Commuting categories for blocks and fusion systems

Adam Glesser and Markus Linckelmann

Abstract. We extend the notion of a commuting poset for a finite group to p-blocks and fusion systems, and we generalize a result, due originally to Alperin and proved independently by Aschbacher and Segev, to commuting graphs of blocks, with a very short proof based on the G-equivariant version, due to Thévenaz and Webb, of a result of Quillen.

Mathematics Subject Classification (2010). 20C20, 20E15, 55P10.

Keywords. Commuting graph, p-blocks, fusion systems.

Let k be a field of prime characteristic p. A block of a finite group G is a primitive idempotent b in Z(kG). A b-Brauer pair is a pair (Q, e) consisting of a p-subgroup Q of G and a block e of C_{G}(Q) satisfying Br_{Q}(b)e ≠ 0, where Br_{Q} : (kG)^{Q} → kC_{G}(Q) is the Brauer homomorphism; the set of b-Brauer pairs is a G-poset with respect to the conjugation action of G (see [10] for more details and background material on block theory). We denote by 𝒜 (b) the G-poset containing all b-Brauer pairs (Q, e) such that Q is nontrivial and elementary abelian.

Two subgroups R, R′ of G are said to commute if they commute elementwise; that is, if [R, R′] = 1. For any nonempty set κ of pairwise commuting subgroups of G we denote by Πκ the product in G of all subgroups belonging to κ; this is clearly a subgroup of G. If all elements of κ are p-subgroups (respectively, abelian subgroups) of G, then Πκ is a p-subgroup (respectively, abelian subgroup) of G. For any abelian subgroup Q of G we denote by c(Q) the set of subgroups of order p of Q.

Definition 1. Let G be a finite group and b a block of G. The commuting poset of b is the G-poset 𝒦 (b) whose elements are pairs (κ, e), where κ is a nonempty set of pairwise commuting subgroups of order p of G and where e is a block of C_{G}(Πκ) such that (Πκ, e) is a b-Brauer pair, with partial order given by

The authors completed much of this work while enjoying the hospitality of MSRI in 2008.

(λ, f) ≤ (κ, e), if \begin{cases}λ ⊆ κ, and \\ (Πλ, f) ≤ (Πκ, e) \end{cases}

for (κ, e), (λ, f) ∈ 𝒦 (b).

If b is the principal block of G then 𝒦 (b) is the clique complex 𝒦_{p}(G) of the commuting graph Λ_{p}(G), where the notation is as in [3]. For nonprincipal blocks, however, 𝒦 (b) need not be the clique complex of a graph (e.g., see Example 5).

Given a G-poset 𝔛 we denote by Δ𝔛 the G-simplicial complex whose set of n-simplices consists of all chains of n proper inclusions in X, where n ≥ 0. For any simplicial complex 𝔜, we denote the geometric realization of 𝔜 by |𝔜|. Two G-spaces X and Y are called G-homotopically equivalent if there are G-equivariant maps f : X → Y, g : Y → X and G-equivariant homotopies h : I × X → X, h′ : I × Y → Y such that h(0, −) = Id_{A}, h(1, −) = f, h′(0, −) = Id_{Y}, and h′(1, −) = g, where the unit interval I = [0, 1] is viewed as a G-space with the trivial G-action. Two G-posets 𝔛 and 𝔜 are called Ghomotopically equivalent if the G-spaces |Δ𝔛| and |Δ𝔜| are G-homotopically equivalent. By the G-equivariant version [11, (1.1)] of [9, 1.3], in order to show that 𝔛 and 𝔜 are G-homotopically equivalent, it suffices to find Gequivariant functors Φ : 𝔛 → 𝔜 and Ψ : 𝔜 → 𝔛 such that there is a natural transformation between Id_{𝔛} and Ψ ∘ Φ (in either direction) and a natural transformation between Id_{𝔜} and Φ ∘ Ψ.

Theorem 2. Let b be a block of a finite group G. The maps:

Φ : \{ \begin{matrix}𝒜 (b) → 𝒦 (b) \\ (Q, e) ↦ (c(Q), e) \end{matrix}  and  Ψ : \{ \begin{matrix}𝒦 (b) → 𝒜 (b) \\ (κ, e) ↦ (Πκ, e) \end{matrix}

are inverse G-homotopy equivalences.

Proof. The maps Φ, Ψ are obviously order preserving and G-equivariant. We have Ψ ∘ Φ = Id_{𝒜 (b)} . There is a natural transformation Id_{𝒦 (b)} → Φ ∘ Ψ given by (κ, e) ≤ (c(Πκ), e), which shows that Ψ is a G-homotopy inverse of Φ. □

Applied to principal blocks, this theorem yields, in particular, a proof of the fact, due independently to Alperin [1, Theorem 3] and to Aschbacher and Segev [4, 9.7], that 𝒦_{p}(G) and 𝒜_{p}(G) have the same homotopy type (see also [3, 5.2]). The G-orbit space of 𝒦 (b) admits a generalization to fusion systems and, in fact, to arbitrary categories on finite p-groups (cf. [7, 2.1]).

Definition 3. Let F be a category on a finite p-group P. The commuting category of F is the category 𝒦 (F) whose objects are the nonempty sets of pairwise commuting subgroups of P of order p, and for objects κ, λ ∈ 𝒦 (F),

Hom_{𝒦 (F)} (κ, λ) = {ψ ∈ Hom_{F}(Πκ, Πλ) | if Q ∈ κ, then ψ(Q) ∈ λ.}

The composition of morphisms in 𝒦 (F) is induced by the usual composition of group homomorphisms. We denote by [𝒦 (F)] the poset consisting of the isomorphism classes [κ] of objects κ of 𝒦 (F) with partial order given by

[κ] ≤ [λ], if Hom_{𝒦 (F)} (κ, λ) ≠ ∅

for κ, λ ∈ 𝒦 (F).

