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

Linckelmann, M.

English

City Research Online

Linckelmann, M. (2007). Blocks of minimal dimension. Archiv der Mathematik, 89(4), 311 - 314.City Research OnlineOriginal citation: Linckelmann, M. (2007). Blocks of minimal dimension. Archiv der Mathematik, 89(4), 311 - 314.Permanent City Research Online URL: http://openaccess.city.ac.uk/1890/ Copyright & reuseCity  University  London has developed City  Research Online  so that  its  users  may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and/ or other copyright holders.  All material in City Research Online is checked for eligibility for copyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web pages. Versions of researchThe version in City Research Online may differ from the final published version. Users are advised to check the Permanent City Research Online URL above for the status of the paper.EnquiriesIf you have any enquiries about any aspect of City Research Online, or if you wish to make contact  with the author(s) of this paper, please email the team at publications@city.ac.uk.BLOCKS OF MINIMAL DIMENSIONMarkus LinckelmannAbstract. Any block with defect group P of a finite group G with Sylow-p-subgroupS has dimension at least |S|2/|P |; we show that a block which attains this bound isnilpotent, answering a question of G. R. Robinson.Mathematics Subject Classification 2000: 20C20Theorem. Let p be a prime and let O be a complete local Noetherian commutativering 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 rkO(OGb) ≥ |S|2/|P |, and if rkO(OGb) = |S|2/|P | then b is a nilpotent block, theblock algebra OGb is isomorphic to the matrix algebra M|S|/|P |(OP ) and the algebraOP is a source algebra of b.Nilpotent blocks were introduced in [2] as a block theoretic analogue of p-nilpotentfinite groups. The proof of the Theorem is based on Puig’s results in [6] on the bimodulestructure of a source algebra of OGb as OP -OP -bimodule and the structure theoryof nilpotent blocks in [7]. Examples of blocks of minimal O-rank include all blocksof p-nilpotent finite groups G with abelian Op′(G) and, with P = 1, the block of theSteinberg module of a finite group of Lie type in defining characteristic.We refer to The´venaz [8] for background material on p-blocks of finite groups. Inparticular, with the notation of the Theorem, by a block of OG we mean a primitiveidempotent b in Z(OG), and a defect group of b is a minimal subgroup P of G such thatOGb is isomorphic to a direct summand of OGb ⊗OPOGb as OGb-OGb-binodule. Thisis equivalent to requiring that P is a maximal p-subgroup of G such that BrP (b) 6= 0,whereBrP : (OG)P −→ kCG(P )is the Brauer homomorphism sending a P -stable element∑x∈Gλxx of the group algebraOG to the element∑x∈CG(P )λ¯xx in the group algebra kCG(P ), where here λ¯x is thecanonical image of the coefficient λx ∈ O in the residue field k. The map BrP iswell-known to be a surjective algebra homomorphism. In particular, BrP (b) is anTypeset by AMS-TEX12 MARKUS LINCKELMANNidempotent in Z(kCG(P )), hence a sum of blocks of kCG(P ). The blocks occurring inBrP (b) are all conjugate under NG(P ). More generally, a b-Brauer pair is a pair (Q, e)consisting of a p-subgroup Q of G and a block e of kCG(Q) such that BrQ(b)e 6= 0.Following [1], the set of b-Brauer pairs admits a canonical structure of partially orderedG-set with respect to the conjugation action of G. This partial order has the propertythat for any b-Brauer pair (Q, e) and any subgroup R of Q there is a unique block f ofkCG(R) such that (R, f) is a b-Brauer pair and such that (R, f) ⊆ (Q, e). The blockb is called nilpotent if NG(Q, e)/CG(Q) is a p-group for any b-Brauer pair (Q, e). Asa consequence of a theorem of Frobenius, the group G is p-nilpotent if and only if theprincipal block of OG is nilpotent, which explains the terminology.Proof of the Theorem. The statement on the minimal possible rank of OGb is well-known, but we include a proof for the convenience of the reader. Choose a Sylow-p-subgroup S of G such that P ⊆ S. Since OGb is a direct summand of OG asOS-OS-bimodule, there is an O-basis X of OGb which is stable under left and rightmultiplication with elements in S. For any subgroup R of