ON THE FOCAL DEFECT GROUP OF A BLOCK, CHARACTERS OF HEIGHT ZERO, AND LOWER DEFECT

GROUP MULTIPLICITIES Geoffrey R. Robinson, Department of Mathematics, University of Aberdeen, Scotland April 2008

ABSTRACT: We discuss the focal subgroup of the defect group D of a pblock B, which we refer to as the focal defect group, and denote by D_{0}. We note that (the character group) of D/D_{0} acts ( in a defect (or height) preserving fashion) on irreducible characters in B, and prove that the action on irreducible characters of height zero is semi-regular. We also prove that all orbits under this action have length divisible by [Z(D) : D_{0} ∩ Z(D)]. As applications, we prove that all Cartan invariants for B are divisible by [Z(D) : D_{0} ∩ Z(D)], that if Out(D) is a p-group ( and D ≠ 1), then the number of irreducible characters of height zero in B is divisible by p and that if Z(D) ≰ D_{0}, then the block B is of Lefschetz type (see [5]).

NOTATION, TERMINOLOGY AND GENERAL BACKGROUND

Let G be a finite group, p be a prime, (K, R, F) be a p-modular system satisfying the “usual properties”, and let B be a block of RG with defect group D, of order p^{d}. Let (D, b_{D}) denote a fixed maximal B-subpair, and for each subgroup Q of D, let (Q, b_{Q}) denote the unique B-subpair contained in (D, b_{D}) with subgroup component Q. Let p^{a} be the order of a Sylow p-subgroup of G. Overlooking slight abuses for ease of notation, we let F denote the fusion system associated to the B-subpairs in G and N_{F}(Q) denote N_{G}(Q, b_{Q}). As usual, we let Irr(B) denote the set of (ordinary) irreducible characters in B, and Irr_{e}(B) denote the set of (ordinary) irreducible characters of defect e in B. We denote |Irr(B)| by k(B), and |Irr_{e}(B)| by k_{e}(B).

We define the subgroup D_{0} of D as follows:

D_{0} = ⟨[D, N_{F}(D)], [U, N_{F}(U)] : U ∈ E(B)⟩,

where E(B) denotes the set of (subgroup components of) the essential B-subpairs contained in (D, b_{D}). Henceforth, we will refer to D_{0} as the focal defect group of B. Then D_{0} ⊲ D, D/D_{0} is Abelian, and by the Alperin- Broué fusion theorem for blocks ( from [1]), we see that whenever λ is a linear character of D which contains D_{0} in its kernel and (x, b_{x}) and (y, b_{y}) are G-conjugate Brauer elements contained in (D, b_{D}), then λ(x) = λ(y). In fact, it follows from [8] that D_{0} = D ∩ G′_{F}, where G_{F} is the iterated amalgam realising the saturated fusion system F associated to B ( see [7]).

SECTION 1: CHARACTER GROUP ACTION ON Irr(B)

We will show that if χ is an irreducible character in B and λ is an irreducible character of D/D_{0}, then λ ∗ χ is also an irreducible character in B, where ( as in the Broué-Puig “star” construction from [2]), λ ∗ χ is defined as follows: set χ = ∑_{(x,b_{x})} Ind_{C_{G}(x)}^{G}(d_{χ,j}^{(x)}ϕ̃_{j}), where (x, b_{x}) runs through a full set of G-conjugacy classes of the Brauer elements contained in (D, b_{D}) and ϕ_{j} runs through the irreducible Brauer characters in b_{x}, and d_{χ,j}^{(x)} is the usual generalized decomposition number. Here, ϕ̃_{j}(xy) = ϕ_{j}(y) for p-regular y ∈ C_{G}(x) and ϕ̃_{j} vanishes on all elements of C_{G}(x) which do not have p-part x.

Then λ ∗ χ = ∑_{(x,b_{x})} Ind_{C_{G}(x)}^{G}(λ(x)d_{χ,j}^{(x)}ϕ̃_{j}), ( retaining the same notation). It is clear that λ ∗ χ is a linear combination of irreducible characters in B, by Brauer’s second main theorem. Clearly, λ ∗ χ is an irreducible character if and only if it is a generalized character, and λ ∗ χ(xy) = λ(x)χ(xy) whenever x ∈ D and y is a p-regular element of C_{G}(x).

