Title: Generalisations of conjectures of Brauer and Olsson

Author: Charles W. Eaton

Address: School of Mathematics and Statistics University of Birmingham Edgbaston Birmingham B15 2TT England email: eatonc@for.mat.bham.ac.uk

Running title: Conjectures of Brauer and Olsson

Generalisations of conjectures of Brauer and Olsson

Charles W. Eaton

May 24, 2002

Abstract We investigate suitable generalisations of the conjectures of Brauer and Olsson bounding numbers of irreducible characters of certain degrees in a block by invariants of the defect group.

1 Introduction

One of the most well-known of the many conjectures of modular representation theory of finite groups is Brauer’s prediction that, for any p-block B with defect group D, we should have k(B) ≤ |D| = p^{d}, where k(B) is the number of irreducible characters ‘belonging’ to B. Later, Olsson conjectured in [10] that we should have k_{d}(B) ≤ |D/D′|, where k_{x}(B) is the number of irreducible characters of defect x (i.e., χ with p^{x}χ(1)_{p} = |G|_{p}). It should be noted that characters with defect d are said to have height zero. At the time Olsson’s conjecture was made the height of a character was the principal object of study, the defect being defined latterly by Dade, and conjectures such as those of Dade ([4]) have drawn our attention to the importance of the defect of a character. It is in this light that we revisit the conjectures of Brauer and Olsson, and predict inequalities involving characters of any given defect of which both of the previous conjectures are special cases.

Based on the case where the defect group of a block is normal, we present a conjecture which naturally includes those of Brauer and Olsson. In practice it is often no stronger than the statement “both Brauer’s and Olsson’s conjectures hold,” the main point being to indicate the general form such a generalisation should take. Then based on evidence from blocks with trivial intersection defect groups, we present a stronger conjecture which really would give us extra information about the number of characters of different heights.

Throughout let B be a block of a finite group G, with defect group D of order p^{d} (we define blocks with respect to a suitable complete local discrete valuation ring R with field of fractions of characteristic zero and residue field algebraically closed of characteristic p). Note that k_{x}(D) is the number of irreducible characters of D of defect x, i.e., degree p^{d−x}, and that k_{x}(B) = k_{x}(D) = 0 when x > d or x < 0.

We give the two conjectures next, first the strong, then the weak:

Conjecture 1.1 For any (non-negative) integer a we have

∑_{x≥a} k_{x}(B) ≤ ∑_{x≥a} p^{2(d−x)}k_{x}(D).

Conjecture 1.2 For any (non-negative) integer a we have

∑_{x≥a} k_{x}(B) ≤ |D/D^{(d−a+1)}|.

Here H^{(i)} denotes the ith term of the derived series of H. For either of the above, Brauer’s conjecture is given by the case a = 0 and Olsson’s conjecture corresponds to a = d.

For convenience we often write k_{≥a}(B) = ∑_{x≥a} k_{x}(B).

We also ask the question, based on Olsson’s observation that k(D) = |D| apparently only when D is abelian, of whether Conjecture 1.1 may be rephrased using some number other than 2, i.e., whether for some fixed l < 2 we have k_{≥a}(B) ≤ ∑_{x≥a} p^{l(d−x)}k_{x}(D) for every B. From the case G = SL_{2}(3) we see that any such l must be at least log_{2}(3), and in all the examples studied here the inequality is indeed satisfied for l = log_{2}(3).

2 Blocks with normal defect groups

In this section we give an elementary generalisation of part of [8], to show that the weaker conjecture for the case where D ◁ G is equivalent to the k(GV) problem. The point of this is to justify the use of the sum ∑_{x≥a} k_{x}(B) in the conjectures (it is clear from numerous examples that it does not make sense to consider each k_{x}(B) separately).

The following well-known result will be useful for this and for later on, so we state it for the convenience of the reader. A proof may be found at [7, 5.12].

Lemma 2.1 Suppose that H is a monomial group. Let {1 = f_{1} < ⋯ < f_{r}} be the set of irreducible character degrees of H. If χ ∈ Irr(H) has χ(1) = f_{i}, then H^{(i)} ≤ ker(χ).

Proposition 2.2 Conjecture 1.2 holds for each block with normal defect group of every finite group if and only if the k(GV) problem holds.

Proof. The ‘only if’ part is trivial. Suppose that the k(GV) problem holds. Let G be a finite group and B ∈ Blk(G) with defect group D ◁ G.

By [12] there is a finite group H with D ∈ Syl_{p}(H) and B_{H} ∈ Blk(H) with normal defect group D such that k_{x}(B) = k_{x}(B_{H}) for each x. Hence we may assume that D ∈ Syl_{p}(G). Note that in this case G is p-solvable.

