Extension groups between simple Mackey functors

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<ul><li><p>Journal of Algebra 321 (2009) 609630</p><p>Contents lists available at ScienceDirect</p><p>E</p><p>M</p><p>EP</p><p>a</p><p>ArReAvCo</p><p>KeMMGrEx</p><p>In</p><p>laprmalMovtore</p><p>00doJournal of Algebra</p><p>www.elsevier.com/locate/jalgebra</p><p>xtension groups between simple Mackey functors</p><p>uriel Nicollerat</p><p>FL, Institut de Gomtrie, Algbre et Topologie, 1015 Lausanne, Switzerland</p><p>r t i c l e i n f o a b s t r a c t</p><p>ticle history:ceived 9 April 2008ailable online 20 November 2008mmunicated by Michel Brou</p><p>ywords:ackey functorackey algebraoup representationtension group</p><p>In order to better understand the structure of indecomposable pro-jective Mackey functors, we study extension groups of degree 1between simple Mackey functors. We explicitly determine thesegroups between simple functors indexed by distinct normal sub-groups. We next study the conditions under which it is possible torestrict ourselves to that case, and we give methods for calculatingextension groups between simple Mackey functors which are notindexed by normal subgroups. We then focus on the case wherethe simple Mackey functors are indexed by the same subgroup. Inthis case, the corresponding extension group can be embedded inan extension group between modules over a group algebra, and wedescribe the image of this embedding. In particular, we determineall extension groups between simple Mackey functors for a p-groupand for a group that has a normal p-Sylow subgroup. Finally, wecompute higher extension groups between simple Mackey functorsfor a group that has a p-Sylow subgroup of order p.</p><p> 2008 Elsevier Inc. All rights reserved.</p><p>troduction</p><p>We study extension groups of degree 1 between simple Mackey functors for a group G . The calcu-tion of these groups is a very important tool in determining the Loewy series of an indecomposableojective Mackey functor. In [7], Thvenaz and Webb proved that Mackey functors may be seen asodules over a nite-dimensional algebra. This allows one to apply the representation theory of suchgebras, such as for example the existence of projective covers. Moreover, the classication of simpleackey functors (see [8]) gives a parametrization of the indecomposable projective Mackey functorser a eld, which enabled Thvenaz and Webb to describe many properties of these projective func-rs. Let us cite some articles about these projective functors. In [3], Bouc deals with some standardsolutions of Mackey functors, in [9] Webb, constructs some specic ltration of projective functors.</p><p>E-mail address:muriel.chaubert@gmail.ch.</p><p>21-8693/$ see front matter 2008 Elsevier Inc. All rights reserved.</p><p>i:10.1016/j.jalgebra.2008.11.002</p></li><li><p>610 M. Nicollerat / Journal of Algebra 321 (2009) 609630</p><p>Inisab</p><p>noWspfugr</p><p>sugeanp</p><p>Mnopare</p><p>ispran(rdeFi</p><p>1.</p><p>meMan</p><p>ansiofposuM</p><p>MK</p><p>KanG[10] and in [11], Yaraneri studies Clifford theory and ination of projective functors. Finally, therea study of the socle of projective functors in [6]. Nevertheless, there is still a lot left to understandout the structure of such Mackey functors.We begin by explicitly calculating extension groups between simple functors indexed by distinctrmal subgroups. We then give some conditions under which it is possible to restrict to this case.hen this restriction is not possible, we obtain some information about extension groups using someecic Mackey functors, called functors T , whose denition is similar to those of the simple Mackeynctors described in the classication. This enables us to give methods for calculating extensionoups between simple Mackey functors which are not indexed by normal subgroups.We also study the case of extension groups between simple Mackey functors indexed by the samebgroup. These groups may be embedded in an extension group between modules over group al-bras. We describe the image of this embedding and we