Clearly 𝒦 (F) is an EI-category. As a consequence of results in [2], any choice of a maximal b-Brauer pair (P, e) of a block b of a finite group G determines a category F_{(P,e)}(G, b) on P that, if k is large enough, is a saturated fusion system (see e.g., [6, §3.3] for details and further references).

Theorem 4. Let b be a block of a finite group G, let (P, e_{P}) be a maximal b-Brauer pair and let F = F_{(P,e_{P})}(G, b). We have an isomorphism of posets

[𝒦 (F)] ≅ 𝒦 (b)/G

mapping the isomorphism class of an object κ ∈ 𝒦 (F) to the G-conjugacy class of the unique Brauer pair (Πκ, e) contained in (P, e_{P}).

Proof. For (κ, e) ∈ 𝒦 (b), let [(κ, e)] denote its G-conjugacy class. For elements (κ, e), (λ, f) ∈ 𝒦 (b), one has [(κ, e)] = [(λ, f)] if and only if there exists g ∈ G such that κ^{g} = λ and e^{g} = f. Define a poset map η : 𝒦 (b)/G → [𝒦 (F)] by setting η([(κ, e)]) = [κ^{g}], where g ∈ G such that (Πκ, e)^{g} ≤ (P, e_{P}). One verifies that this map is the inverse of the given map in the statement. □

Example 5. The following example was communicated to the authors by R. Kessar. Suppose p = 2. Set G = S_{n}, where n ≥ 6 is an integer such that kG has a block b with a dihedral defect group P ≅ D_{8} of order 8. By results in [8], b is of principal type; that is, for any 2-subgroup Q of G either Br_{Q}(b) = 0 or Br_{Q}(b) is a block of kC_{G}(Q). Moreover, P may be chosen as a Sylow 2-subgroup of S_{4}, canonically embedded into G and such that P contains the involutions x = (1 2), y = (3 4). Setting z = (5 6), we have x, z ∈ P^{(3 5)(4 6)} and y, z ∈ P^{(1 5)(2 6)}. Since b is of principal type, there are unique blocks e_{x}, e_{y}, e_{z} of kC_{G}(x), kC_{G}(y), kC_{G}(z), respectively, and unique blocks e_{xy}, e_{xz}, e_{yz} of kC_{G}(⟨x, y⟩), kC_{G}(⟨x, z⟩), kC_{G}(⟨y, z⟩), respectively, giving the following inclusions of b-Brauer pairs:

(⟨x, y⟩, e_{xy})  (⟨x, z⟩, e_{xz})  (⟨y, z⟩, e_{yz}) (⟨x⟩, e_{x})  (⟨y⟩, e_{y})  (⟨z⟩, e_{z}) (1, b)

4 Adam Glesser and Markus Linckelmann

Suppose that Γ is a graph whose clique complex is 𝒦 (b). The b-Brauer pairs (⟨x⟩, e_{x}), (⟨y⟩, e_{y}), and (⟨z⟩, e_{z}) are minimal in the poset 𝒦 (b) and are pairwise contained in a common b-Brauer pair, implying that the graph Γ has a clique of the form:

(⟨x⟩, e_{x}) (⟨z⟩, e_{z}) (⟨y⟩, e_{y})

However, the corresponding clique is not an element of the poset 𝒦 (b) because the group ⟨x, y, z⟩ is not contained in a defect group of b. This contradiction shows that there is no graph whose clique complex yields 𝒦 (b) and explains why we have refrained from defining a commuting graph of b in this way.

References

[1] J. L. Alperin, A Lie approach to finite groups, Lecture Notes in Math. 1456, Springer, Berlin (1990), 1–9.

[2] J. Alperin, M. Broué, Local methods in block theory, Ann. Math. 110 (1979), 143–157.

[3] M. Aschbacher, Simple connectivity of p-group complexes, Israel J. Math. 82 (1993), 1–43.

[4] M. Aschbacher, Y. Segev, The uniqueness of groups of Lyons type, J. Amer. Math. Soc. 5 (1992), 75–98.

[5] C. Broto, R. Levi, B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779–856.

[6] R. Kessar, Introduction to Block Theory, in: Group Representation Theory (edts. M. Geck, D. Testerman, J. Thévenaz), EPFL Press, Lausanne (2007), 47–77.

[7] M. Linckelmann, Introduction to Fusion systems, in: Group Representation Theory (edts. M. Geck, D. Testerman, J. Thévenaz), EPFL Press, Lausanne (2007), 79–113.

[8] L. Puig, The Nakayama conjecture and the Brauer pairs, Sminaire sur les groupes finis, Tome III, ii, 171189, Publ. Math. Univ. Paris VII, 25, Univ. Paris VII, Paris (1986).

[9] D. Quillen, Homotopy properties of the Poset of Nontrivial p-Subgroups of a Group, Advances Math. 28 (1978), 101–128.

[10] J. Thévenaz, G-Algebras and Modular Representation Theory, Oxford Science Publications, Clarendon Press, Oxford (1995).

[11] J. Thévenaz, P. J. Webb, Homotopy equivalence of posets with a group action, J. Combin. Theory Ser. A 56 (1991), no. 2, 173–181.

Commuting categories for blocks and fusion systems 5

Adam Glesser Suffolk University, 8 Ashburton Place, Boston, MA 02108, USA

e-mail: aglesser@suffolk.edu

Markus Linckelmann Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, UK

e-mail: m.linckelmann@abdn.ac.uk