Fixed-point functors for symmetric groups and Schur algebras

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<ul><li><p>Journal of Algebra 280 (2004)</p><p>Fixed-point functors for symmetric groupsand Schur algebras</p><p>David J. Hemmer 1</p><p>Department of Mathematics, University of Toledo, 2801 W. Bancroft, Toledo, OH 43606, USAReceived 16 March 2004</p><p>Available online 2 July 2004</p><p>Communicated by Gordon James</p><p>Abstract</p><p>Let d be the symmetric group. For 1 &lt; m &lt; d let Fm be the functor which takes a d -module U to the space of fixed points Um , which is naturally a module for dm. This functorwas previously used by the author to study cohomology of the symmetric group, but little isknown about it. This paper initiates a study of Fm. First, we relate it to James work on rowand column removal and decomposition numbers for the Schur algebra. Next, we determine theimage of dual Specht modules, permutation and twisted permutation modules, and some Youngand twisted Young modules under Fm. In particular, Fm acts as first row removal on dual Spechtmodules S with 1 = m and as first column removal on twisted Young and twisted permutationmodules corresponding to partitions with m parts. Finally, we prove that determining Fm on theYoung modules is equivalent to determining the decomposition numbers for the Schur algebra. 2004 Elsevier Inc. All rights reserved.</p><p>1. Notation and preliminaries</p><p>We will assume familiarity with the representation theory of the symmetric group dand of the Schur algebra S(n, d) as found in [3,7,11]. Let k be an algebraically closed fieldof characteristic p &gt; 2. We write d for = (1, . . . , r ) a partition of d and |= d fora composition of d . Let +(n, d) denote the set of partitions of d with at most n parts and</p><p>E-mail address:</p><p>1 Research of the author was supported in part by NSF grant DMS-0102019.</p><p>0021-8693/$ see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2004.04.014</p></li><li><p>296 D.J. Hemmer / Journal of Algebra 280 (2004) 295312let (n,d) denote the set of compositions of d with at most n parts. We do not distinguishbetween and its Young diagram:</p><p> = {(i, j) N N j i}.A partition is p-regular if there is no i such that i = i+1 = = i+p1 = 0. It isp-restricted if its conjugate partition, denoted , is p-regular. We write for the usualdominance order on partitions. For = (1, 2, . . . , r ) we write for with its first rowremoved, i.e.,</p><p> = (2, . . . , r ) d 1.We write for with its first column removed, i.e.,</p><p> = (1 1, 2,1, . . . , r 1) d r.The complex simple d -modules are the Specht modules {S | d}. Simple kd -</p><p>modules can be indexed by p-restricted partitions or by p-regular partitions. Both{D := S/rad(S) is p-regular} and {D = soc(S) is p-restricted}</p><p>are complete sets of nonisomorphic simple kd -modules. The two indexings are relatedby D = D sgn, where sgn is the one-dimensional signature representation. We recallthat</p><p>S sgn = S , (1.1)where S denotes the dual of the Specht module S.</p><p>We will also consider the Young modules {Y | d}, the permutation modules{M | (n,d)}, and their twisted versions obtained by tensoring with sgn. If isp-restricted, then Y is the projective cover of D. All these modules, and the S(n, d)-modules below, are described in [11].</p><p>Let V = kn be the natural module for the general linear group GLn(k). Then V d isa GLn(k) -module with the diagonal action and a kd -module by place permutation. TheSchur algebra S(n, d) is defined by</p><p>S(n, d) := Endkd(V d</p><p>).</p><p>The actions of GLn(k) and d commute, so we get a map GLn(k) S(n, d). Thismap identifies the category mod-S(n, d) with the category of homogeneous polynomialrepresentations of GLn(k) of degree d .