r.robinson@abdn.ac.uk *Dedicated to the memory of my cat William

INTRODUCTION: In an earlier note [2], we proved that if p is a prime, P is a finite p-group, and θ is a generalized character of P which does not vanish identically on P^{#}, then ∑_{x∈P^{#}} |θ(x)|^{2} ≥ |P | − μ(1)^{2}, where μ is an irreducible character of maximal degree of P. Clearly, taking θ = μ shows that this result is best possible for all finite p-groups P. Here, we complete the proof of the result which eluded us in [2]. As in [2], we give applications to estimating the number of irreducible characters of a finite group G which do not vanish identically on non-identity elements of N, and we give a related estimate for the number of characters in a p-block.

THEOREM 1: Let N be a non-trivial finite nilpotent group, let θ be a generalized character of N which does not vanish identically on N^{#} and let χ be an irreducible character of maximal degree of N. Then

∑_{n∈N^{#}} |θ(n)|^{2} ≥ |N| − χ(1)^{2}.

REMARKS: Notice that |N| − χ(1)^{2} ≥ \frac{|N|}{2}, since χ(1)^{2} ≤ [N : Z(N)].

In the direction opposite to the theorem, let p be any prime and let n be any positive integer. Let G be a Frobenius group of order p^{n}(p^{n}−1) with elementary Abelian kernel of order p^{n} and complement of order (p^{n} − 1). Let ψ be the class function of G with ψ(1) = 1 − p^{n}, ψ(g) = 1 if g has order p and ψ(g) = 0 otherwise. Then ψ is a generalized character of G (for example, by Brauer’s characterization of characters), and we have ∑_{g∈G^{#}} |ψ(g)|^{2} = p^{n} − 1 < |G|^{\frac{1}{2}} .

It is tempting to ask whether, for a general finite group G, and a generalized character θ of G which does not vanish identically on G^{#}, we have ∑_{g∈G^{#}} |θ(g)|^{2} ≥ \frac{|G|}{χ(1)} − 1, where χ is an irreducible character of G of maximal degree ( a bound which is attained by the above Frobenius groups). If so, then we would always have ∑_{g∈G^{#}} |θ(g)|^{2} > |G|^{\frac{1}{2}} − 1 for such a θ.

NOTATION: If a finite group X acts by conjugation on another finite group Y (possibly containing X), we let k_{X}(Y) denote the number of X-conjugacy classes of Y. If A is a finite group and B is a subgroup of A, then we let k_{+}(A, B) denote the number of irreducible characters of A which do not vanish identically on B^{#}.

When S and T are finite groups, and α, β are complex characters of S, T respectively, we let α⊗β denote the character of S×T with α⊗β(s, t) = α(s)β(t). We let ρ_{S} denote the regular character of S, which takes values |S| on 1_{S} and 0 elsewhere.

COROLLARY 2: Let N be a nilpotent subgroup of the finite group G and let χ be an irreducible character of maximal degree of N and let C = ∩_{g∈G}N^{g}. Then: i)

k_{+}(G, N) ≤ \frac{|N|}{|N| − χ(1)^{2}} (k_{N}(G) − [G : N])

≤ \frac{|N|(k(N) − 1)}{|N| − χ(1)^{2}}max_{x∈N^{#}}[C_{G}(x) : C_{N}(x)].

Furthermore,

k_{+}(G, N) ≤ \frac{|N| − 1}{|N| − χ(1)^{2}} max_{x∈N^{#}}|C_{G}(x)|.

ii) If C ≠ 1_{G}, then k(G) ≤ |C_{G}(z)| for some z ∈ Z(C)^{#} and

k(G) ≤ \frac{|N|}{|N| − χ(1)^{2}} (k_{N}(G) − [G : N])

≤ \frac{|N|(k(N) − 1)}{|N| − χ(1)^{2}}max_{x∈N^{#}}[C_{G}(x) : C_{N}(x)].

REMARK: We notice that [C_{G}(x) : C_{N}(x)] ≤ [G : N] for any x ∈ N^{#} and if equality holds for any such x, then we have G = NC_{G}(x) and

Z(N) ≤ core_{G}(C_{G}(x)).

We also prove the following result which combines the results and methods of [1] and [2] with those used to prove Corollary 2:

THEOREM 3: Let G be a finite group, p be a prime, and let B be a p-block of G with defect group D > 1. Let μ be an irreducible character of maximal degree of D. Then we have

k(B) ≤ \frac{|D|(k(D) − 1)}{(|D| − μ(1)^{2})[C_{G}(D) : Z(D)]}max_{x∈D^{#}}[C_{G}(x) : C_{D}(x)].

