* Schur-De-Rham complex and its cohomology Marcin Cha_lupnik Instytut Matematyki, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland Abstract We associate to a Young diagram ~ the Schur-De-Rham complex S~. We show that this complex is exact when the p-core of ~ is nontrivial and compute its cohomology when the p-core is trivial and the p-quotient of ~ consists of a single diagram. Keywords: Young diagram; Schur complex; cohomology. 1 Introduction One can associate to a map f : F - ! G of free R-modules the Koszul com- plex by a well known construction. Under some regularity assumptions this complex is a free resolution of an R-module S*(G=im (f)). This construction was generalized in [ABW] by putting it into the context of representation theory (see also [La], [Ni]). The resulting complex: the Schur complex S~(f) (depending additionally on a Young diagram ~) provided a free resolution for a much larger class of R-modules and became a standard tool in repre- sentation theory. The Koszul complex however, turned out to be useful also in the situation when it was exact (eg. for f being an identity map). As an example we may point out the whole string of works concerning the so-called Koszul duality ______________________________ *The author was partially supported by Bia_lynicki-Birula Subsydium of Found* *ation of Polish Science. E-mail address: mchal@mimuw.edu.pl 1 ([BGG], [GKM], [MW] etc.). Another striking application of the Koszul complex (for f = id) appeared in the work [FLS] on homological algebra in the category of functors. The main result of that work was a computation of some Ext-groups in this category. The crucial part of the proof was an analysis of hiperExt spectral sequences with coefficients in the Koszul complex and in the De-Rham complex (which is the same graded space as the Koszul complex but has a differential going into another direction). The ideas of that important paper have inspired many further works on the functor category (eg. [B1], [FS], [FFSS], [K1], [K2], [K3]). In particular, there was established a tight connection between the category of functors and the category of GLn(R)-modules (at least for R being a field). This connec- tion suggests the possibility of generalization of computations of Ext-groups begun in [FLS] by using ideas from representation theory. I started this pro- gram in [C1] where I extended calculations of Ext-groups to a large class of functors important in representation theory. But to push the computa- tions further we need appropriate generalizations of the Koszul and De-Rham complexes. Although it turns out that the Schur complex is exactly this gen- eralization of the Koszul complex we need, it seems, surprisingly enough, that nobody has tried to generalize the De-Rham complex in a similar man- ner. This generalization, which I call the Schur-De-Rham complex is the objective of the present paper. I focus on the most fundamental case of the complex associated to the identity map (this is the case important for the applications I have in mind). Thus our complex is associated to an R-module V and a Young diagram ~ (we denote it by S~(V )). In the parallel case, the "classical" Schur complex is exact but the Schur-De-Rham complex is not. Its cohomology reflects deep properties of the Young diagram (if R is a field of positive characteristic). It turns out that the cohomology of S~ may be nonzero if ~ has a trivial p-core (cf. Fact 4.3). The main result of this paper (Theorem 5.3) describes H*(S~) for ~ with a trivial p-core and p-quotient consisting of a single diagram. This connection between the cohomology of the Schur-De-Rham complex and combinatorics introduced for description of blocks of modular representations was quite surprising for me. I hope it will be developed and better understood in the future. I introduced the Schur-De-Rham complex in order to generalize compu- tations of Ext-groups initiated in [FLS] where the De-Rham complex played an important role. But this complex turned out to be interesting for its own. I hope it will also be useful in other branches of representation theory. For 2 this reason I decided to devote a separate paper to the study of the Schur- De-Rham complex and I tried to make this paper independent of the other parts of my work on the functor category ([C1],[C2]). The applications of the results of the present paper to computations of Ext-groups will appear in [C2]. 2 Basic combinatorics We recall here some standard facts concerning Young diagrams mainly in order to fix notation and terminology. We will usually think of partitions of d (ie. weakly decreasing sequences of positive integers with sum d) as Young diagrams of weight d. The Young diagram of weight d is an arrangement of d squares (boxes) associated to a given partition in the obvious way. Given a diagram ~ we may form its conjugate e~by reflecting the diagram ~ in its diagonal: __________ ________||||| __________|||||| ________ ~ = (4, 3, 1)_______||||| e~= (3, 2, 2, 1)_____|||| ___||| ____|||_|||| We will also consider skew partitions ~=~ for ~ ~ (ie. ~i ~i for all i). We get the Young diagram for a skew partition ~=~ by removing the boxes belonging to ~ from the diagram for ~. For example, for (4, 3, 1)=(2, 1) we get _____ ________|||| _______|||| ___||| Sometimes to emphasize the difference between diagrams nad skew diagrams we will call the former solid diagrams. We say that a skew diagram ~=~ is a disconnected sum of diagrams ff=ff0 and fi=fi0 if the set of boxes of ~=~ is a disjoint sum of the sets of boxes of ff=ff0and fi=fi0, and each box in ff=ff0lies above and to the right of each box in fi=fi0. For example, for diagrams ff = (ff1, . .,.ffk) and fi = (fi1, . .,.