</p><p>For +(n, d) we denote the irreducible S(n, d)-module with highest weight by L(). We also write () and () (sometimes written V () and H 0()) for thestandard and costandard modules with highest weight , respectively. P() will be theprojective cover of L(), I () the injective hull, and T () the corresponding tilting</p><p>module. Finally, S(V ) and (V ) will be the symmetric and exterior powers.</p></li><li><p>D.J. Hemmer / Journal of Algebra 280 (2004) 295312 297Let I (n, d) denote the set of all d-tuples i = (i1, i2, . . . , id) with is {1,2, . . . , n}. Thesymmetric group d acts on the right on I (n, d) by</p><p>i = (i(1), i(2) . . . , i(d)).This action extends to an action on I (n, d) I (n, d), and we write (i, j) (k, l) if k = iand l = j for some d . Let be a set of equivalence class representatives under .Recall [3] that S(n, d) has a basis {i,j} indexed by . For i I (n, d) define the weightof i by wt(i) = (1, 2, . . . , n) (n,d), where s is the number of times s appears in i.Then i,i is an idempotent, usually denoted .</p><p>1.1. The Schur and adjoint Schur functor</p><p>Henceforth assume n d . Let = (1d) (n,d) and let e denote the idempotent S(n, d). Then eS(n, d)e = kd , and the Schur functor F : mod-S(n, d) mod-kdis defined by F(U) := eU . This is an exact, covariant functor with</p><p>F(())= S, F(())= S, F(L())= D or 0,F(P())= Y, F(I ())= Y, F(T ())= Y sgn. (1.2)</p><p>The Schur functor admits a right adjoint functor G : mod-kd mod-S(n, d) definedby</p><p>G(N) := Homkd(V d ,N</p><p>)= HomeS(n,d)e(eS(n, d),N).The two definitions are equivalent since eS(n, d)e = kd and eS(n, d) = V d . Themodule V d is not injective as a kd -module. Thus the functor G is only left exact, andso has higher right derived functors</p><p>RiG(N) = Extikd(V d ,N</p><p>).</p><p>We now collect a few results about G and R1G. Recall that we are assuming p &gt; 2throughout:</p><p>Proposition 1.1.</p><p>(i) [10, 3.2] G(S) = (). In particular, G(k) = (d).(ii) [5, 3.8.2] G(Y ) = P().</p><p>(iii) [10, 6.4] For p &gt; 3, R1G(S) = 0.</p><p>1.2. Decomposition numbers, Specht filtrations, and dual Specht filtrations</p><p>We collect here material that will be used repeatedly in later sections. For a thorough</p><p>treatment see Martins book [11].</p></li><li><p>298 D.J. Hemmer / Journal of Algebra 280 (2004) 295312We say a kd -module U has a Specht filtration if it has a filtration with successivequotients isomorphic to Specht modules. Similarly, we will say U has a dual Spechtfiltration. For S(n, d)-modules we refer to a Weyl filtration (by ()s) or a good filtration(by ()s). The multiplicities in a good or Weyl filtration are independent of the choice offiltration. The same holds for Specht and dual Specht filtrations in mod-kd when p &gt; 3(but not for p = 2 or 3) by work in [5].</p><p>The Young modules Y are self-dual and are known to have both Specht and dual Spechtfiltrations. If [Y : S] denotes the multiplicity of S in a dual Specht filtration of Y and[() : L()] denotes a decomposition number for S(n, d), then a well-known reciprocitytheorem [11, p. 118] gives[</p><p>() : L()]= [I () : ()]= [Y : S]= [Y : S]. (1.3)So knowledge of the decomposition numbers for S(n, d) is equivalent to knowledge ofmultiplicities in dual Specht filtrations of Young modules, a fact we will use repeatedlylater.</p><p>The category mod-S(n, d) is a highest weight category. In particular, [() : L()] = 1and [() : L()] = 0 unless . This triangular structure in the decomposition matrix,together with reciprocity (1.3), will be very useful to us, since it gives a triangular structureto the matrix of filtration multiplicities [Y : S]. In particular, suppose we know a kd -module is a direct sum of Young modules. If we know the multiplicities in a dual Spechtfiltration of the module, then we can determine the multiplicities of the Young modulesummands.</p><p>The permutation module M is a direct sum of Young modules</p><p>M = Y</p><p>KY.</p><p>The p-Kostka numbers K are not known. However, Youngs rule (see [7, Chapter 14])gives a nice formula for the multiplicities in a Specht or dual Specht filtration of M.Thus if we know the decomposition numbers for S(n, d), then Youngs rule together withreciprocity let us determine the p-Kostka numbers.