BLOCKS OF MINIMAL DIMENSION

Markus Linckelmann

February 2006

Abstract. Any block with defect group P of a finite group G with Sylow-p-subgroup S has dimension at least |S|^{2}/|P |; we show that a block which attains this bound is nilpotent, answering a question of G. R. Robinson.

Theorem. Let p be a prime and let O be a complete local Noetherian commutative ring with algebraically closed residue field k of characteristic p. Let G be a finite group, let b be a block of OG with a defect group P and let S be a Sylow-p-subgroup of G. Then rk_{O}(OGb) ≥ |S|^{2}/|P |, and if rk_{O}(OGb) = |S|^{2}/|P | then b is a nilpotent block, the block algebra OGb is isomorphic to the matrix algebra M_{[S:P]}(OP) and the algebra OP is a source algebra of b.

Nilpotent blocks were introduced in [2]. The proof of the Theorem is based on Puig’s results in [6] on the bimodule structure of a source algebra of OGb as OP -OP - bimodule and the structure theory of nilpotent blocks in [7]. Examples of blocks of minimal O-rank include all blocks of p-nilpotent finite groups G with abelian O_{p′}(G) and, with P = 1, the block of the Steinberg module of a finite group of Lie type in defining characteristic.

We refer to Thévenaz [8] for background material on p-blocks of finite groups. In particular, with the notation of the Theorem, by a block of OG we mean a primitive idempotent b in Z(OG), and a defect group of b is a minimal subgroup P of G such that OGb is isomorphic to a direct summand of OGb ⊗_{OP} OGb as OGb-OGb-binodule. This

is equivalent to requiring that P is a maximal p-subgroup of G such that Br_{P} (b) ≠ 0, where

Br_{P} : (OG)^{P} ⟶ kC_{G}(P)

is the Brauer homomorphism sending a P -stable element ∑_{x∈G} λ_{x}x of the group algebra

OG to the element ∑_{x∈C_{G}(P)}λ̄_{x}x in the group algebra kC_{G}(P), where here λ̄_{x} is the

canonical image of the coefficient λ_{x} ∈ O in the residue field k. The map Br_{P} is

Typeset by AMS-TEX

well-known to be a surjective algebra homomorphism. In particular, Br_{P} (b) is an idempotent in Z(kC_{G}(P)), hence a sum of blocks of kC_{G}(P). The blocks occurring in Br_{P} (b) are all conjugate under N_{G}(P).

Proof of the Theorem. The statement on the minimal possible rank of OGb is wellknown, but we include a proof for the convenience of the reader. Choose notation such that P ⊆ S. Since OGb is a direct summand of OG as OS-OS-bimodule, there is an O-basis X of OGb which is stable under left and right multiplication with elements in S. For any subgroup R of S, the set of “diagonal” fixpoints

X^{R} = {x ∈ X | uxu^{−1} = x for all u ∈ R}

is mapped by Br_{R} to a k-basis in Br_{R}((OGb)^{R}) = kC_{G}(R)Br_{R}(b). Since P is maximal such that Br_{P} (b) ≠ 0, the set X^{P} is in particular non empty. Also, OGb has vertex ΔP and trivial source as O(G×G)-module, hence is a direct summand of Ind_{ΔP}^{G×G}(O), where ΔP = {(u, u) | u ∈ P }. Mackey’s formula implies that every indecomposable direct summand of OGb as OS-OS-bimodule is of the form Ind_{Q}^{S×S}(O) for some subgroup Q of S × S of the form S × S ∩^{(x,y)}ΔP with x, y ∈ G; in particular, Q has order at most |P |. In other words, the stabiliser of any element x ∈ X in S × S has at most order |P |.

Let x ∈ X^{P}. The stabiliser of x in S × S contains ΔP but has at most order |P |, hence is equal to ΔP. Thus the OS-OS-bimodule O[SxS] generated by x is a direct summand of OGb as OS-OS-bimodule isomorphic to OS ⊗_{OP} OS. In particular,

rk_{O}(OGb) ≥ rk_{O}(OS ⊗_{OP} OS) = |S|^{2}/|P |.