* *fil), 3 the (skew) diagram ~=~ = (fi1 + ff1, . .,.fi1 + ffk, fi1, . .,.fil)=(fik1) is t* *heir dis- connected sum __________| || ___| | ff___| _______________||| | | | fi | || _____| _____|| There is a natural total ordering of the set of solid diagrams called the lexi- cographic order. We say that ~ is lexicographically smaller than ~ (notation: ~ < ~) if for the smallest i such that ~i 6= ~i we have ~i > ~i. The last in- equality looks like a misprint. I explained the reason for taking this strange convention in ([C1], sect. 2). The best way to memorize it is to remem- ber that the diagram (d) is the smallest among diagrams of weight d, while (1, . .,.1) (we will denote this diagram by (1d)) is the largest. At last, I would like to distinguish certain types of diagrams which will be important later. We call a diagram of shape (n, 1m ) a hook. We call its first row the arm of a hook, its first column _ the leg of a hook. The last box in the arm is called the hand, and the lowest box in the leg is called the foot. The box in which the leg and the arm intersect is called the corner. I would like to recall a combinatorial algorithm which is important in block theory (see [JK] pp. 75-76). Let ~ be a diagram and a be its box lying in the ith row and the jth column. We consider a set consisting of boxes in the ith row lying to the right of a and boxes in the jth column lying below a. Of course, such an arrangement of boxes looks like a hook with the corner a and the hand and foot lying on the rim of ~. We call such an arrangement of boxes an n-hook in ~, where n is the number of its boxes. Observe that if we remove from ~ a part of rim contained between the hand and foot of some n-hook in ~, we get a new diagram, say ~0 (of course, |~0| = |~| - n, where "| |" means the weight of a diagram). We call this removed part of a diagram a rim n-hook, and we say that ~0was obtained from ~ by removing the rim n-hook. The following picture shows: a diagram ~ with some n-hook marked off, ~ with the corresponding rim n-hook, the diagram ~0 4 |______________ |______________ |______________ | ____________|_ | ___|__ | _____| | |_________a_||||||| | _____|_ | _____| | ||__ ____|_ | ___|_____||||||__ | ___| | ||___|_ | ____|||_ | | ||_ | __|_ | __| |__|_|| |___|_|| |__| We call a diagram an n-core if it does not contain any n-hook. For an arbitrary diagram ~ we may perform the following procedure. We find an n-hook in it and get rid off its rim n-hook. In the obtained diagram we again find some n-hook and remove its rim n-hook etc. We continue the procedure until we get an n-core (perhaps being the empty diagram). It is a highly nontrivial fact that this n-core does not depend on the order of removals of rim n-hooks ([JK], Th. 2.7.16). We call this diagram the n-core of ~ and denote it by c(~). Here are presented two ways of reaching the 3-core of (4, 3) which is (1): __________ _____ ___ |__________|||||7!______||||7! ___||| |________|||| _____|||| ; and the second way __________ __________ ___ |__________|||||7!__________||||||7!___||| |________|||| . 3 The Schur-De-Rham complex Although all definitions given in this article work over any ground commu- tative ring R, some theorems hold (or are interesting) for fields of positive characteristic only. Therefore from now on, the term space will mean a finite dimensional vector space over a fixed field k of positive characteristic p. The Schur-De-Rham complex S~ is a complex of functors. By this we mean that for any space V we have a complex S~(V ) and the construction is natural with respect to linear maps V -! W . In fact it is a complex of öh mogeneous strict polynomial functors of degree |~|" in the sense of [FS]. I will use the language of strict polynomial functors (SP-functors for short) throughout this article because it is best adapted for the applications in [C2* *]. But all results of the present work could be, with only minor changes, stated 5 and proved without appealing to this concept. The reader not familiar with strict polynomial functors may think of them just as functors in a usual sense. In a few places where the difference between these notions is substantial I will explain what is really going on. We now turn to defining the Schur-De-Rham complex. Our construction may be thought of as an analogue of the construction of the Schur functor in the category of complexes. To make this analogy clear I shall briefly recall the construction of the Schur functor S~=~ (cf. [ABW], II.1). Given a skew diagram ~=~ of weight d (ie. d = |~| - |~|), we set ~=~(V ) := ~1-~1(V ) . . . ~l-~l(V ). Then the natural embedding ~=~(V ) -! V d may be seen as an inclusion of the invariants of the alternating action of the horizontal Young subgroup ~=~ of d (see [JK] p. 29) on the d-th tensor power of V (the alternating action is given by the formula: oe.(v1 . . .vd) = (-1)|ff|vff(1) . . .vff(d)). Similarly, we define S~f=~and we have an epimorphism V d -! S~f=~which is a projection to the coinvariants of the permutative action of the vertical Young subgroup ~=~ of d on the d-th tensor power of V . The Schur functor S~=~(V ) is defined to be the image of the composition ~=~(V ) -! V d -! S~f=~(this definition needs a modification for p = 2, I discuss this point in length in ([C1], sect. 3)). It comes with two natural transformations which we call the structural arrows: the epimorphism OE~=~ : ~=~ -! S~=~ (which for ~=~ = (d) yields an isomorphism d ' S(d)) and the monomorphism _~=~ : S~=~ -! S~f=~(which for ~=~ = (1d) gives S(1d)' Sd). To repeat this construction in the category of complexes we take the sequence K(V ) = {0 -! V = V -! 0} regarded as a cohomological com- plex concentrated in degrees 0 and 1 and consider its d-th tensor power (K(V )) d. There is a natural "permutativeä ction of d on this complex (the transposition (j, j + 1) sends an element v1 . . .vj vj+1 . . .vd to (-1)|vj||vj+1|v1 . . .vj+1 vj . . .vd, ( 1 is necessary to obtain an action in the category of complexes)). The Schur-De-Rham complex S~=~(V ) is simply the image of the composition of the maps of complexes (((K(V )) d)alt) ~=~ -! (K(V )) d - ! ((K(V )) d) ~f=~(for any d-module M, Maltstands for M sgn). In fact, this construction is only a slight modification of the one which was known earlier. Namely, it is easy to see that if we apply the above construction to the K(V ) considered as a homological complex we get nothing but the Schur complex introduced in ([ABW], sect. V.1). This purely formal 6 difference in definition, which looks quite innocent, affects the homology dramatically. While the Schur complex is exact (([ABW], Cor. V.1.5), we recall that in the terminology of that paper we deal with a Schur complex associated to the identity map V = V ), the cohomology of the Schur-De- Rham complex is complicated and interesting. We now list some basic properties of the Schur-De-Rham complex. Let us first look at the simplest cases. For ~=~ = (1)d we have S~=~ = (K d) d, ie. the complex of coinvariants of the action. We will denote this complex by Sd(V ) for it is an analogue of the symmetric power of a space. We can identify the ith degree component of K d(V ) with the d-module (V d-i (V i)alt) k[ d-ix i]k[ d]. Hence (Sd)i(V ) = Sd-i(V ) i(V ). It is easy to write down thePdifferential explicitly. We get the formula ffi(x1 . .x.d-i y1 ^ . .^.yi) = d-ij=1x1 . .x.j-1xj+1. .x.d-i yj ^ y1 ^ . .^.yi, which is nothing but the differential in the algebraic De-Rham complex (cf. [FLS], sect. 3): 0 -! Sd(V ) -ffi!Sd-1(V ) 1(V ) -ffi!. .