give some conditions under which it isisomorphism. In particular, this is the case when G is a p-group or when G has a normal Sylow</p><p>-subgroup.Extension groups of higher degree are closely related to minimal projective resolutions of simple</p><p>ackey functors. We exhibit some of these resolutions in the case of a group G whose order ist divisible by p2, which is the case where the Mackey algebra has nite representation type. Inrticular, we show that every simple Mackey functor for such a group possesses a minimal projectivesolution which is periodic.Throughout this paper, G denotes a nite group and k an algebraically closed eld of character-</p><p>tic p. We write H G (respectively H &lt; G) to indicate that H is a subgroup of G (respectively aoper subgroup of G). If H and K are subgroups of G , K =G H means that K is G-conjugate to H ,d similarly we write K G H (respectively K </p></li><li><p>M. Nicollerat / Journal of Algebra 321 (2009) 609630 611</p><p>Infopr</p><p>Prev</p><p>gicoan</p><p>Prfo</p><p>M</p><p>FiPr</p><p>[8</p><p>Prmw</p><p>po</p><p>w</p><p>isprbyfuthcoevfuation has a right adjoint, denoted + , and a left adjoint, denoted . If M is a Mackey functorr G , these two functors are dened by M+(K/N) = M(K )/ JK , JN I KJ (M( J )) and L(K/N) =JK , JN Ker(R</p><p>KJ ) (for more details about the previous denitions and results, see [8]). Using these</p><p>operties of adjunction, we obtain the following result.</p><p>oposition 1.1. Let M be a Mackey functor for a group G, H G and V a kNG(H)-module. If M(K ) = 0 forery proper subgroup K of H, then there exists an isomorphism of k-vector spaces</p><p> : Homk(G)(M,(InfNG (H)</p><p>NG (H)F PV</p><p>)GNG (H)) HomkNG (H)</p><p>(M(H), V</p><p>)</p><p>ven by evaluation at H. Moreover, if Hom(M(H), V ), then 1( ) = ( (K ))KG , where (K ) = 0 if Kntains no conjugate of H, and (K )(a) = ( (cxRKHx (a)))xT otherwise, where T = [NG(H)\TG (H, K )/K ]d a M(K ) (using the formula of evaluation at K of an induced Mackey functor described above).</p><p>oof. Using properties of adjunction given by Propositions 4.2, 5.1 and 6.1 of [8], we obtain thellowing bijections:</p><p>Homk(G)(M,(InfNG (H)</p><p>NG (H)F PV</p><p>)GNG (H))= Homk(NG (H))(MGNG (H), InfNG (H)NG (H) F PV</p><p>)= Homk(NG (H))</p><p>((MGNG (H)</p><p>)+, F PV</p><p>)= Homk(NG (H))</p><p>((MGNG (H)</p><p>)+(1), V</p><p>).</p><p>oreover, using the hypothesis on M , one obtains</p><p>(MGNG (H)</p><p>)+(1) = M(H)/</p><p>J</p></li><li><p>612 M. Nicollerat / Journal of Algebra 321 (2009) 609630</p><p>ThtoS</p><p>Pris</p><p>PrHof</p><p>2.</p><p>ExcuMof</p><p>ThW</p><p>wof</p><p>PrExIn</p><p>by</p><p>anSSco</p><p>ofQthsuonth</p><p>mpoaneorem 1.3. The multiplicity of PH,V as a summand of BKGK is dim(SH,V (K )); that is BKGK=(H,V ) dimk(SH,V (K )) PH,V , where the sum is taken over pairs (H, V ) where H is a subgroup of G, upconjugation, and V is a simple kNG(H)-module, up to isomorphism. In particular, if G is a p-group, then</p><p>H,k= BHGH .</p><p>Let us nally remark that these projective Mackey functors are stable under induction:</p><p>oposition 1.4. Let P be a projective Mackey functor for a subgroup H of G. Then, the induced functor PGHalso projective.</p><p>oof. One can verify that the functor Homk(G)(PGH , ) is exact, using the fact that the functoromk(H)(P , ) is exact and that the functor of induction is both left and right adjoint of the functorrestriction (see [8], Proposition 4.2). Extension groups of degree 1</p><p>We x a nite group G and an algebraically closed eld k, of characteristic p. We want to studyt(SQ ,W , SH,V ) := Ext1k(G)(SQ ,W , SH,V ), for any simple Mackey functors SQ ,W and SH,V . The cal-lation of these groups is indeed a very important tool in determining the Loewy series of projectiveackey functors. We begin by focusing on the case where Q and H are non-conjugate subgroupsG . The main result deals with the case where they are normal.