In order to show that b is nilpotent we use a result of Puig [6, 3.1] in the form as described in [4, 7.8]. Let i ∈ (OGb)^{P} be a primitive idempotent in the algebra of fixpoints in OGb with respect to the conjugation action by P on OGb such that Br_{P} (i) ≠ 0; that is, i is a source idempotent for b and the algebra iOGi is a source algebra of b. Since i commutes with the action of P, the source algebra iOGi is also a direct summand of OGb ≅ OS ⊗_{OP} OS as OP -OP -bimodule. As a consequence of results

in [1], the choice of the source idempotent i determines a fusion system F = F_{(P,e)}(G, b) on P, where e is the unique block of kC_{G}(P) such that Br_{P} (i)e = Br_{P} (i); this makes sense as Br_{P} (i) is a primitive idempotent in kC_{G}(P). See e.g. [3] or [5], for more details on fusion systems of blocks. Now let Q be a subgroup of P and let φ ∈ Aut_{F}(Q). Denote by _{φ}OQ the OQ-OQ-bimodule which is, as O-module, equal to OQ but with u ∈ Q acting on the left by multiplication with φ(u) and on the right by multiplication with u. By [4, 7.8], the OQ-OQ-bimodule _{φ}OQ is isomorphic to a direct summand of iOGi as OQ-OQ-bimodule. Thus_{φ}OQ is isomorphic to a direct summand of OS ⊗_{OP} OS

as OQ-OQ-bimodule. This forces φ to be induced by conjugation with an element in N_{S}(Q). In particular, φ is a p-automorphism of Q. Thus Aut_{F}(Q) is a p-group for all subgroups Q of P, and hence b is a nilpotent block.

By the general structure theory of nilpotent blocks [7], the block algebra OGb is isomorphic to a matrix algebra M_{n}(OP); in particular, the block b has a unique isomorphism class of simple modules. If V is a simple OGb-module then V has the defect group P as vertex and an endo-permutation kP -module W as source. This source is trivial if and only if the source algebra iOGi is isomorphic to OP. Dimension counting yields rk_{O}(OGb) = n^{2}|P | = |S|^{2}/|P |, hence dim_{k}(V) = n = [S : P]. Now V is a direct summand of Ind_{P}^{G}(W), hence by Mackey’s formula, Res_{S}^{G}(V) is a direct sum of direct summands of Ind_{S∩^{x}P}^{S} (^{x}W) with x ∈ G. Green’s indecomposability theorem [8, (23.6)] forces S ∩^{x}P = ^{x}P and dim_{k}(W) = 1, hence W is the trivial kP -module. □

Remark. If OGb has O-rank |S|^{2}/|P | then the first part in the proof of the Theorem says that OGb ≅ OS ⊗_{OP} OS as OS-OS-bimodules for any defect group P of b contained

in S. Thus, if x ∈ G such that ^{x}P ⊆ S then OS ⊗_{OP} OS ≅ OS ⊗_{O^{x}P} OS, which forces

xP = ^{u}P for some u ∈ S. In other words,

Hom_{G}(P, S) = Hom_{S}(P, S) ∘ Aut_{G}(P)

or equivalently, N_{G}(P, S) = SN_{G}(P), where N_{G}(P, S) = {x ∈ G | ^{x}P ⊆ S}.

References

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2. M. Broué, L. Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), 117–128.

3. R. Kessar, M. Linckelmann, G. R. Robinson, Local control in fusion systems of p-blocks of finite groups, J. Algebra 257 (2002), 393–413.

4. M. Linckelmann, On splendid derived and stable equivalences between blocks of finite groups, J. Algebra 242 (2001), 819–843.

5. M. Linckelmann, Simple fusion systems and the Solomon 2-local groups, preprint (2003).

6. L. Puig, Local fusion in block source algebras, J. Algebra 104 (1986), 358–369.

7. L. Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1986), 77–116.

8. J. Thévenaz, G−Algebras and Modular Representation Theory, Oxford Science Publications, Clarendon Press, Oxford, 1995.

Markus Linckelmann Department of Mathematical Sciences Meston Building Aberdeen, AB24 3UE

United Kingdom