Projective modules and involutions

ARTICLE IN PRESS UNCORRECTED PROOF S0021-8693(05)00338-8/FLA AID:10633 Vol.•••(•••) [DTD5] P.1 (1-7) YJABR:m1 v 1.39 Prn:23/06/2005; 8:51 yjabr10633 by:Gi p. 1 Journal of Algebra ••• (••••)•••–••• www.elsevier.com/locate/jalgebra 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 Projective modules and involutions John Murray Mathematics Department, National University of Ireland, Maynooth, Co. Kildare, Ireland Received 30 March 2005 Communicated by Michel Broué Abstract Let Gbe a finite group, and let Ω:= {t∈G|t2=1}.ThenΩis a G-set under conjugation. Let kbe an algebraically closed field of characteristic 2. It is shown that each projective indecomposable summand of the G-permutation module kΩ is irreducible and self-dual, whence it belongs to a real 2-block of defect zero. This, together with the fact that each irreducible kG-module that belongs to a real 2-block of defect zero occurs with multiplicity 1 as a direct summand of kΩ, establishes a bijection between the projective components of kΩ and the real 2-blocks of Gof defect zero. 2005 Published by Elsevier Inc. Keywords: Involutions; Blocks of defect zero; Green correspondence; Burry–Carlson–Puig theorem Let Gbe a finite group, with identity element e, and let Ω:= {t∈G|t2=e}. Then Ωis a G-set under conjugation. In this note we describe the projective components of the permutation module kΩ, where kis an algebraically closed field of characteristic 2. By a projective component we mean an indecomposable direct summand of kΩ that is also a direct summand of a free kG-module. We show that all such components are irreducible, self-dual and occur with multiplicity 1. This gives an alternative proof of Remark (2) on p. 254 of [5], and strengthens Corollaries 3 through 7 of that paper. In addition, we can give the following quick proof of Proposition 8 in [5]: E-mail addresses: [email protected], [email protected].ie. 0021-8693/$ – see front matter 2005 Published by Elsevier Inc. doi:10.1016/j.jalgebra.2005.05.032 ARTICLE IN PRESS UNCORRECTED PROOF S0021-8693(05)00338-8/FLA AID:10633 Vol.•••(•••) [DTD5] P.2 (1-7) YJABR:m1 v 1.39 Prn:23/06/2005; 8:51 yjabr10633 by:Gi p. 2 2J. Murray / Journal of Algebra ••• (••••)•••–••• 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 Corollary 1. Suppose that His a strongly embedded subgroup of G. Then kH↑G∼ = kG⊕[ s i=1Pi]where s⩾0and the Piare pairwise nonisomorphic self-dual projective irreducible kG-modules. Proof. That His strongly embedded means that |H|is even and |H∩Hg|is odd, for each g∈G\H.Lett∈Hbe an involution. Then clearly CG(t) ⩽H.SokH↑Gis isomorphic to a submodule of (kCG(t))↑G. Mackey’s theorem implies that every component of kH↑G, other than kG, is a projective kG-module. Being projective, these modules must be components of (kCG(t))↑G. The result now follows from Theorem 8. 2 Consider the wreath product GΣof Gwith a cyclic group Σof order 2. Here Σis generated by an involution σand GΣis isomorphic to the semidirect product of the base group G×Gby Σ. The conjugation action of σon G×Gis given by (g1,g 2)σ=(g2,g 1), for all g1,g 2∈G. The elements of GΣwill be written (g1,g 2),(g1,g 2)σ or σ. We shall exploit the fact that kG is a kG Σ-module. For, as is well known, kG is an k(G ×G)-module via: x·(g1,g 2):= g−1 1xg2, for each x∈kG, and g1,g 2∈G. The action of Σon kG is induced by the permutation action of σon the distinguished basis Gof kG: gσ:= g−1, for each g∈G. Clearly σacts as an involutary k-algebra anti-automorphism of kG. It follows that the actions of G×Gand Σon kG are compatible with the group relations in GΣ. Byablock of kG, or a 2-block of G, we mean an indecomposable k-algebra direct summand