-.ffi! d(V ) -! 0. Let us compare this complex with a corresponding Schur complex. Of course, we have the same graded space but the differentialPchanges direction. It is given by the formula @(x1 . .x.d-i y1^. .^.yi) = ij=1(-1)j-1x1 . .x.d-ixj y1^. .^.yj-1^yj+1^. .^.yi, hence this time we obtained the Koszul complex (see again [FLS], sect. 3). Already at this stage we can see the difference between the homological and cohomological complex. Sd(V ) equipped with the Koszul differential is acyclic, but equipped with the De-Rham has nontrivial cohomology for d di- visible by p. Namely, Cartier's theorem ([Ca]; [FS], Th. 4.1) says that there is an isomorphism of graded functors H*(Spd) ' (Sd)(1), where (-)(1)means the Frobenius twist of a functor. Here is the first place where we use the assumption on k and also the language of strict polynomial functors plays some (positive) role. I shall describe this point in some detail. Perhaps I should begin with explaining what the "Frobenius twist" means in this con- text. The Frobenius twist V (1)of a space V is the same space but with the action of scalars induced by the Frobenius automorphism (c.v := cp. v). The assignment V - ! V (1)is (a strict polynomial) functor and, strictly speaking, F (1)means the composition of functors (-)(1)O F (cf. [FS], sect. 1). Thus of course, this concept makes sense for fields of positive characteristic only (in fact, the De-Rham complex over a field of characteristic 0 is acyclic (=ex- act)). Viewing F (1)as a SP-functor may be sometimes profitable. To see 7 this, let us assume for the moment that k has cardinality p. Then the func- tors F and F (1)are isomorphic, but they are not as SP-functors. Thus the language of SP-functors provides some additional information about func- tors. For example, there is a notion of degree of an SP-functor ([FS], Lemma 2.2) refining the notion of the Eilenberg-LacLane degree of a functor. The principal advantage of the SP-degree is that there are no transformations be- tween SP-functors of different degree (to be precise: between homogeneous functors in the terminology of [FS]). It will be important for us that S~=~ con- sists of (homogeneous) SP-functors of degree |~|-|~| and that the Frobenius twist of a functor of degree d has degree pd, which enables us to compute the SP-degree of functors appearing in Cartier's Theorem. This information, for example, allows to exclude the possibility of existence of some maps as we will see in Section 6. Now I would like to discuss the case ~=~ = (d) which will be even more important than (1d). We will denote the corresponding Schur-De-Rham complex by d(V ). This time in the ith degree component we have d-i(V ) Di(V ), where Di(V ) stands for the i-th divided power of a space ie. Di(V ) = (V i) d for the permutative action. In order to describe the differential it is convenient to use a multiplicative structure on D*(V ). Since this structure is less popular than that on the symmetric power, I shall recall it briefly (cf. [ABW], I.4). Let Y = {y1, . .,.yn} be a basis of V . Then, like in the symmetric power, the set of weakly increasing sequences of elements of Y of length n forms a basis of Dn(V ). There exists a multiplication Dn(V ) Dm (V ) -! Dn+m (V ). The point is, that it does not act on basis sequences just as concatenation, but some_multiplicities!occur when we take a n + m n+m power of an element. Namely: yn .ym = y . Thus for example: n y1y22y4 . y32y3y4 = 20y1y52y3y24. The cohomological differential is given by the P d-i-1 d-i-j formula: ffi(x1 ^ . .^.xd-i yk11. .y.ki0i0) = j=0 (-1) x1 ^ . .^.xj-1 ^ xj+1^. .^.xd-i yj.yk11. .y.ki0i0. It turns out that d is a dual complex to Sd * *in the sense of the Kuhn duality. This duality (known in representation theory as öc ntravariant duality") is a contravariant involution on the category of functors which sends a functor F to F # which is defined by the formula F #(V ) := (F (V *)*. Then it is easy to see that (Sd(V ))i = (( d(V ))d-i)# and that the differentials are mutually dual. Thus the üd al Cartier's theorem" holds: H*( pd) = d(1)[(p - 1)d] (the shift of grading is caused by the fact that the duality üt rns the complex upside down"). Similarly, d equipped 8 with a homological differential is dual to the Koszul complex. Luckily, many basic properties of the Schur-De-Rham complex may be obtained by repeating word for word proofs of the corresponding properties of the Schur complex. The most important for us will be a description of elements of S~=~ in terms of tableaux, which we recall in the moment, but in the forthcoming sections we will also use the Decomposition Formula ([ABW], Th. V.1.13), the Littlewood-Richardson rule [Bo] etc. We now give the promised description of a basis of S~=~(V ). Let X = {x1, . .,.xn} be a basis of a copy of V placed in degree 0 of K and Y = {y1, . .,.yn} a basis of that in degree 1. We order totally the set X [Y putting x1 < y1 < x2 < . .<.yn and assume that the differential in K takes xi to yi. By a tableau of shape ~=~ with values in X [ Y we mean a function from the set of boxes of ~=~ to X [ Y . We say that a tableau is row Y -standard if we have in each row a nondecreasing sequence and repetitions occur only among elements of Y , we say it is column Y -standard if we have in each column a nondecreasing sequence and repetitions occur only among elements of X. A tableau whichLis both row and column Y -standard is called Y -standard. Since d(V ) = di=0 d-i(V ) Di(V ), the set of row Y -standard tableaux of shape ~=~ with values in X [ Y forms its basis. The main result of [ABW] (Th. V.1.10) says that the the structural epimorphism OE~=~ : ~=~(V ) -! S~=~(V ) takes the set of Y -standard tableaux of shape ~=~ with values X [Y to a basis of S~=~(V ). This theorem was proved by establishing an effective algorithm called the Straightening Formula allowing to express any tableau (regarded as an element of S~=~(V )) as a linear combination of Y -standard tableaux. We will use this algorithm in the next section. 