We have

k_{≥a}(B) = ∑_{μ∈Irr(D)/G} k_{≥a}(B, μ) = ∑_{μ∈Irr_{≥a}(D)/G} k(B, μ).

Let {1 = f_{1} < ⋯ < f_{r}} be the set of irreducible character degrees of D, a monomial group. If μ ∈ Irr_{x}(D), then μ(1) = f_{i} where i ≤ d − x + 1. So by Lemma 2.1 D^{(d−x+1)} ≤ ker(μ). Hence

k_{≥a}(B) ≤ ∑_{μ∈Irr(D,1_{D^{(d−a+1)}})/G} k(B, μ) = k(B, 1_{D^{(d−a+1)}}).

By [8, Corollary 4] the natural epimorphism RG → R(G/D^{(d−a+1)}) gives a 1-1 correspondence between blocks of G and blocks of G/D^{(d−a+1)}, and this correspondence respects inflation of characters. Hence there is an unique block \overline{B} of G/D^{(d−a+1)} such that k(B, 1_{D^{(d−a+1)}}) = k(\overline{B}). Note that \overline{B} has normal defect group D/D^{(d−a+1)}.

Nagao in [9] proved that the Brauer k(B)-conjecture for p-solvable groups is equivalent to the k(GV)-problem. Hence since we are assuming that the k(GV)-problem holds, then k(\overline{B}) ≤ D/D^{(d−x+1)} and the result follows. □

It also follows very quickly from Lemma 2.1 that Conjecture 1.1 is indeed stronger than Conjecture 1.2:

Proposition 2.3 If Conjecture 1.1 holds for a block B, then Conjecture 1.2 also holds for B.

Proof. As before, if μ ∈ Irr_{x}(D), then μ(1) is at most the (d − x + 1)th character degree of D, so by Lemma 2.1 D^{(d−x+1)} ≤ ker(μ), and ∑_{x≥a} k_{x}(D)p^{2(d−x)} is at most

∑_{x≥a} k_{x}(D, 1_{D^{(d−a+1)}})p^{2(d−x)} = ∑_{x≥a′} k_{x}(D/D^{(d−a+1)})p^{2(d′−x)},

where y′ := y − log_{p}(|D^{(d−a+1)}|). This is at most |D/D^{(d−a+1)}|, so we are done. □

3 Blocks with trivial intersection defect groups

We say that a block B ∈ Blk(G) is a TI defect block if it has a defect group D which is trivial intersection (TI), i.e., D^{g} ∩ D = 1 whenever g ∈ G − N_{G}(D). We assume here that D is not normal in G. In [1] it is proved that B is Morita equivalent to one (or more) of: (a) a block with cyclic, generalised quaternion or Klein-four defect groups;

(b) a 2-block of maximal defect of some X with ^{2}B_{2}(2^{2m+1}) ≤ X ≤ Aut( ^{2}B_{2}(2^{2m+1})), m ≥ 1, and 2 |̸[X :^{2}B_{2}(2^{2m+1})]; (c) a block with elementary abelian defect groups of order 9 in 2.A_{6}, 2.S_{6}, O′N, Aut(O′N), 2.Suz, Aut(Suz), M_{11} or a group X with 4.P SL_{3}(4) ≤ X ≤ Aut(4.P SL_{3}(4)); (d) the principal 3-block of ^{2}G_{2}(3); (e) a 3-block of maximal defect of some X with ^{2}G_{2}(3^{2m+1}) ≤ X ≤ Aut( ^{2}G_{2}(3^{2m+1})), m ≥ 1, and 3 |̸[X :^{2}G_{2}(3^{2m+1})]; (f) a 5-block of maximal defect of 3.McL, Aut(McL), ^{2}F_{4}(2)′, 2F_{4}(2) or Aut( ^{2}B_{2}(2^{5})); (g) the principal 11-block of J_{4}; (h) a p-block of maximal defect of a group X with N ≤ X ≤ Aut(N), where p |̸[X : N] and N is a perfect p′-central extension of P SL_{2}(p^{m}) or P SU_{3}(p^{m}) with m > 1. When D ∈ Syl_{p}(G), the conjectures of Brauer and Olsson are verified in [2], and further the inequality of Theorem 3.1 holds when in addition D is abelian (since in [2] it is also proved that Brauer’s height zero conjecture holds when D ∈ Syl_{p}(G) is TI).