of kG. Each block has associated to it a primitive idempotent in Z(kG), a Brauer equivalence class of characters of irreducible kG-modules and a Brauer equivalence class, modulo 2, of ordinary irreducible characters of G. A block has defect zero if it is a simple k-algebra, and is real if it contains the complex conjugates of its ordinary irreducible characters. Theorem 8 establishes a bijection between the real 2-blocks of Gthat have defect zero and the projective components of kΩ. We could equally well work over a complete discrete valuation ring Rof characteristic 0, whose field of fractions Fis algebraically closed, and whose residue field R/J(R) is k.SoweuseOto indicate either of the commutative rings kor R. All our modules are right-modules. We denote the trivial OG-module by OG.IfMis an OG-module, we use M↓Hto denote the restriction of Mto H.IfHis a subgroup of Gand Nis an OH-module, we use N↑Gto denote the induction of Nto G. Whenever g∈G,we write gfor (g, g) ∈G×G, and we set X:= {x|x∈X}, for each X⊂G. Other notation and concepts can be found in a standard textbook on modular representation theory, such as [1] or [4]. If Bis a block of OG, then so too is Bo={xσ|x∈B}. We call Ba real block if B=Bo. Our first result describes the components of OGas OGΣ-module. Lemma 2. There is an indecomposable decomposition of OGas OGΣ-module: OG=B1⊕···⊕Br⊕Br+1+Bo r+1⊕···⊕Br+s+Bo r+s+1. Here B1,...,B rare the real 2-blocks and Br+1,Bo r+1,...,B r+s,Bo r+sare the nonreal 2-blocks of G. ARTICLE IN PRESS UNCORRECTED PROOF S0021-8693(05)00338-8/FLA AID:10633 Vol.•••(•••) [DTD5] P.3 (1-7) YJABR:m1 v 1.39 Prn:23/06/2005; 8:51 yjabr10633 by:Gi p. 3 J. Murray / Journal of Algebra ••• (••••)•••–••• 3 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 Proof. This follows from the well-known indecomposable decomposition of OG,asan O(G ×G)-module, into a direct sum of its blocks, and the fact that Bσ i=Bifor i= 1,...,r, and Bσ r+j=Bo r+jfor j=1,...,s.2 An obvious but useful fact is that OGis a permutation module: Lemma3.The OGΣ-module OGis isomorphic to thepermutationmodule (OG×Σ)↑GΣ. Proof. The elements of Gform a GΣ-invariant basis of OG. Moreover if g1,g 2∈G, then g2=g1·(g1,g 2).SoGis a transitive GΣ-set. The stabilizer of e∈OGin GΣis G×Σ. The lemma follows from these facts. 2 Let Cbe a conjugacy class of G. Set Co:= {c∈G|c−1∈C}. Then Cois also a conjugacy class of G, and C∪Cocan be regarded as an orbit of G×Σon the GΣset G. As such, the corresponding permutation module O(C ∪Co)is a OG×Σ-direct summand of OG.IfC=Co, we call Ca real class of G. In this case for each c∈Cthere exists x∈Gsuch that cx=c−1. The point stabilizer of cin G×Σis CG(c)xσ.So OC∼ =(OCG(c)xσ)↑G×Σ. If C= Co, we call Ca nonreal class of G. In this case the point stabilizer of c∈C∪Coin G×Σis CG(c).So O(C ∪Co)∼ =(OCG(c))↑G×Σ. Suppose now that the real classes are C1,...,C tand that the nonreal classes are Ct+1,Co t+1,...,C t+u,Co t+u. Then we have: Lemma 4. There is a decomposition of OGas an OG×Σ-permutation module: OG=OC1⊕···⊕OCt⊕OCt+1∪Co t+1⊕···⊕OCt+u∪Co t+u+1. Proof. This follows from Lemma 3 and the discussion above. 2 By a quasi-permutation module we mean a direct summand of a permutation module. Our next result is Lemma 9.7 of [1]. We include a proof for the convenience of the reader. Lemma 5. Let Mbe an indecomposable quasi-permutation OG-module and suppose that His a subgroup of Gsuch that M↓His indecomposable. Then there is a vertex Vof M such that V∩His a vertex of M↓H.IfHis a normal subgroup of G, then this is true for all vertices of M. Proof. Let UbeavertexofM.AsOU|M↓Uwe have OU∩H|(M↓H)↓U∩H.ButU∩H is a vertex of OU∩H. So Mackey’s theorem implies that there exists a vertex Wof M↓H such that U∩H⩽W. ARTICLE IN PRESS UNCORRECTED PROOF S0021-8693(05)00338-8/FLA AID:10633 Vol.