4 Cohomology: the first steps We start with the simplest diagrams. We say that a skew diagram ~=~ is a skew hook if it is connected (ie. it cannot be presented as a disconnected sum of nontrivial diagrams) and it does not contain a subdiagram isomorphic to (2, 2). The set of boxes of a skew hook may be totally ordered in an obvious manner: we say that a box (~=~)ij (ie. belonging to the ith row and jth column) is greater than (~=~)i0j0if i < i0 or j > j0. The smallest box of a skew hook will be called the foot while the largest _ the hand. Of course, our terminology agrees with that concerning hooks and rim hooks, which are also skew hooks. The following easy observation explains the importance of 9 this class of diagrams. Fact 4.1 If dim (V ) = 1, then S~=~(V ) 6= 0 if and only if ~=~ is a discon- nected sum of skew hooks. Proof: It is easy to see that there are no Y -standard tableaux of shape (2, 2) with values in the set X [ Y = {x, y}. Thus if ~=~ contains (2, 2) then S~(V ) = 0 (we recall that x spans 1(V ) and y spans D1(V )). But a diagram does not contain (2, 2) if and only if it is a disconnected sum of skew hooks. To conclude the proof we should construct a Y -standard tableau for any disconnected sum of skew hooks. Since a Schur-De-Rham complex associ- ated to a disconnected sum is a tensor product of Schur-De-Rham complexes associated to factors, it suffices to consider the case when ~=~ is a skew hook. In order to construct a tableau we will fill subsequent boxes of ~=~ with x or y going from the foot to the hand according to the total order. We start by putting x or y to the foot. Observe that in the next box the condition of Y -standardness already determines an element we should put. Namely if this box lies above the foot then we must put x while if it lies to the right of the foot then we put y. Turning to the next box we must use this rule again etc. Thus we see that there exist exactly two Y -standard tableaux of shape ~=~ with values in {x, y} which only differ by an element put to the foot. We denote them by (x) and (y) respectively: ___ ___ |x| |x| |_|_ |_|_ ________||_x ________||_x (x) = |x|y |y| (y) = |x |y|y| |_|_____|||||||__ |__|____|||||||__ |_|_x |__|x |x| |y | |_|__ |__|_ || The next natural question concerns the behaviour of a differential for a one- dimensional space. The following fact is responsible for the complexity of the cohomology of the Schur-De-Rham complex. Fact 4.2 For any skew hook ~=~ of weight n, we have ffi( (x)) = n (y). Proof: Let us first recall the action of ffi in terms of tableaux. Writing down a tableau is just taking the preimage of an element in ~=~ = ~1-~1 . . . ~l-~l.Thus it suffices to remember the formula for the differential in 10 k, which is ffi(x yk-1) = kyk. Throughout the proof we will identify a tableau t of shape ~=~ with the sequence (t1, . .,.tl) consisting of tableaux filling successive rows of ~=~ (l is the number of rows of ~=~). It is easy to see thatPwe associate to (x) the sequence ( (x), . .,. (x)). Therefore ffi( ) = lj=1(-1)h(j)(~j - ~j)sj, where sj stands for the tableau ( (x), . .,. (y), . .,. (x)) in which (y) is placed in the jth row and h(j) is a sum of degrees of elements standing in the earlier rows. This sum is equal to the number of appearings of y in the earlier rows or to the number of columns lying to the left of the hand of the jth row. Thus we got a sum of exactly n tableaux (with different signs). The proof will be finished if we show that all factors in this sum are in fact equal in S~=~(V ). But observe that if we apply the Straightening Formula ([ABW], p. 264) to the jth and (j + 1)th row of ~=~, we get the ralation sj+1 = (-1)~j-~j-1sj. Therefore (-1)h(j+1)sj+1 = (-1)h(j)sj, which completes the proof. || These elementary observations lead to a general necessary condition for the cohomological nontriviality of the Schur complex. To express it we should slightly generalize the notion of the n-core of a diagram. Namely, we call a n-hook in a skew diagram ~=~ any n-hook in ~ whose rim n-hook is contained in ~ \ ~ (we do not demand the original hook to be contained in ~ \ ~). We alert the reader that the notion of the n-core of a skew diagram is not well defined. for example, by removing rim 2-hooks from the digram (3, 3)=(2) we may get the empty diagram ___ _________|||7! _____ 7! ________||||| _____|||| ;, or the diagram (3, 1)=(2) ___ ___ _________|||7! ___ ___||| ________||||| ___||| . Nevertheless, we say that ~=~ has a trivial n-core if it is possible to reach the empty diagram by the removing rim n-hooks (hence, according to our definition, the diagram (3, 3)=(2) has a trivial 2-core). Fact 4.3 If H*(S~=~) is nontrivial then ~=~ has a trivial p-core (we recall 11 that p is the characteristic of our ground field). Proof: We proceed by induction on dim (V ). If dim (V ) = 1 then, according to Fact 4.1, the only diagrams for which our claim is nontrivial are discon- nected sums of skew hooks. Therefore, thanks to Fact 4.2, we should only show that a skew hook of weight divisible by p has a trivial p-core. Thus let us take such a skew hook and try to find a rim p-hook in it. When we take the first p boxes (counting from the foot), then they form a rim p-hook if and only if the (p + 1)th box lies above the pth. But if the (p + 1)th box lies to the right of the pth, then it can be a foot of a hook in ~=~. Thus, the boxes from the (p + 1)th to the (2p)th form a rim p-hook if and only if the (2p + 1)th lies above the (2p)th. If we are still unlucky then we can try the next p boxes etc. In the worst case it will turn out that the last p boxes form a rim p-hook. Thus we see that any skew hook of weight divisible by p contains a rim p-hook. Moreover, it follows from the construction that this rim p-hook may be chosen in such a way that after removing it our skew hook breaks up into two skew hooks of weights divisible by p. This finishes the proof in the case dim (V ) = 1. We now assume our assertion for all spaces of dimension smaller than dim(V ). We take V = W L where dim (L) = 1. By the Decomposition Formula ([ABW], Th. V.1.10) we have a filtration on S~=~(W L) with the associated graded object M Sff=~(W ) S~=ff(L). ~ ff ~ It suffices to show that all factors in the sum are acyclic. We may restrict our attention to the factors where ~=ff is a disconnected sum of skew hooks of weights divisible by p, since otherwise S~=ff(L) is acyclic. We will show that in such a situation Sff=~(W ) is acyclic. Thanks to the induction assumption it suffices to show that ff=~ has a nontrivial p-core. But observe that ff=~ may be obtained from ~=~ by removing rim p-hooks, since ~=ff has a trivial p-core (we use here the obvious fact that a p-hook in ~=ff is also a p-hook in ~=~). Thus if ff=~ would have a trivial p-core then the same would hold for ~=~. This contradiction completes the proof. || We finish this section by examining the simplest situation where the coho- mology is nontrivial. Fact 4.4 H*(S(k+1,1p-k-1)= S(1)(1)[k]. 