By the Broué-Puig induction theorem (again from [2]), it suffices to prove that

⟨Res_{QC_{G}(Q)}^{G}(λ ∗ χ), μ ∗ Φ⟩ ∈ ℤ

whenever Q is a subgroup of D, and Φ is the character of a Projective indecomposable b_{Q}-module ( where μ ∗ Φ(tu) = μ(t)Φ(u) for t ∈ Q p-regular u ∈ C_{G}(Q) and μ ∗ Φ(tu) = 0 if the p-part of u lies outside Z(Q) ( but u ∈ C_{G}(Q) and t ∈ Q). But it is clear that

⟨Res_{QC_{G}(Q)}^{G}(λ ∗ χ), μ ∗ Φ⟩ = ⟨Res_{QC_{G}(Q)}^{G}(χ), Res_{Q}^{D}(\overline{λ})μ ∗ Φ⟩,

which is certainly an integer.

We next observe that if χ has height zero in B and λ is a non-trivial linear character of D/D_{0}, then λ ∗ χ ≠ χ. For choose an element x ∈ D\(ker λ). There is a p-regular element y ∈ G such that D ∈ Syl_{p}(C_{G}(y)) and

\frac{[G : C_{G})]χ(y)}{χ(1)} ∉ J(R).

Then χ(y) ∉ J(R) so that χ(xy) ∉ J(R). In particular χ(xy ≠ 0, so that λ ∗ χ(xy) ≠ χ(xy) as λ(x) ≠ 1. (This is an argument which really dates back to R. Brauer).

When |D| > 1, it has long been known that the number of irreducible characters of height zero in B at least 2, is divisible by p if p < 5, and is divisible by 4 if p = 2 and |D| > 2. In the case that D_{0} < D, this can be strengthened somewhat.

We proved above that (the character group of) D/D_{0} acts semi-regularly on Irr_{d}(B) (which is also the set of irreducible characters of height zero in B). If [D : D_{0}] = p and |D| > p, then we have p ∑_{i∈I} χ(1)_{p′}^{2} ≡ 0 (mod p^{2}), where I is a set of representatives for the orbits of D/D_{0} on the irreducible characters of height 0 in B. By a standard argument, this implies that in ℤ/pℤ, we may express 0 as a sum of |I| non-zero squares. If p < 5, this implies that |I| is divisible by p and if p ≡ 3 ( mod 4), it implies that |I| ≥ 3. Hence we have:

THEOREM 1: If D ≠ D_{0}, then the number of irreducible characters of height zero in B is a multiple of [D : D_{0}] so is, in particular, divisible by p. If p = 2 or 3 and we also have |D| > p (as well as the previous hypothesis), then k_{d}(B) ≡ 0 (mod p^{2}).

For irreducible characters of positive height, the action of D/D_{0} obtained via the ∗-construction need not be semi-regular. However, we do have the following result ( which proves, in particular, that the above action is semi-regular on all irreducible characters in B in the case that the defect group is Abelian, even though (at present, at least), we do not know that all irreducible characters in B have height 0.

THEOREM 2: For every non-negative integer e, k_{e}(B) is divisible by [Z(D) : D_{0} ∩ Z(D)].

PROOF: We know that (the character group of ) D/D_{0} acts on Irr_{e}(B) via χ → λ ∗ χ. We first claim that if the linear character λ of D (with D_{0} in its kernel) fixes an irreducible character χ in this action, then Z(D)D_{0} ≤ kerλ. For if z ∈ Z(D)\kerλ, and λ ∗ χ = χ, then χ vanishes identically on the p-section of z, contrary to a theorem of Brauer, as z is central in D. Hence the number of elements of D/D_{0} which fix any irreducible character at all in B in this action is at most [D : Z(D)D_{0}], so, in particular, the length of any orbit is at least

\frac{[D : D_{0}]}{[D : Z(D)D_{0}]} = [Z(D) : D_{0} ∩ Z(D)]

(so is divisible by this integer, as both are powers of p).

SECTION 2 : FURTHER CONSEQUENCES AND APPLICATIONS

The first application may have some independent interest, and is easily seen to be in accordance with the predictions of the Alperin-McKay conjecture (although in somewhat restrictive circumstances).

COROLLARY 3: Suppose that |D| > 1, and that N_{F}(D) = DC_{G}(D). Then the number of irreducible characters of height zero in B is divisible by p.

PROOF: If p ∈ {2, 3}, then (as Brauer knew), we have that

∑_{μ∈Irr(B)} \frac{μ(1)^{2}}{p^{2a−2d}}

is divisible by p^{d}, but not by p^{d+1}. Hence the number of irreducible characters of height zero in B is divisible by p in each case. Thus we may suppose that p ≥ 5.

Set K = K_{∞}(D) ( the characteristic subgroup defined by Glauberman in [4]). From the recent article (Diaz,Glesser, Mazza, Park, [3]), we have [U, N_{F}(U)] ≤ D ∩ N_{F}(K)′ for each subgroup component U of a centric B-subpair contained in (D, b_{D}).