</p><p>eorem 2.1. Let H and Q be normal subgroups of G with H = Q , let V be a simple kNG(H)-module and letbe a simple kNG(Q )/Q -module. The k-vector space Ext(SQ ,W , SH,V ) is isomorphic to</p><p>k if H &lt; Q , [Q : H] = p, V Q /H = V and V = W (as kG/Q -module),k if Q &lt; H, [H : Q ] = p, W H/Q = W and W = V (as kG/H-module),0 otherwise,</p><p>here V Q /H is the set of elements of V xed by the action of Q /H, and similarly W H/Q is the set of elementsW xed by the action of H/Q .</p><p>oof. Let us rst suppose that H &lt; Q . Given Theorem 14.3 of [7] the dimension over k oft(SQ ,W , SH,V ) equals the multiplicity of SQ ,W in the second socle layer of the functor M =fGG/H F PV . By Lemma 8.1 of [8], the socle of M equals S = SH,V , the subfunctor of M generatedS(H) = V = M(H). Moreover, as H is a normal subgroup, this subfunctor is dened by InfGG/H SG/H1,Vd S(K ) = I KH (V ), for every subgroup K of G containing H . We have to determine the multiplicity of</p><p>Q ,W in M/Soc(M) = M/S . In order to do so, we will prove that M/S has a subfunctor isomorphic toQ ,W only if [Q : H] = p, V Q /H = V and W = V . Later on, we will see that under the three previousnditions, there exists a unique subfunctor of M/S isomorphic to SQ ,W .Let T be a simple subfunctor of M/S , isomorphic to SQ ,W . In particular, W = T (Q ) is a submoduleM(Q )/S(Q ) = V Q /H/I QH (V ) and V Q /H possesses a structure of kG/H-module due to the fact thatis a normal subgroup. Hence, V Q /H is a kG/H-submodule of V . Consequently, by simplicity of V ,ere are two possible cases: either V Q /H = 0, or V Q /H = V . In the rst case, there is no suchbfunctor T in M/S . So, we can assume that V Q /H = V , or in other words, that Q acts triviallyV . This implies that if x V , we have I QH (x) = [Q : H] x. Therefore, if p does not divide [Q : H],</p><p>en (M/S)(Q ) = V /V = 0 and there is again no subfunctor isomorphic to SQ ,W in M/S .Let us suppose then that V Q /H = V and that p divides [Q : H]. As Q acts trivially on V , this</p><p>odule naturally possesses a structure of simple kG/Q -module. As W (M/S)(Q ) = V , the onlyssibility is W = V . Let us next assume that there exists a subgroup J such that H &lt; J &lt; Q</p><p>Qd p | [ J : H]. Hence, on the one hand, R J (T (Q )) = V , since the application of restriction is an</p></li><li><p>M. Nicollerat / Journal of Algebra 321 (2009) 609630 613</p><p>in</p><p>M</p><p>haof[Q</p><p>mfuFoJm(oth</p><p>M</p><p>inthcoitSfuotdi</p><p>usofspon</p><p>H</p><p>thfucl</p><p>Dsu</p><p>Lede</p><p>Prclusion. Indeed, as S( J ) = 0 = S(Q ), the application RQJ of the functor T comes from those of= InfGG/H F PV , which is an inclusion by denition of the xed point functor. However, on the othernd, RQJ (T (Q )) T ( J ) = 0, since T is a subfunctor and the evaluation of T at proper subgroupsQ is zero. Therefore, in the latter case, there cannot exist such a subfunctor T ; or in other words,: H] must be equal to p.To sum up, we can assume that [Q : H] = p, V Q /H = V and W = V , because otherwise the</p><p>ultiplicity of SQ ,W in M/S is zero. Under these assumptions, (M/S)(Q ) = V . Let N be the sub-nctor of M/S generated by N(Q ) = V . We will show that N is equally a subfunctor of InfGG/Q F PV .r this purpose, remark that if J &lt; Q , then either J = H , in which case (M/S)( J ) = V /V = 0, or= H , in which case M( J ) = 0 and thus (M/S)( J ) = 0. Consequently, N( J ) = 0 if J &lt; Q . Further-ore, due to the denition of N , the module N( J ) is zero if the subgroup J does not contain Qne can see this using Proposition 2.4 of [7], with being the set of subgroups of Q ). If Q &lt; K ,en S(K ) = I KH (V ) = I KQ (I QH (V )) = 0, and so (M/S)(K ) = M(K ). Therefore, N(K ) is a submodule of(K ) = V K/H = V K/Q = InfGG/Q F PV (K ), since Q acts trivially on V . Moreover, the applications ofduction, restriction and conjugation of M/S are the same as those of the functor InfGG/Q F PV (usinge denition of ination and the fact that the functor S is zero when evaluated at the subgroupsntaining Q ). Hence, N can be identied to the subfunctor of InfGG/Q F PV generated by N(Q ) = V ;is thus isomorphic to SQ ,W . Consequently, if T is a simple subfunctor of M/S , isomorphic