PROOF OF THEOREM 1: By [2], we may suppose that |N| has more than one prime divisor, and we do so. Hence we may write N = A × B, where A and B are non-trivial normal subgroups of N of coprime order.

Let θ be a generalized character of N which minimizes

∑_{n∈N^{#}} |θ(n)|^{2}.

If θ agrees with ±μ on N^{#} for an irreducible character μ of N, we are done, so we may suppose that this is not the case. By subtracting integer multiples of the regular character of N from θ, then replacing θ by −θ if necessary (neither of which change ∑_{n∈N^{#}} |θ(n)|^{2}) we may suppose that 0 < θ(1) ≤ \frac{|N|}{2}. ( We note that θ(1) ≠ 0, otherwise ∑_{n∈N^{#}} |θ(n)|^{2} ≥ |N|).

From part of the analysis in [2], we know that for each irreducible character ϕ of N, we have

|⟨θ, ϕ⟩ − \frac{θ(1)ϕ(1)}{|N|}| < \frac{1}{2}

(this particular observation just relies on the fact that for ε = ±1, we have

∑_{n∈N^{#}} |(θ − εϕ)(n)|^{2} ≥ ∑_{n∈N^{#}} |θ(n)|^{2}

by the choice of θ, and does not depend on the fact that N is nilpotent).

Since 0 < θ(1) ≤ \frac{|N|}{2}, this leads easily to

−\frac{1}{2} < ⟨θ, ϕ⟩ < \frac{1 + ϕ(1)}{2},

so certainly

0 ≤ ⟨θ, ϕ⟩ < ϕ(1).

For each irreducible character μ of B, let e_{μ} = \frac{μ(1)}{|B|} ∑_{b∈B} μ(b^{−1})b, a central idempotent in the group algebra ℂB. Let e^{μ} be a primitive (not necessarily central) idempotent of ℂB with e_{μ}e^{μ} ≠ 0.

Let b be an element of B and let C_{b} denote its class sum in the group algebra ℂB, so that C_{b}e_{μ} = \frac{[B:C_{B}(b)]μ(b)}{μ(1)}e_{μ}. We note then that for any a ∈ A^{#}, we have

θ(ab) = ∑_{μ∈Irr(B)} θ(e^{μ}a)μ(b),

since e_{μ} is a sum of μ(1) (ℂB)^{×}-conjugates of e^{μ}.

Hence we have

∑_{b∈B} |θ(ab)|^{2} = |B| ∑_{μ∈Irr(B)} |θ(e^{μ}a)|^{2}.

There are two cases to consider :

i) For each a ∈ A^{#}, we have θ(ab) = 0 for all b ∈ B. In that case, since (for any b ∈ B), a → θ(ab) is an algebraic integer combination of characters of A), we see that \frac{θ(b)}{|A|} is an algebraic integer for each b ∈ B. Hence Res_{B}^{N}(θ) = |A|ψ for some generalized character ψ of B. Furthermore, ψ does not vanish on all non-identity elements of B by the hypotheses on θ. Hence (by induction)

∑_{b∈B^{#}} |ψ(b)|^{2} ≥ \frac{|B|}{2},

so that

∑_{n∈N^{#}} |θ(n)|^{2} ≥ \frac{|A||N|}{2} ≥ |N|,

contradicting the choice of θ.

ii) For some a ∈ A^{#} and some b ∈ B, we have θ(ab) ≠ 0. Then for some μ ∈ Irr(B), we have θ(e^{μ}a) ≠ 0 for some a ∈ A^{#}. Now (for any such μ) a → θ(e^{μ}a) is a generalized character of A which does not vanish identically on A^{#}, so that ( by the main result of [2]), we have

∑_{a∈A^{#}} |θ(ae^{μ})|^{2} ≥ |A| − ϕ(1)^{2},

where ϕ(1) is an irreducible character of A of maximal degree. In this case, μ is unique, for otherwise we have

∑_{a∈A^{#}} ∑_{b∈B} |θ(ab)|^{2} ≥ 2|B|(|A| − ϕ(1)^{2}) ≥ |N|,

contrary to the choice of θ.