12 Proof: This fact may be easily derived from Fact 4.2 and the Decomposition Formula. The proof which will be presented is slightly more complicated but gives some additional information which will be useful later. We proceed by induction on k. For k = 0 we get Cartier's Theorem. In order to make the induction step we apply the Littlewood-Richardson rule (see [Bo]) to the complex k Sp-k. We get the short exact sequence 0 -! S(k+1,1p-k-1)-! k Sp-k -! S(k,1p-k)-! 0, in which the middle term is acyclic by Cartier's Theorem. Thus we get the required shift of the cohomological grading. || We now derive some consequences of the proof. Namely we would like to obtain a more explicit description of the cohomology. We start with the case k = 0. Then the isomorphism I(1)' H*(Sp) is realized by the formulae x 7! xp, y 7! xp-1 y (see [FLS], p. 520), where the second formula be- comes linear only after taking cohomology. Observe that in the language of tableaux this is the map which we defined at the beginning of the section. To describe explicitly the cohomology of other hooks we recall that the connecting homomorphism in the sequence which we used in the proof ffik,k-1: H*(S(k,1p-k)) -! H*+1(S(k+1,1p-k-1)) is an isomorphism. Com- puting its values directly we get the following relations: ffik,k-1( k-1(x)) = k k(x), and ffik,k-1( k-1(y)) = (-1)kk k(x), where k is a map which sends an element to a tableau of shape (k + 1, 1p-k-1). Thus the composition ffik,k-1O . .O.ffi1,0O 0 yields the isomorphism k : I(1)[k] ' H*(S(k+1,1p-k-1* *)), which, up to a nonzero scalar factor (depending on k and cohomological degree), is equal to k. 5 Cores, quotients and enlargement of a dia- gram We now should look closer at diagrams with a trivial p-core. As we saw in Facts 4.2 and 4.3, in a sense, cohomology is concentrated in rim p-hooks. Thus we should understand how a diagram may be divided into rim p-hooks. A combinatorial structure which controls this process is p-quotient. A Young diagram ~ is determined by its p-core c(~) and the ordered family od dia- grams {q0(~), . .,.qp-1(~)} called the p-quotient of ~. When we remove from a diagram a rim p-hook then we do not change its core but we remove one box 13 from its quotient. The precise algorithm may be found in ([JK], sect. 2.7). For our purposes the most important thing will be to recognize from which diagram in the quotient we should remove a box. The answer is that if the hand of a hook belongs to the ith row and jth column then we choose the diagram qk(~), for k = (j - i)(mod p ) (this number is called the p-residue of a box). Let us illustrate this algorithm by a simple example. We consider the diagram ~ = (2, 2) for p = 2. Then c(~) = ;, q0(~) = q1(~) = (1). According to what was just said, there should be two ways of reaching the empty diagram from (2, 2). The first when we start with removing the (only) box from q0(~) and the second when we start with q1(~). Indeed: in the first case we remove from (2, 2) the second row (whose hand has residue 0) and we are left with the diagram (2) which is a 2-hook with a hand of residue 1. In the second case we do the same with the columns of ~. Already this ex- ample suggests that the cohomology of the Schur-De-Rham complex should be particularly simple for diagrams with a quotient consisting of only one nonempty diagram. From now on we focus on this particular situation. We start with introducing another bit of notation. For a given diagram ~ we will denote by Fk(~) the diagram with qk(Fk(~)) = ~ and c(Fk(~) = q0(Fk(~)) = . .=.qk-1(Fk(~)) = qk+1(Fk(~)) = qp-1(Fk(~)) = ;. Our first task will be to understand how Fk(~) looks like, which is not at all clear relying on its definition from [JK] only. We first consider the case of a hook ~ = (n, 1m ). Then it is easy to check (eg. with the aid of a "star diagram" ([JK], p. 85)) that Fk(~) = (p(n - 1) + k + 1, 1p(m+1)-k-1). Moreover, in the case of hooks, the process of removing of rims p-hooks is, in a sense, uniquely determined. Assume first that ~ is not of the form (n) or (1m ). Then there are two p-hooks in Fk(~): the first consists of the first p boxes (counting from the foot) in the leg of Fk(~) and corresponds to the foot of ~, the second consists of the last p boxes in the arm of Fk(~) and corresponds to the hand. Thus of course, we may remove rim p-hooks in many different ways (until a p-quotient becomes one-rowed or one-columned). Uniqueness means the following: if we divide the set of boxes of Fk(~) into p-elemented subsets consisting of successive rim p-hooks, then we always get the same family of subsets (no matter in which order we removed boxes from the p- quotient). Moreover, to a given box in ~ there corresponds the same set of boxes in Fk(~). Thus we get a bijection between the set of boxes in ~ and some family of disjoint p-elemented subsets of the set of boxes of Fk(~). It is also easy to see that in this bijection we assign to the boxes from the leg of Fk(~) (except the corner) the subsets in Fk(~) of shape (1)p, to the boxes 14 from the arm (except the corner again) the subsets of shape (p), and finally we assign to the corner of ~ the subset of shape (k + 1, 1p-k-1) = Fk((1)). Here we draw F0((2, 1)), F1((2, 1)), F2((2, 1)) for p = 3, dividing diagrams into appropriate arrangements of boxes |__|______ |____________ _______________ | |______|_ | __|______|_ |______|______|_ | | | | | | | |__| | | | | | | | | |__| | | | | | | | | | | | | | | |_|_ | | |__| | | |__| Let us turn to the case of an arbitrary diagram ~. We say that ~ has the principal diagonal of length e if ~e e but ~e+1 e. We assign to ~ a sequence of hooks (Ø 1, . .,.Øe) in the following manner. We take as Ø1 a hook consisting of the first row and the first column of ~, as Ø2 a hook consisting * *of the first row and the first column of ~\Ø1 etc. It is clear that after e steps * *we are left with the empty diagram. We call this procedure the decomposition of a diagram into hooks. The picture presents the decomposition of (5, 4, 3, 1) into hooks |____________ | __________| | | | | _____| | |__|_| | | |_|_ This point of view allows to describe Fk(~) in a very convenient way Fact 5.1 Let (Ø 1, . .,.