to</p><p>Q ,W , then N must be a non-zero subfunctor of T , as T (Q ) = V and as N is the smallest sub-nctor of M/S whose evaluation at Q equals V . Since T is simple, it follows that T = N; or inher words, there is exactly one copy of SQ ,W in the second socle layer of M . Thus, in this case,mk(Ext(SQ ,W , SH,V )) = 1.Let us next assume that Q &lt; H . Then</p><p>Extk(G)(SQ ,W , SH,V ) = Extk(G)op((SH,V )</p><p>, (SQ ,W ))= Extk(G)(SH,V , SQ ,W )</p><p>ing the fact that for every simple Mackey functor S J ,U , we have (S J ,U ) = S J ,U (by Proposition 4.1[7]) and that k(G)op = k(G) (by Chapter 4 of [7]). Using the case where H &lt; Q , this k-vectorace equals k if and only if [H : Q ] = p, (W )H/Q = W and W = V . Yet (W )H/Q = W if andly if W H/Q = W and, similarly, W = V if and only if V = W .Finally, if Q H and H Q , the group Ext(SQ ,W , SH,V ) is trivial by Theorem 14.3 of [7], as= Q by hypothesis. The problem now is to determine when it is possible to restrict the general case to the case where</p><p>e subgroups indexing the simple functors are normal. To answer this question, we will introducenctors, called functors T , whose denition is similar to those of simple functors determined in theassication, except that they are not indexed by simple modules.</p><p>enition 2.2. Let H be a subgroup of G and V a kNG(H)-module. The Mackey functor TH,V is thebfunctor of M = (InfNG (H)</p><p>NG (H)F PV )GNG (H) that is generated by TH,V (H) = M(H) = V .</p><p>These functors may be characterized in the following way:</p><p>mma2.3. Let V be a kNG(H)-module, where H G. Let TV be the subfunctor of F PV , for the group NG(H),ned by TV ( J ) = I J1 (V ). The functor TH,V is then isomorphic to (InfNG (H)NG (H) TV )</p><p>GNG (H)</p><p>.</p><p>oof. The proof is similar to the one of Lemma 8.1 of [8]. </p><p>The functors TH,V may be dened using G-sets (with the denition of Dress).</p></li><li><p>614 M. Nicollerat / Journal of Algebra 321 (2009) 609630</p><p>Pr</p><p>w</p><p>PrX</p><p>Repr</p><p>InNm</p><p>Th</p><p>Co</p><p>Th</p><p>Moposition 2.4. Let V be a kNG(H)-module, where H G, and let X be a G-set. Then,</p><p>TH,V (X) = ING (H)1(Homk</p><p>(k(XH), V))</p><p>here ING (H)1 is the relative trace map.</p><p>oof. First suppose that H = 1. We can assume that X is a transitive G-set; or in other words= G/ J and we shall show that</p><p>I J1 (V )= IG1</p><p>(Homk(kG/ J , V )</p><p>).</p><p>mark that I J1 (V ) V J and IG1 (Homk(kG/ J , V )) (Homk(kG/ J , V ))G . Moreover, the Frobenius reci-ocity theorem leads us to the following isomorphism:</p><p> : V J Homk J (k, V ) HomkG(kG/ J , V ) (Homk(kG/ J , V )</p><p>)G.</p><p>particular, if x = j J jv I J1 (V ) with v V , then (x) is the kG-linear map dened by (x)(u) =(x)(1 J ) = x.ext, let us show that (x) belongs to IG1 (Homk(kG/ J , V ). In order to do so, let us dene the k-linearap f : kG/ J V by</p><p>f (g J ) ={x si g J ,0 sinon.</p><p>en,</p><p>IG1 ( f )(u) =gG</p><p>g1 f (gu) =j J</p><p>t[G/ J ]</p><p>j1t1 f (t ju) =j J</p><p>j1( </p><p>t[G/ J ]t1 f (t J )</p><p>)=j J</p><p>j1v = x.</p><p>nsequently, (x) = IG1 ( f ), and so (I J1 (V )) IG1 (Homk(kG/ J , V )).Moreover, if f Homk(kG/ J , V ), then</p><p>IG1(f (u)</p><p>)=gG</p><p>g1 f (gu) =j J</p><p>t[G/ J ]</p><p>j1t1 f (t j J ) =j J</p><p>j1( </p><p>t[G/ J ]t1 f (t J )</p><p>)</p><p>= ( </p><p>t[G/ J ]t1 f (t J )</p><p>).</p><p>us, induces an isomorphism from I J1 (V ) onto IG1 (Homk(kG/ J , V )).</p><p>Using the previous result, we obtain, for every G-set X ,</p><p>TH,V (X) = InfNG (H)NG (H)(T1,V</p><p>(ResGNG (H)(X)</p><p>))= T1,V ((ResGNG (H)(X))H)</p><p>= ING (H)1(Homk</p><p>(k((ResGNG (H)(X)</p><p>)H), V))</p><p>= ING (H)1(Homk</p><p>(k(XH), V))</p><p>. These functors T possess properties that entirely characterize them (in a similar way as simpleackey functors).</p></li><li><p>M. Nicollerat / Journal of Algebra 321 (2009) 609630 615</p><p>PrthM</p><p>(a</p><p>(b</p><p>Prbyco</p><p>an</p><p>Fi</p><p>isH</p><p>m(w</p><p>InH</p><p>wbe</p><p>vaoposition 2.5. Let M be a Mackey functor for G and H a min...</p></li></ul>