This means that there is a character β of B and a character α of A which does not vanish identically on non-identity elements, such that ⟨β, μ⟩ = 0 and

θ = (α ⊗ μ) + (ρ_{A} ⊗ β).

By a similar argument, there is a unique irreducible character λ of A such that

θ = (λ ⊗ γ) + (δ ⊗ ρ_{B}),

where γ is a character of B which does not vanish identically on non-identity elements and δ is a character of A with ⟨λ, δ⟩ = 0.

Now let η ≠ λ be an irreducible character of A and ζ ≠ μ be an irreducible character of B. Then we have

⟨θ, η ⊗ ζ⟩ = η(1)⟨β, ζ⟩ = ⟨δ, η⟩ζ(1),

which is divisible by η(1)ζ(1). But each irreducible character τ of N occurs with non-negative multiplicity less than τ(1) in θ, so we must have ⟨θ, η ⊗ ζ⟩ = 0.

This yields that ⟨β, ζ⟩ = 0 for each such ζ, and we already have ⟨β, μ⟩ = 0, so we have β = 0. Likewise, we obtain δ = 0.

Now we have

θ = α ⊗ μ = λ ⊗ γ,

so that θ is a positive integer multiple of λ⊗μ by the uniqueness of its expression as a linear combination of irreducible characters of N. The choice of θ now forces θ = λ ⊗ μ, contrary to our assumption that θ does not identically agree with an irreducible character on non-identity elements of N.

PROOF OF COROLLARY 2: From Theorem 1 and the orthogonality relations for group characters, we have

∑_{x∈N^{#}} |C_{G}(x)| ≥ k_{+}(G, N)(|N| − χ(1)^{2}).

Now by Burnside’s counting theorem,

k_{N}(G) = [G : N] + \frac{1}{|N|} ∑_{x∈N^{#}} |C_{G}(x)|.

Also,

∑_{x∈N^{#}} |C_{N}(x)| = |N|(k(N) − 1),

so part i) follows. ii) Suppose that C ≠ 1_{G}. Then by Clifford’s theorem, no irreducible character of G vanishes identically on Z(C)^{#}, so k(G) = k_{+}(G, N) in this case. For the first inequality of part ii), we apply the last statement of part i) with Z(C) in the role of N.

PROOF OF THEOREM 3: From [1], we know that ( for an appropriate complete discrete valuation ring R of characteristic 0 with R/J(R) of characteristic p) there is a primitive idempotent e of RC_{G}(D) which lies in a root,b say, of B such that

k(B)(|D| − μ(1)^{2}) ≤ ∑_{x∈D^{#}} rank_{R}(eB(x)e),

where B(x) is the sum of the Brauer correspondent blocks of B for RC_{G}(x). Then certainly we have ( for each such x),

rank_{R}(eB(x)e) ≤ rank_{R}(eRC_{G}(x)e)
= rank_{R}(Hom_{RC_{G}(x)}(eRC_{G}(x), eRC_{G}(x))).

Now, as right RC_{G}(x)-module, we have

eRC_{G}(x) ≅ Ind_{C_{G}(D)}^{C_{G}(x)}(eRC_{G}(D)).

From the main result of [1], we have

rank_{R}(eB(x)e) ≤ [C_{G}(x) : C_{G}(D)]rank_{R}(Hom_{RC_{G}(D)}(eRC_{G}(D), eRC_{G}(D))).

But the block b of RC_{G}(D) has defect group Z(D), so we have

rank_{R}(Hom_{RC_{G}(D)}(eRC_{G}(D), eRC_{G}(D))) = |Z(D)|.

Hence we have

k(B)(|D| − μ(1)^{2}) ≤ \frac{1}{[C_{G}(D) : Z(D)]} ∑_{x∈D^{#}} |C_{G}(x)|.

Since

∑_{x∈D^{#}} |C_{D}(x)| = |D|(k(D) − 1),

the result follows.

REMARK: We note, for example, that if C_{G}(x) ≤ DC_{G}(D) for all x ∈ D^{#}, then the above bound simplifies to

k(B) ≤ \frac{|D|(k(D) − 1)}{|D| − μ(1)^{2}}.

References

[1] G.R. Robinson A remark on Mackey’s formula and Cartan invariants, Journal of Algebra,274, 267-270,(2004).

[2] G.R.Robinson, More on bounds on norms of generalized characters with applications to p-local bounds and blocks, Bull. London Math. Soc.,37,(4), 555-565,(2005).