•••(•••) [DTD5] P.4 (1-7) YJABR:m1 v 1.39 Prn:23/06/2005; 8:51 yjabr10633 by:Gi p. 4 4J. Murray / Journal of Algebra ••• (••••)•••–••• 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 As M↓His a component of the restriction of Mto H, Mackey’s theorem shows that there exists g∈Gsuch that W⩽Ug∩H.NowUgis a vertex of M. So by the previous paragraph, and the uniqueness of vertices of M↓Hup to H-conjugacy, there exists h∈H such that Ug∩H⩽Wh. Comparing cardinalities, we see that W=Ug∩H.SoUg∩H is a vertex of M↓H. Suppose that His a normal subgroup of G. Then U∩H⩽Wand W=Ug∩H= (U ∩H)gimply that U∩H=W.2 R. Brauer showed how to associate to each block of OGaG-conjugacy class of 2-subgroups, its so-called defect groups. It is known that a block has defect zero if and only if its defect groups are all trivial. J.A. Green showed how to associate to each indecomposable OG-module a G-conjugacy class of 2-subgroups, its so-called vertices. He also showed how to identify the defect groups of a block using its vertices as an indecomposable O(G ×G)-module. Corollary 6. Let Bbe a block of OGand let Dbe a defect group of B.IfBis not real then Dis a vertex of B+Bo,asOGΣ-module. If Bis real, then there exists x∈NG(D), with x2∈D, such that Dxσ is a vertex of B,asOGΣ-module. In particular, Σis a vertex of B+Boif and only if Bis a real 2-block of Gthat has defect zero. Proof. J.A. Green showed in [2] that Dis a vertex of B, when Bis regarded as an indecomposable O(G ×G)-module. Suppose first that Bis not real. Then B+Bo= (B↓G×G)↑GΣ, for instance by Corollary 8.3 of [1]. It follows that B+Bohas vertex D, as an indecomposable OGΣ-module. Suppose then that B=B+Bois real. Lemma 3 shows that Bis G×Σ-projective. So we may choose a vertex Vof Bsuch that V⩽G×Σ. Moreover, Bis a quasi-permutation OGΣ-module, and its restriction to the normal subgroup G×Gis indecomposable. Lemma 5 then implies that V∩(G ×G) =V∩Gis a vertex of B↓G×G. So by Green’s result, we may choose Dso that V∩G=D.NowG×Ghas index 2 in GΣ. So Green’s indecomposability theorem, and the fact that B↓G×Gis indecomposable, implies that V⊆ (G ×G). It follows that there exists x∈NG(D), with x2∈D, such that V=Dxσ . If Bhas defect zero, then D=e.Sox2=e. In this case, xσ =Σ(e,x) is GΣconjugate to Σ.SoΣis a vertex of B. Conversely, suppose that Σis a vertex of B+Bo. The first paragraph shows that Bis a real block of G. Moreover Bhas defect zero, as Σ∩G=e.2 We quote the following result of Burry, Carlson and Puig [4, 4.4.6] on the Green correspondence: Lemma 7. Let V⩽H⩽Gbe such that Vis a p-group and NG(V ) ⩽H. Let fdenote the Green correspondence with respect to (G,V,H). Suppose that Mis an indecomposable OG-module such that M↓Hhas a component Nwith vertex V. Then Vis a vertex of M and N=f(M). ARTICLE IN PRESS UNCORRECTED PROOF S0021-8693(05)00338-8/FLA AID:10633 Vol.•••(•••) [DTD5] P.5 (1-7) YJABR:m1 v 1.39 Prn:23/06/2005; 8:51 yjabr10633 by:Gi p. 5 J. Murray / Journal of Algebra ••• (••••)•••–••• 5 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 We can now prove our main result. Part (ii) is Remark (2) on p. 254 of [5], but our proof is independent of the proof given there. Theorem 8. (i) Let t∈G, with t2=e. Suppose that Pis an indecomposable projective direct summand of (OCG(t))↑G. Then Pis irreducible and self-dual and occurs with multiplicity 1as a component of (OCG(t))↑G. In particular Pbelongs to a real 2-block of Gthat has defect zero. (ii) Suppose that Mis a projective indecomposable OG-module that belongs to a real 2-block of Gthat has defect zero. Then there exists s∈G, with s2=e, such that M is a component of (OCG(s))↑G. Moreover, sis uniquely determined up to conjugacy in G. Proof. If t=ethen P=OG.SoPis irreducible and self-dual. The assumption that Pis projective and the fact that dimO(P ) =1 implies that |G|is odd. So all blocks of OG,in particular the one containing P, have defect zero. Now suppose that t= e.LetTbe the conjugacy class of Gthat contains t. The permutation module OTis a direct summand of the restriction of OGto G×Σ.RegardPas an OG-module. Let I(P) be the inflation of this module to G×Σ. Then I(P) is a component of OT.AsΣis contained in the kernel of I(P), and Pis a projective OG-module, it follows that I(P)has vertex Σas an indecomposable OG×Σ-module. By Lemma 2, and the Krull–Schmidt theorem, there exists a 2-block Bof Gsuch that I(P) is a component of the restriction (B +Bo)↓G×Σ. An easy computation shows that NGΣ(Σ) =G×Σ. It then follows from Lemma 7 that (B +Bo)has vertex Σand also that I(P)is the Green correspondent of (B +Bo)with respect to (G Σ,Σ,G ×Σ).We conclude from Corollary 6 that Bis a real 2-block of Gthat has defect zero. Let ˆ Bbe the 2-block of GΣthat contains B. Then ˆ Bis real and has defect group Σ. Let ˆ Abe the Brauer correspondent of ˆ B. Then ˆ Ais a real 2-block of G×Σthat has defect group Σ.Now ˆ A=A⊗OΣ, where Ais a real 2-block of OGthat has defect zero. In particular Ahas a unique indecomposable module, and this module is projective, irreducible and self-dual. Corollary 14.4 of [1] implies that I(P) belongs to ˆ A.SoP belongs to A. We conclude that Pis irreducible and self-dual and belongs to a real 2-block of Gthat has defect zero. Now Boccurs with multiplicity 1 as a component of OG, and I(P) is the Green correspondent of Bwith respect to (G Σ,Σ,G ×Σ).SoI(P) has multiplicity 1 as a component of the restriction of OGto G×Σ. It follows that Poccurs with multiplicity 1 as a component of (OCG(t))↑G, and with multiplicity 0 as a component of (OCG(r))↑G, for r∈Gwith r2=e,butrnot G-conjugate to t. This completes the proof of part (i). Let Rbe a real 2-block of Gthat has defect zero. Then Rhas vertex Σas indecomposable OGΣ-module. So its Green correspondent f(R), with respect to (GΣ,Σ,G×Σ), is a component of the restriction of OGto G×Σthat has vertex Σ. Lemma 4 and the Krull–Schmidt theorem imply that f(R)is isomorphic to a component of O(C ∪Co),for some conjugacy class Cof G.NowΣis a central subgroup of G×Σ.SoΣmust be a subgroup of the point stabilizer of C∪Coin G×Σ. It follows that s2=e, for each s∈C. ARTICLE IN PRESS UNCORRECTED PROOF S0021-8693(05)00338-8/FLA AID:10633 Vol.•••(•••) [DTD5] P.6 (1-7) YJABR:m1 v 1.39 Prn:23/06/2005; 8:51 yjabr10633 by:Gi p. 6 6J. Murray / Journal of Algebra ••• (••••)•••–••• 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 Let Ndenote the restriction of f(R) to G, and consider Nas an OG-module. We have just shown that Nis a component of (OCG(s))↑G. Arguing as before, we see that Nis an indecomposable projective OG-module that belongs to a real 2-block of Gthat has defect zero. The last paragraph establishes an injective map between the real 2-blocks of Gthat have defect zero and certain projective components of OΩ. As each block of defect zero contains a single irreducible OG-module, this map must be onto. It follows that the module M in the statement of the theorem is a component of some permutation module (OCG(s))↑G, where s∈Gand s2=e. The fact that sis determined up to G-conjugacy now follows from the last statement of the proof of part (i). This completes the proof of part (ii). 