Øe) be the decomposition of ~ into hooks. Then the sequence (Fk(Ø 1), . .,.Fk(Ø e)) is the decomposition into hooks of Fk(~). To prove this claim it suffices to draw a "star diagram" ([JK] p. 85) for a diagram with decomposition into hooks (Fk(Ø 1), . .,.Fk(Ø e)). || Moreover, we observe that we still have a bijection between the set of boxes in ~ and certain dissection of the set of boxes in Fk(~) into p-elemented subsets. We call this dissection the slicing of Fk(~), and we call these p-elemented subsets the slices. We obtain this bijection just by assembling the bijections for all hooks in the decomposition of ~ into hooks. Again the best way to understand how it works is to draw a picture. Here we have the slicings of 15 the diagrams F0((2, 2)), F1((2, 2)), F2((2, 2)) for p = 3 |__|______ |____________ _______________ | |______|_ | __|______|_ |______|______|_ | | | | | | | | |__| ___| | |______|_ | | | | | | | |__| | | |_|_ | | | | | | | | | | |_|_ | | |_|_ | | |__| | | |__| Similarly to the case of hooks, the set of slices (and also the correspondence between the slices in Fk(~) and the boxes in ~) does not depend on the order of removals of rim p-hooks. This time however, to avoid any confusion, I make this statement precise: Fact 5.2 Let Fk(~) = ff0 ff1 . . .ffd = ; be a sequence of diagrams in which each diagram ffj is obtained from ffj-1 by removing a rim p-hook. Let ~ = fi0 fi1 . . .fid = ; be the corresponding sequence of p-quotients. Then for every 0 j < d the set of boxes ffj \ ffj+1 is the slice associated to the box fij \ fij+1 in the bijection we just described. Proof: In order to understand that this fact requires any proof one should recall the diagram (2, 2) which was not of the form Fk(~). For that diagram, as we remember, a different order of removing boxes from the p-quotient led to different "sets of slices" (which consisted either of rows or of columns of the diagram). But in our situation, thanks to detailed knowledge about the structure of a diagram Fk(~), the proof is easy. In fact, the only nontrivial thing is to check that each rim p-hook in Fk(~) is a slice. But this follows immediately from the fact that the lengths of two consecutive rows whose ends lie above the principal diagonal of Fk(~) differ by -1(mod p ), and the analogous fact for columns with ends above the diagonal. || Thus the operation Fk may be thought of as a kind of p-times enlargement of a diagram, for we replace each box of ~ by a p-hook. Taking into account Fact 4.4 it is quite reasonable to expect the following description of the cohomology of the Schur-De-Rham complex Theorem 5.3 For any skew diagram ~=~ and 0 k < p H*(SFk(~)=Fk(~)) = S(1)~=~[hk(~=~)], 16 where the shift of grading is given by the formula hk(~=~) = (p - 1)f~=~ + ke~=~, where e~=~ is the number of boxes of ~=~ lying on the principal diagonal and f~=~ is the number of boxes of ~=~ lying above it. Unfortunately the proof of this theorem is quite involved and undirect. We must start by better understanding relationship between enlargement of a diagram and the Decomposition Formula. 6 Homological Decomposition Formula and compatible families of transformations Since some formulae in this section will be quite complicated, I would like to simplify notation. Namely from now on, we will denote skew diagrams just by ~ etc. not specifying, if not necessary, a subdiagram we divide through. For example, the expression ff ~ is an abbreviation for saying that we have skew diagrams ~=~, ff=~ for which ff ~. Also eg. ~j means in fact the skew diagram (~1, . .,.~j)=(~1, . .,.~j-1, ~j). To get used to this convention we recall the Decomposition Formula ([ABW], sect. II.4) for a skew diagram ~. It is an SP-filtration {Mff}ff ~of a functor in two variables S~(V W ) with the graded object M Sff(V ) S~=ff(W ). ff Here we made further simplification of the notation not writing that the sum is taken over ff contained in ~. We will assume it tacitly whenever in our formulae skew diagrams ~=ff appear. The ordering in the filtration is the lexicographic order. In general this filtration does not split. Nevertheless, dividing it into parts of different SP-degree we get a splitting in the category of bifunctors M S~(V W ) = jS~(V W ), 0 j |~| (of course, |~| means the weight of a skew diagram) and each jS~(V W ) has a filtration with the graded object M Sff(V ) S~=ff(W ). |ff|=j 17 Here we benefit from the fact that there are no transformations between homogeneous SP-functors of different degrees. To prove the existence of this splitting without appealing to SP-functors one should observe that the filtration is defined over the integers and investigate the effect of extension of scalars. I leave the details to the (interested) reader. From this splitting we derive useful consequences: if ff is the lexicograph- ically smallest subdiagram of a given weight in ~, then we have a transfor- mation Sff(V ) S~=ff(W ) -! S~(V W ), and similarly, if fi is the largest subdiagram of a given weight in ~, then we have S~(V W ) -! Sfi(V ) S~=fi(W ). We now turn to diagrams enlarged by the operation Fk. Also here we will use a simplified notation. First, recall that we have not considered the notion of the p-quotient of a skew diagram. Thus when ~ is skew (ie. "~ = ~=~") then Fk(~) stands for the diagram Fk(~)=Fk(~). Next, by Fk(~)we do not mean (Fk(~))1 . . . (Fk(~))lwhich complex never appears in our consid- erations but rather SFk((~1)) . . .SFk((~l)). I should alert the reader that even if ~ is solid, the diagram Fk((~1)) need not to be a horizontal hook (for k < p - 1), and that Fk((~j)) for j > 1 is just a skew hook. The picture presents F1((2, 2)) for p = 3 divided into skew hooks corresponding to the rows of (2, 2) |____________ | __________| | |_|_ __|_ | | ___| | | | | |_|_ We take a similar convention for SFk(e~)(one should also remember that Fgk(~)= Fp-1-k(e~)). Let us look at the Decomposition Formula for SFk(~), but we will be interested only in factors cohomologically nontrivial. Notice a simple but important combinatorial fact. Fact 6.1 If fi Fk(~) is such that Fk(~)=fi has a trivial p-core then fi = Fk(ff) for some ff ~. 18 Proof: Readily, it suffice to consider the situation when ~ is solid. Then, since Fk(~)=fi has a trivial p-core, we may obtain fi from Fk(~) by removing rim p-hooks. Therefore fi must, like Fk(~), have a trivial p-core and the p-quotient contained in the p-quotient of Fk(~). This completes the proof. || Thanks to this fact and Fact 4.3, if we neglect acyclic factors in the Decom- position Formula for SFk(~), we get the graded object labeled by the same set of diagrams as in the formula for S~ (Fk preserves the lexicographic order): M M SFk(ff)(V ) SFk(~=ff)(W ), 0 j |~||ff|=j but one should remember that the cohomology of the total object need not to have a filtration with the quotients being the cohomlogy of the quotients of the filtration, because the spectral sequence of the filtration may have nontrivial differentials. Next, we observe that if ff is the (lexicographically) smallest subdiagram of a given weight in ~, then we still have a well defined transformation H*(SFk(ff))(V ) H*(SFk(~=ff))(W ) -! H*(SFk(~))(V