Now we claim ( still in the case p ≥ 5) that if N_{F}(D) = DC_{G}(D), then D_{0} < D. Using [3] again, it suffices to deal with the case that G = N_{F}(K). If K = D, then D_{0} ≤ [D, G] ≤ D_{0}, so that D_{0} = D′, and the claim clearly holds. Otherwise, we may work with the Külshammer-Puig extension L(K), which is an extension of of G/KC_{G}(K) by K. The group L(K) has Sylow p-subgroup D and has the same fusion on p-subgroups as G does on B-subpairs ( see [6]). From (Glauberman, [4]), L(K) has a normal subgroup T of index p. Let D_{1} be a Sylow p-subgroup of T. Then (using the focal subgroup theorem ( for groups) in L(K), and translating the fusion information back to G) means that

D_{0} = ⟨[D, N_{F}(D)], [U, N_{F}(U)] : U ∈ E(B)⟩ ≤ D_{1} < D,

as required.

REMARKS: It follows from these arguments that if D_{0} < D and |D| ≥ p^{2}, then k_{0}(B) ≥ 2p. For we obtain

∑_{χ∈Irr_{0}(B)} χ(1)_{p′}^{2} ≡ 0

( mod p^{2}). If there were only p irreducible characters of height zero, then they would all have the same degree, say p^{a−d}t. Then we would have pt^{2} ≡ 0 ( mod p^{2}), a contradiction. Notice that if p is odd and Out(D) is a p-group ( but D ≠ 1), then must have |D| ≥ p^{3}. Hence:

COROLLARY 4: If D ≠ 1 and Out(D) is a p-group, then k_{0}(B) ≡ 0 ( mod p). If also p > 3, and |D| > p, then k_{0}(B) ≥ 2p, and if p ≡ 3 ( mod 4) ( still p > 3), then we have k_{0}(B) ≥ 3p.

In general, it need not be the case that the Cartan invariants of the block B with defect group D are divisible by [D : D_{0}]. For example, when p = 2 and B is the principal 2-block of S_{4}, we have D_{0} < D, but there are odd Cartan invariants, since there is a unique 2-block, but 1 occurs as an elementary divisor of the Cartan matrix, as there is a 2-regular class of defect 0.

In general, it seems to be relatively difficult to provide p-locally determined conditions which guarantee that 1 does not occur as an elementary divisor for the defect group of the block B, so we consider the following result in that direction worth remarking on. We also take the opportunity to include an application related to Alperin’s conjecture. We recall that in [5], the block B was said to be of Lefschetz type if the alternating sum occurring in the reformulation of Alperin’s conjecture vanishes ( see [5]) for a more precise statement).

COROLLARY 5 : i) All Cartan invariants for B are divisible by [Z(D) : D_{0} ∩ Z(D)]. In particular, if Z(D) ≰ D_{0}, then all Cartan invariants of B are divisible by p.

ii) If Z(D) ≰ D_{0}, then B is a block of Lefschetz type.

PROOF: i) If θ and ψ are characters of projective indecomposable B-modules, then all irreducible constituents of θ in a given orbit under the above action occur with equal multiplicity in θ, and likewise for ψ. Since each such orbit has length divisible by [Z(D) : D_{0} ∩ Z(D)], we see that ⟨θ, ψ⟩ is divisible by [Z(D) : D_{0} ∩ Z(D)].

ii) Using i), we see that whenever σ is a chain of B-subpairs as in Knörr-Robinson

[5] ( and arguing as in Proposition 5.1 of [5]), we have (in the notation of [5]) ℓ_{0}(B_{σ}) = 0, so the result follows by Corollary 4.3 of [5].

REFERENCES

[1] J. L. Alperin & M. Broué, Local methods in block theory, Annals of Mathematics,2, (1979),1,143-157.

[2] M. Broué & L. Puig,A Frobenius Theorem for blocks, Inventiones Mathematicae,56,(1980),2,117-128.

[3] A. Diaz, A.Glesser,N. Mazza, S. Park, Control of Transfer and Weak Closure in Fusion Systems, in preparation, 2008.

[4] G. Glauberman, “Global and local properties of finite groups”, in, Finite Simple Groups, eds Powell and Higman, Academic Press, London, 1971, 1-64.

[5] R. Knörr and G.R. Robinson, Some Remarks on a Conjecture of Alperin, Journal of London Mathematical Society, (2), 39,(1989),1,48-60.

[6] B. Külshammer & L. Puig, Extensions of nilpotent blocks. Inventiones Mathematicae,102,(1990),1,17-71.

[7] G.R. Robinson, Amalgams, blocks,weights, fusion systems and finite simple groups, Journal of Algebra, 314, 2 (2007),912-923.

[8] G.R. Robinson, Reduction ( mod q) of fusion system amalgams, in preparation, April 2008