2 It is possible to simplify the above proof by showing that if Bis a real 2-block of Gthat has defect zero, then its Green correspondent, with respect to (G Σ,Σ,G ×Σ) is MFr, where MFr is the Frobenius conjugate of the unique irreducible OG-module that belongs to B. Suppose that Ris a complete discrete valuation ring and that Lis an RCG(t)-module, where Lhas R-rank 1 and O2(CG(t)) acts trivially on L. Then the 2-modular reduction of Lis the trivial kCG(t)-module, although Lis not necessarily the trivial RCG(t)-module. Now each projective irreducible kG-module lifts to a projective irreducible RG-module. So the conclusions of part (i) of the above theorem apply to L↑G: all of its projective components are irreducible and self-dual. We thank the referee for pointing out this extension of our result. The proof of Theorem 8 hints at the fact that we have some 2-local control over all the components of (OCG(t))↑G. The investigation of special properties of such components is continued in [3]. Corollary 9. Let Ω={t∈G|t2=e}. Then there is a bijection between the real 2-blocks of Gthat have defect zero and the projective components of OΩ. Here is a sample application. It was suggested to me by G.R. Robinson. Corollary 10. Let n⩾1and let tbe an involution in the symmetric group Σn.If n=m(m +1)/2is a triangular number, and tis a product of (m2+1)/4commuting transpositions, then there is a single projective irreducible OΣn-module, and this module is the unique projective component of (OCΣn(t))↑Σn. For all other values of nor nonconjugate involutions t, the modules (OCΣn(t))↑Σnare projective free. Proof. We give a proof of the following result in [3, Corollary 8.4]: Let Gbe a finite group, let Bbe a real 2-block of Gof defect zero, and let χbe the unique irreducible character in B. Then there exists a 2-regular conjugacy class Cof Gsuch that C=Co,|CG(c)|is odd, for c∈C, and χ(c) is nonzero, modulo a prime ideal containing 2. Moreover, there exists an involution t∈Gsuch that ct=c−1, and for this twe have χCG(t),1CG(t)=1. The existence of twas shown in [5]. The identification of tusing the class Cwas first shown by R. Gow (in unpublished work). ARTICLE IN PRESS UNCORRECTED PROOF S0021-8693(05)00338-8/FLA AID:10633 Vol.•••(•••) [DTD5] P.7 (1-7) YJABR:m1 v 1.39 Prn:23/06/2005; 8:51 yjabr10633 by:Gi p. 7 J. Murray / Journal of Algebra ••• (••••)•••–••• 7 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 Suppose that (OCΣn(t))↑Σnhas a projectivecomponent.Then Σnhas a 2-block ofdefect zero, by Theorem8. The 2-blocks of Σnare indexed bytriangularpartitions µ=[m, m−1, ...,2,1], where mranges over those natural numbers for which n−m(m +1)/2 is even. Moreover, the 2-block corresponding to µhas defect zero if and only if n=m(m +1)/2. In particular, we can assume that n=m(m +1)/2, for some m⩾1. Let Bbe the unique 2-block of Σnthat has defect zero, let χbe the unique irreducible character in Band let g∈Σnhave cycle type λ=[2m−1,2m−5,...]. Then |CΣn(g)| is odd. As the parts of λare the “diagonal hooklengths” of µ, the Murnaghan–Nakayama formula shows that χ(g)=1. Now λhas (m −1)/2nonzero parts. So gis inverted by an involution tthat is a product of (n −(m −1)/2)/2=(m2+1)/4commuting transpositions. It follows from Theorem 8 and the previous paragraph that the unique irreducible projective B-module occurs with multiplicity 1 as a component of (OCΣn(t))↑Σn.Thelast statement of the corollary now follows from Theorem 8. 2 References [1] J.L. Alperin, Local Representation Theory, Cambridge Stud. Adv. Math., vol. 11, 1986. [2] J.A. Green, Blocks of modular representations, Math. Z. 79 (1962) 100–115. [3] J. Murray, Extended defect groups and extended vertices, Osaka J. Math, in press. [4] H. Nagao, Y. Tsushima, Representations of Finite Groups, Academic Press, 1989. [5] G.R. Robinson, The Frobenius–Schur indicator and projective modules, J. Algebra 126 (1989) 252–257.