Table 1: Irreducible character degrees for tame blocks B with non-abelian defect group D, where |D| = p^{d} with d ≥ 3

\begin{matrix}k_{d}(B) & 4 & k_{d}(D) & 4 \\ k_{≥d−1}(B) & 2^{d−2} + 3 & ∑_{x≥d−1} p^{2(d−x)}k_{x}(D) & 2^{d} \\ k_{≥d−2}(B) & ≤ 2^{d−2} + 5 & ∑_{x≥d−2} p^{2(d−x)}k_{x}(D) & 2^{d} \end{matrix}

Theorem 3.1 Let D be a TI defect block with defect group D of order p^{d}. Then k_{≥a}(B) ≤ ∑_{x≥a} p^{l(d−x)} for each a, where l = log_{2}(3).

Proof. Recall that if B_{1} and B_{2} are Morita equivalent blocks, then they have isomorphic defect groups and k_{x}(B_{1}) = k_{x}(B_{2}) for each x. Hence in verifying Conjecture 1.1 for all TI defect blocks B it suffices to verify it for the blocks listed above. Throughout we do not list k_{x}(B) and k_{x}(D) when they are zero.

Case (a) follows by the cyclic defect theory and Table 1 (which is taken from [11]). Consider case (b). Let N ◁ G, where N ≅ ^{2}B_{2}(q), with q = 2^{2m+1} for some m ≥ 1,

and G/N with order an odd divisor of 2m + 1. The principal block B_{0} is the unique 2block of positive defect (d = 4m+2) of G. Let b_{0} be the principal block of N. From [6] k_{4m+2}(b_{0}) = q and k_{3m+2}(b_{0}) = 2. Also k_{4m+2}(D) = q and k_{3m+2}(D) = 2(q − 1). Then k_{4m+2}(B_{0}) ≤ (2m + 1)q and k_{3m+2}(B_{0}) ≤ 2(2m + 1) by Clifford theory (using the fact that the quotient group is cyclic, so that every irreducible character extends to its stabiliser). The case a = d holds by [2]. It thus suffices to consider a = 3m + 2. We have k_{≥3m+2}(B_{0}) ≤ (2m + 1)(q + 2) ≤ q + 2^{lm}(2(2m + 1)) = ∑_{x≥a} 2^{l(d−x)}k_{x}(D) as required.

Cases (c), (d), (f) and (g) are easily checked in [3].

Case (e) is similar to (b). Let N ◁ G, where N ≅ ^{2}G_{2}(q), with q = 3^{2m+1} for some m ≥ 1, and G/N with order a divisor of 2m + 1 prime to 3. The principal block B_{0} is the unique 3-block of positive defect (d = 6m + 3) of G. Let b_{0} be the principal block of N. From [5] k_{6m+3}(b_{0}) = q, k_{5m+3}(b_{0}) = 6, k_{4m+2}(b_{0}) = 1, k_{6m+3}(D) = q, k_{5m+3}(D) = 3(q − 1), k_{4m+2}(D) = q − 1. The case a = d holds by [2], and by a similar argument to that for (b) k_{≥5m+3}(B_{0}) ≤ (q + 6)(2m + 1) ≤ q + 3^{lm}(3(q − 1)) = ∑_{x≥5m+3} 2^{l(d−x)}k_{x}(D), and k_{≥4m+2}(B_{0}) ≤ (q + 6)(2m + 1) + 1 ≤ q + 3^{lm}(3(q − 1)) + q^{l}(q − 1) = ∑_{x≥4m+2} 2^{l(d−x)}k_{x}(D), as required.

It remains to check case (h). The case where N covers P SL_{2}(p^{m}) follows from [2]. Let N ◁ G, where N is a perfect p′-central extension of P SU_{n}(q) with q = p^{m} for some m > 1, and G/N has order not divisible by p. Note that G/N is either cyclic of order dividing m or a direct product of a cyclic group L_{1} of order dividing m with a group L_{2} of order 3. Hence G/N has Schur multiplier dividing 3. Let |L_{1}|_{3} = 3^{t}. Note that a central extension L of L_{1} × L_{2} with Z(L) ≤ L′ has |Z(L)| = 3 and k(L) = 11|L_{1}|_{3′}3^{t−1} ≤ 11m3^{t−1} ≤ 4m. By [6] we have k_{3m}(D) = q^{2} and k_{2m}(D) = q−1. We have (see [2]) k_{3m}(N) ≤ q + 9 and k_{2m}(N) ≤ q + 1. The case a = d holds by [2]. Also, by arguments similar to those for (b) and the above discussion concerning L (taking into account all of the forms the inertia subgroup of an irreducible character of N may take) k_{2m}(G) ≤ 4m(q + 1) ≤ q^{3/2}(q − 1) = p^{3/2}k_{2m}(D) for p ≥ 3 with q ≠ 3, and for p = 2 with m ≥ 3. In the case q = 4 we have G = N, which is covered by [2]. The result follows. □

ACKNOWLEDGEMENTS

I thank Burkhard Külshammer and Geoffrey Robinson for some helpful conversations.

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