S, the set of “diagonal”fixpointsXR = {x ∈ X | uxu−1 = x for all u ∈ R}is mapped by BrR to a k-basis in BrR((OGb)R) = kCG(R)BrR(b). Since P is maximalsuch that BrP (b) 6= 0, the set XP is in particular non empty. Also, OGb has vertex ∆Pand trivial source asO(G×G)-module, hence is a direct summand of IndG×G∆P (O), where∆P = {(u, u) | u ∈ P}. Mackey’s formula implies that every indecomposable directsummand of OGb as OS-OS-bimodule is of the form IndS×SQ (O) for some subgroup Qof S × S of the form S × S ∩ (x,y)∆P 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 ∈ XP . The stabiliser of x in S × S contains ∆P but has at most order|P |, hence is equal to ∆P . Thus the OS-OS-bimodule O[SxS] generated by x is adirect summand of OGb as OS-OS-bimodule isomorphic to OS ⊗OPOS. In particular,rkO(OGb) ≥ rkO(OS ⊗OPOS) = |S|2/|P |.In order to show that b is nilpotent we use a result of Puig [6, 3.1] in the formas described in [4, 7.8]. Let i ∈ (OGb)P be a primitive idempotent in the algebraof fixpoints in OGb with respect to the conjugation action by P on OGb such thatBrP (i) 6= 0; that is, i is a source idempotent for b and the algebra iOGi is a sourcealgebra of b. Since i commutes with the action of P , the source algebra iOGi is also adirect summand of OGb ∼= OS ⊗OPOS asOP -OP -bimodule. As a consequence of resultsin [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 kCG(P ) such that BrP (i)e = BrP (i); this makessense as BrP (i) is a primitive idempotent in kCG(P ). More precisely, for any subgroupQ of P we have AutF (Q) ∼= NG(Q, eQ)/CG(Q) where eQ is the unique block of kCG(Q)such that (Q, eQ) ⊆ (P, e). See e.g. [3] or [5], for more details on fusion systems ofblocks. Now let Q be a subgroup of P and let ϕ ∈ AutF (Q). Denote by ϕOQ theBLOCKS OF MINIMAL DIMENSION 3OQ-OQ-bimodule which is, as O-module, equal to OQ but with u ∈ Q acting onthe left by multiplication with ϕ(u) and on the right by multiplication with u. By[4, 7.8], the OQ-OQ-bimodule ϕOQ is isomorphic to a direct summand of iOGi asOQ-OQ-bimodule. Thus ϕOQ is isomorphic to a direct summand of OS ⊗OPOS asOQ-OQ-bimodule. This forces ϕ to be induced by conjugation with an element inNS(Q). In particular, ϕ is a p-automorphism of Q. Thus AutF (Q) is a p-group for allsubgroups Q of P , and hence b is a nilpotent block.By the general structure theory of nilpotent blocks [7], the block algebra OGb isisomorphic to a matrix algebra Mn(OP ); in particular, the block b has a unique iso-morphism class of simple modules. If V is a simple OGb-module then V has the defectgroup P as vertex and an endo-permutation kP -module W as source. This source istrivial if and only if the source algebra iOGi is isomorphic to OP . Dimension countingyields rkO(OGb) = n2|P | = |S|2/|P |, hence dimk(V ) = n = [S : P ]. Now V is a directsummand of IndGP (W ), hence by Mackey’s formula, ResGS (V ) is a direct sum of directsummands of IndSS∩xP (xW ) with x ∈ G. Green’s indecomposability theorem [8, (23.6)]forces S ∩ xP = xP and dimk(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 Theoremsays that OGb ∼= OS ⊗OPOS as OS-OS-bimodules for any defect group P of b containedin S. Thus, if x ∈ G such that xP ⊆ S then OS ⊗OPOS ∼= OS ⊗OxPOS, which forces xP =uP for some u ∈ S. It follows that the set HomG(P, S) of group homomorphisms fromP to S induced by conjugation with elements in G is equal to HomS(P, S) ◦ AutG(P )or equivalently, NG(P, S) = SNG(P ), where NG(P, S) = {x ∈ G |xP ⊆ S}. In otherwords, the fact that P is a defect group of a block of minimal O-rank has implicationsfor the fusion system of the group itself.References1. J. L. Alperin, M. Broue´, Local methods in block theory, Ann. Math. 110 (1979), 143–157.2. M. Broue´, 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 finitegroups, 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, J. Algebra 296 (2006),385–401.6. 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. The´venaz, G−Algebras and Modular Representation Theory, Oxford Science Publications,Clarendon Press, Oxford, 1995.Markus LinckelmannDepartment of Mathematical SciencesMeston BuildingAberdeen, AB24 3UEUnited Kingdom

Blocks of minimal dimension

Abstract

Similar works

Full text

Available Versions

City Research Online