</p><p>2. Relating the fixed-point functor to James idempotent</p><p>In [4] we proved some theorems on extensions between simple modules for thesymmetric group when the first row or column of the corresponding partitions is removed.The proof used the fixed-point functor, which we now define. Throughout the paper we willconsider m as the subgroup of d fixing {m+1,m+2, . . . , d} and dm as the subgroupof d fixing {1,2, . . . ,m}. Then dm commutes with m d . So for a kd -module U ,the space of m-fixed points is a dm-submodule of U . Thus we can defineFm : mod-kd mod-kdm</p></li><li><p>D.J. Hemmer / Journal of Algebra 280 (2004) 295312 299by</p><p>Fm(U) := Um = Homkm(k,U) = Homkd(M(m,1</p><p>dm),U).</p><p>The proofs in [4] were motivated by, but did not really use, James work in [8]. In thispaper we will first relate James paper to the functor Fm. It will often be convenient tothink of Fm as first restricting to m dm d , and then taking the largest subspaceon which m acts trivially.</p><p>In order to describe James work, we need some more notation. Fix 1 &lt; m &lt; d and letI(n, d) be the subset of I (n, d) consisting of those d-tuples with idm+1 = idm+2 = = id = 1 and ik = 1 for 1 k d m. James defined an idempotent = i,i, thesum being over distinct elements i,i with i I(n, d). James also defined</p><p>S1 := span{i,j</p><p> i, j I(n, d)} S(n, d).Then S1 is a subalgebra of S(n, d), and James proved [8] that S1 = S(n 1, d m).Thus we have the following (only partly commutative) diagram of functors, where F andG are as above while F and G are the corresponding Schur and adjoint Schur functors forthe smaller symmetric group dm. Also we use J to denote multiplication by followedby restriction to S1.</p><p>mod-S(n, d)</p><p>J</p><p>Fmod-kd</p><p>G</p><p>Fmmod-S(n, d)</p><p>res</p><p>mod-S1F</p><p>mod-kdmG</p><p>Our first theorem is that James functor J is closely related to the fixed-point functor Fm.</p><p>Theorem 2.1. Let U mod-kd and let the functors be as in the diagram above. Then</p><p>Fm(U) = F(J (G(U))).</p><p>Proof. We begin the proof with a well-known lemma:</p><p>Lemma 2.2. Recall that S(n, d) = Endkd (V d ). Then</p><p>(i) V d =(n,d)M as kd -modules.(ii) i,j S(n, d) corresponds to an element in Homkd (M,M) where = wt(i) and = wt(j).</p></li><li><p>300 D.J. Hemmer / Journal of Algebra 280 (2004) 295312(iii) corresponds to projection onto M.(iv) corresponds to projection onto </p><p>(n,d)1=m</p><p>M.</p><p>Proof. To see (i), recall that V d has standard basis {ei := ei1 ei2 eid |i I (n, d)}. Under this identification, M is spanned by the basis vectors {ei | wt(i) = }.Parts (ii) and (iii) are immediate from the description in [3, 2.6a] of the action of i,jon V d . Part (iv) follows from (iii). </p><p>To prove the theorem, let U mod-kd . Let i = (m + 1,m + 2, . . . , d,1,1, . . . ,1) I(n, d) and let e = i,i. Then e is an idempotent, eS1e = kdm, and the Schur functorF is multiplication by e.</p><p>Now G(U) = Homkd (V d ,U) is a left S(n, d)-module with action obtained from theright action of S(n, d) on V d . Thus the action of S(n, d) (and hence of S(n, d) andof S1) is given by precomposing functions. That is if f : V d V d is in S(n, d) andg : V d U is in G(U), then fg = g f : V d U . So Lemma 2.2(iv) gives</p><p>G(U) = Homkd( </p><p>(n,d)1=m</p><p>M,U</p><p>).</p><p>But e is projection onto M(m,1dm), so we get</p><p>eG(U) = Homkd(M(m,1</p><p>dm),U)=Fm(U). </p><p>3. Some general properties of Fm</p><p>In this brief section we collect a few general properties of Fm which will be useful indetermining how Fm acts on specific modules. We first remark that Fm has a left adjointfunctor Gm : mod-kdm mod-kd given by</p><p>Gm(U) := Inddmdm(k U).The functor Gm is exact, but Fm is exact only when m &lt; p, and is left exact in general.