W ), and an analogous map exists for the largest subdiagram. The main difficulty in the proof of Theorem 5.3 is to construct a transfor- mation ~ : S(1)~[hk(~)] - ! H*(SFk(~)) with some good properties . Once we have it at our disposal we will prove in rather formal manner that it is an isomorphism. We postpone its construction to the next section. Now, we make it precise what we mean by ög od propertiesö f a transformation and how to use them to prove Theorem 5.3. We will prove that the transformation is an isomorphism inductively us- ing the Decomposition Formula. Therefore we should, together with ~, define transformations ff, ~=fffor all ff ~ in a way compatible with the Decomposition Formula. The following definition extracts properties of transformations we need for a proof. Definition 6.2 A family of transformations ff=ff0: S(1)ff=ff0[hk(ff=ff0)] -! H*(SFk(ff=ff0)), defined for all pairs ff, ff0 such that ff0 ff ~ is called a compatible fami* *ly of transformation for ~, if it satisfies for all ff0 ff00 ff the following conditions: 19 o There exists a morphism e ff=ff0,ff00: M*(1)ff00(V W )[hk(ff=ff0)] -! H*(* *MFk(ff00))(V W ) making the diagram e ff=ff0,ff00 M*(1)ff00(V W )[hk(ff=ff0)]___-H*(MFk(ff00))(V W ) X X X XX X | X X X | ff=ff0|M*(1) X XX | ff00 X X XXz ||? H*(SFk(ff=ff0))(V W ) commutative. o Therefore if ff000is the lexicographic predecessor of ff00, then we obtain the commutative diagram e ff=ff0,ff000 M*(1)ff000(V W )[hk(ff=ff0)]__________- H*(MFk(ff000))(V W ) | | | | |? e |? ff=ff0,ff00 M*(1)ff00(V W )[hk(ff=ff0)]___________- H*(MFk(ff00))(V W ) | | | | |? |? S(1)ff00=ff0(V ) S(1)ff=ff00(W )[hk(ff=ff0)]H*(SFk(ff00=ff0))(V ) H*(* *SFk(ff=ff00))(W ), which may be completed to the commutative diagram by exactly one bottom arrow. We requires this arrow to be ff00=ff0(V ) ff=ff00(W ). The meaning of the first condition is that a compatible family takes the filtration M*(1)ff00[hk(ff=ff0)] to H*(MFk(ff00)). But since we do not know a p* *riori whether the map induced on the cohomology by the embedding M*Fk(ff00)-! SFk(ff=ff0)is a monomorphism we were forced to express this condition in such an awkward way. We should also remember that the second column in the diagram in the second condition need not to be a short exact sequence. But the existence and uniqueness of the bottom arrow follows easily by a diagram chasing. Also, perhaps at first glance it is not clear why we introduced yet another diagram ff00. The reason is that it makes our definition hereditary ie. a compatible family for ~ restricts to a compatible family for any ff=ff0 such that ff0 ff ~. 20 Now we show, as we have promised, that the very existence of a compat- ible family is almost sufficient to prove Theorem 5.3. Lemma 6.3 If in a compatible family for ~ all transformations ff=ff0for ff0 ff ~ such that ff=ff0 consists of a single box are isomorphisms, then ~ is an isomorphism. Proof: We proceed by induction on dim (V ). Let us first assume that dim(V ) = 1. Since Fk(~) is a skew hook if and only if so is ~, we may assume that ~ is a skew hook. We start another induction, this time with respect to the weight of diagrams. More precisely: we are going to show that ff=ff0(V ) is an isomorphism inductively on the weight of ff=ff0. Thanks to the assumption of the lemma we may begin our induction. We consider an arbitrary ff=ff0 assuming an isomorphism for all smaller diagrams. Since the definition of a compatible family is hereditary, we may assume that ff=ff0= ~ which simplifies notation. Now, I shall describe some general construction which will be used repeat- edly. We say that ~ = ff|vfi if the diagram ff consists exactly of these boxes of ~ which belong to at most jth column for some number j, while fi con- sists of these which belong to at least (j + 1)th. For example (4, 4, 1)=(3) = (3, 1)|v(12): ___ ___ __________||__ _______ ||_|| |_|_______|||||7! |_|____|||| |__|_ | | | | |_|__ |_|__ In such a situation there exists an embedding S~ -! Sff Sfi. To see this, consider a diagram ~ - ! Se~ # k ff fi - ! Seff Sefi, in which the horizontal arrows are compositions of structural arrows in re- spective Schur complexes, while the left vertical arrow is a comultiplication in the Hopf algebra * (see [ABW], sect. V.1). Thus we see that the im- age of the top arrow (ie. S~) is contained in the image of the bottom one (ie. Sff Sfi), which yields the required embedding. Analogously, we write 21 ~ = ff|hfi if ff consists of the first j rows of ~, and fi of the rest. In this situation there exists an epimorphism Sff Sfi-! S~. It will turn out to be plausible to describe these transformations in terms of the Decomposition Formula. Observe that if ~ = ff|vfi, then ff is the largest subdiagram of a given weight in ~. Since ~=ff = fi, we have a morphism S~(V W ) - ! Sff(V ) Sfi(W ). It is easy to check on tableaux that if we compose this morphism with a map S~(V ) - ! S~(V V ) induced by the diagonal we get the map S~ - ! Sff Sfiwhich we defined earlier in terms of structural maps. The drawback of this new description is that it is not clear from it that the map is a monomorphism. In a similar manner we can interpret the epimorphism Sff Sfi- ! S~ when ~ = ff|hfi. The only differences are that this time we have the smallest diagram and we compose with a map induced by an addition V V - ! V . We come back to the proof of the lemma. Let us first assume that ~ consists of at least two columns. If so, we can write it as ~ = ff|vfi where fi is the l* *ast column of ~. Thus we have a monomorphism S~(V ) -! S~(V V ) -! Sff(V ) Sfi(V ). Now we will try to lift this morphism to the level of Fk. Observe that Fk(~) = Fk(ff)|vFk(fi). It is because the foot of a slice corresponding to the first box in the last column of ~ lies to the right of the hand of a slice corresponding to the last box in the last but one column (we count boxes from the foot to the hand). Hence we get the composition SFk(~)(V ) -! SFk(~)(V V ) -! SFk((ff)(V ) SFk(fi)(V ). We apply cohomology to this sequence and consider the diagram ~ S(1)~(V )[hk(~)]_________________________- H*(SFk(~))(V ) | | | | |? |? ~ S(1)~(V V )[hk(~)]_____________________-H*(SFk(~))(V V ) | | | | |? |? ff fi S(1)ff(V ) S(1)fi(V )[hk(~)]________-H*(SFk(ff))(V ) H*(SFk(fi))(V ). This diagram commutes: the commutativity of the first square follows from the naturality of ~, and the commutativity of the second from the compati- bility of a family. Since the bottom arrow is an isomorphism by the induction 22 assumption and the composition of the left vertical arrows is monic, ~(V ) is monic. But by Fact 4.2, the