</p><p>Thus Fm has higher right derived functors</p><p>RiFm(U) = Extikd(M(m,1</p><p>dm),U).</p><p>A key fact in understanding Fm is that R1Fm vanishes on dual Specht modules.A closely related fact (that the first higher right derived functor of the adjoint Schur functor</p><p>G vanishes on dual Specht modules) played a key role in [5]. Some of the results below</p></li><li><p>D.J. Hemmer / Journal of Algebra 280 (2004) 295312 301require the assumption p &gt; 3. This seems to be similar to what was observed in [5].Specifically when p = 3 it is possible for H 1(d,S) to be nonzero, and this has a dramaticeffect on the results.</p><p>Proposition 3.1. Let p &gt; 3 and d . Then R1Fm(S) = 0.</p><p>Proof. We have</p><p>R1Fm(S) = Ext1kd(M(m,1</p><p>dm), S)= Ext1kd (S,M(m,1dm)).</p><p>But the permutation modules M are direct sums of Young modules, and Ext1kd (S,Y)</p><p>is always zero when p &gt; 3 [10, 6.4b], so the result follows. Proposition 3.1 is false when p = 3. For example if d = 5 and m = 3, then</p><p>R1F3(S(15)</p><p>)= Ext1k3(k,Res3(S(15)))= Ext1k3(k, sgn) = 0.Finally we recall that Fm was used in [4], where its image was determined on the modulesS, S, and D if 1 m &lt; p, i.e., when Fm is exact. In the case when m = 1 &lt; p, thefunctor Fm removes the first row from S, S, and D (i.e., maps S to S, etc.). In thegeneral case considered here (with m arbitrary), we will see thatFm acts as first row or firstcolumn removal only on the twisted modules, namely S sgn, Y sgn, and M sgn.</p><p>4. Fm on dual Specht modules</p><p>In this section we show that Fm behaves very nicely on dual Specht modules S. Wewill need the notions of semistandard -tableaux of type and the basis of M given by-tabloids, which are described in the book [7]. First we need a lemma.</p><p>Lemma 4.1. The dimension of Fm(S) is the number of semistandard -tableaux of type(m,1dm). In particular, if 1 &lt; m, then Fm(S) = 0.</p><p>Proof. Since</p><p>Fm(S) = Homkd(M(m,1</p><p>dm), S)= Homkd (S,M(m,1dm)),</p><p>the result follows from [7, Theorem 13.13], where a basis for Homkd (S,M) isdescribed for arbitrary and . In particular, if 1 &lt; m then there is no such tableau,so Fm(S) = 0. </p><p>Thus we must consider Fm(S) for 1 m. The case 1 = m follows from our work inSection 2 together with work of James:Theorem 4.2. Let 1 = m. Then Fm(S) = S.</p></li><li><p>302 D.J. Hemmer / Journal of Algebra 280 (2004) 295312Proof. James showed in [8] that J (()) = (). So</p><p>Fm(S) = F(J (G(S))) by Theorem 2.1</p><p>= F(J (())) by Proposition 1.1(i)= F(())= S by (1.2). </p><p>In order to describe Fm(S) in general, we must discuss skew diagrams and thecorresponding skew Specht modules. Suppose = (1, 2, . . . , r ) d and = (1,2,. . . ,s) t for t &lt; d . Suppose i i for all i (where i is interpreted as 0 for i &gt; s).Then the skew diagram \ is defined as</p><p>\ := {(i, j) N N 1 i r, i j i}.To each skew diagram there is associated a skew Specht module S\ and its dual, whichwe denote S\. Their construction can be found in [9], where it is shown that S\ has aSpecht filtration, and the filtration multiplicities are determined combinatorially. We willnot describe the details of the construction here, but we will need the following from thepaper of James and Peel.</p><p>Proposition 4.3 [9, Theorem 3.1]. Let d . When restricted to m dm, the moduleS has a chain of submodules with factors isomorphic to S S\ , where each partition of m such that \ exists occurs exactly once. If m 1 (so \(m) exists), the filtrationcan be chosen so that S S\(m) occurs on the top.</p><p>James and Peel constructed a filtration of S, but of course taking duals gives a filtrationof S by modules S S\ with S(m) S\(m) as a submodule. We also need the followingwell-known lemma.</p><p>Lemma 4.4 [7, 13.17].</p><p>Homd (k, S) ={k, if = (d),0, otherwise. (4.1)</p><p>We can now determine Fm(S) in general.</p><p>Theorem 4.5. Let d . Then</p><p>Fm(S) ={</p><p>S\(m),...</p></li></ul>