dimensions of the source and target are equal. Hence ~(V ) is an isomorphism. Thus we finished the proof in the case dim(V ) = 1 for a diagram consisting of at least two columns. To cover all diagrams of weight greater than 1 it suffices to consider the symmetric case of a diagram consisting of at least two rows. We argue in a similar manner. This time we write ~ in the form ff|hfi, where fi is the last row of ~. Then we consider the commutative diagram ff fi S(1)ff(V ) S(1)fi(V )[hk(~)]________-H*(SFk(ff))(V ) H*(SFk(fi))(V ) | | | | |? |? ~ S(1)~(V )[hk(~)]_________________________-H*(SFk(~))(V ). This time the situation is slightly more complicated, because in order to conclude that the bottom arrow is onto (hence iso) we should show that the right vertical arrow is epic. But according to Fact 4.2, the complexes under consideration have trivial differentials. Therefore the right vertical arrow is just the epimorphism SFk(ff)(V ) SFk(fi)-! SFk(~)(V ), provided by the decomposition Fk(~) = Fk(ff)|hFk(fi). This completes the proof for a one-dimensional space V . We now turn to the induction step (with respect to dim(V )). Let V = W L. We shall show by induction on the lexicographic order among ff contained in ~ that ~|M(1)ffinduces an isomorphism: M*(1)ff(W L) ' H*(MFk(ff))(W L). We can start this induction thanks to the assumption of the external induc- tion (with respect to dimension). Let ff0 be the lexicographic predecessor of ff. We consider the commutative diagram 0- !M*(1)ff0(W L)[hk(~)] -! M*(1)ff(W L)[hk(~)] -! S*(1)ff(W ) S*(1)~=ff(* *W )[hk(~)]- !0 # ~|M*(1) # ~| *(1) # ff ~=ff ff0 Mff H*(MFk(ff0))(W L) -! H*(MFk(ff))(W L) -! H*(SFk(ff))(W ) H*(SFk(~=ff))(W * *). The top row in this diagram is exact and the left and right vertical arrows are isomorphisms by the induction assumption. Therefore the right bottom arrow is epic. But we have in the bottom row a long exact sequence. Thus 23 we have just shown that this long exact sequence splits into short exact sequences. Hence by the Five-Lemma, the middle vertical arrow is an iso- morphism. This completes the proof of Lemma 6.3. || 7 Construction of a compatible family Thus our task is to construct a compatible family satisfying the assumptions of Lemma 6.3. We start with a special case when ~ consists of a single row. Let Ø1, . .,.Ød be the set of slices of Fk(~) ordered from the one corresponding to the foot of ~ to the one corresponding to the hand (I recall that Fk(~) may be a skew diagram). Since Fk(~) = Ø1|vØ2|v . .|.vØd, we have the embedding SFk(~)- ! Sffl1 . . .Sffld. The required transformation ~ will close (ie. make commutative) a diagram d[hk(~)] -! I(1)[h(Ø1)] . . .I(1)[h(Ød)] | | |? H*(SFk(~)) -! H*(Sffl1 . .S.ffld). in which the bottom arrow is the tensor product h(ffl1) . . . h(ffld)of isomorphisms h(ffls): I(1)[h(Øs)] -! H*(Sffls) described in remarks after the proof of Fact 4.4 (h(Øs) is equal to the number of columns of Øs minus 1). To close this diagram we should understand the behaviour of its bottom arrow H*(SFk(~)) - ! H*(Sffl1 . .S.ffld). To describe the image of this map we endow H*(Sffl1 . .S.ffld) with a structure of permutative d-module pulling it by the isomorphisms h(Ø1) ... h(Ød) I d(1)' I(1)[h(Ø1)] . . .I(1)[h(Ød)] -! H*(Sffl1 . .S.ffld). These isomorphisms shift the grading but we regard them as morphisms in the ungraded category. Thus when we consider the alternating action on H*(Sffl1 . .S.ffld), then, independently of the parity of h(Øi), we always get (H*(Sffl1 . .S.ffld)) d = d(1)[hk(~)]. This subcomplex turns out to be the desired image. We show this even in a slightly greater generality. Lemma 7.1 Let ff = Ø1|v . .|.vØd for p-hooks Ø1, . .,.Ød placed in such a way that the foot of the next hook lies to the right of the hand of the previous 24 one (in such a situation we say that the hooks are placed horizontally). Then the map Sff-! Sffl1 . . .Sffldinduces an isomorphism H*(Sff) ' (H*(Sffl1 . . .Sffld)) d, with respect to the alternating action on H*(Sffl1 . .S.ffld) which we just described. Proof: We proceed by induction on the number of rows of ff. When ff = (pd), then Sff= pd is the Kuhn dual of the De-Rham complex. Thus our description follows from Cartier's Theorem. Now we turn to the induction step. Let ff1 be the first row of ff and let ff2 = ff=ff1. Then ff = ff1|hff2. * *We form a new diagram: ff0 = ff2|vff1 in such a way that the foot of ff1 lies to the right of the hand of ff2: _____ _____ _________|||| _______ _____|||| ____________ |_|____|||| |_|____|||| |_|_________|||||| | | | | | | |_|__ |_|__ |_|__ ff ff2 ff1 ff0 We consider the sequence 0 -! Sff0-! Sff1 Sff2-! Sff-! 0. We shall show that this sequence is exact. The fact that the composition of arrows is trivial may be checked directly on tableaux. Therefore it suffices to check that the sum of dimensions of the first and the last term is equal to the dimension of the middle term. To see this, we use the Littlewood- Richardson rule (cf. [Bo]). According to it, we should check that the number of Yamanouchi's words of shape fi being a disconnected sum of ff1 and ff2 of a given content is equal to the sum of the number of Yamanouchi's words of shape ff and the number of Yamanouchi's words of shape ff0 (of this given content). To this end, let us take a word of shape fi. If the number placed in the hand of ff2 is greater than or equal to the number placed in the foot of ff1, then we can associate to it in the obvious manner a word of shape ff of the same content. Otherwise (if this number is smaller), we can form 25 a word of shape ff0. It is easy to see that in this way we get a bijection between considered sets of Yamanouchi's words. This finishes the proof of the exactness of the sequence. We now observe that the weight of ff1 cannot be divisible by p, because the slices of ff are placed horizontally. Hence the complex Sff1is acyclic. Let |ff1| = sp + j for some 0 < j < p. We have a morphism of short exact sequences with acyclic middle terms 0 0 # # Sff0 -! Sffl1 . . .S(ffld-s)0 . . .Sffld # # Sff1 Sff2 -! Sffl1 . . . j Sp-j . . .Sffld # # Sff -! Sffl1 . . .Sffld-s . . .Sffld # # 0 0, where arrows in the right column come from the sequence 0 -! S(ffld-s)0-! j Sp-j -! Sffld-s-! 0 inducing shift in grading of the cohomology of hooks (see remark after Fact 4.4), tensored with identities. Taking the connecting homomorphisms in the long sequences of cohomology we get the commutative diagram H*(Sff) - ! I[h(Ø1)] . . .I[h(Ød-s)] . . .I[h(Ød)] # @ # @0 H*(Sff0)[1]- ! I[h(Ø1)] . . .I[h(Ø0d-s)] . . .I[h(Ød)], in which the bottom row, by the induction assumption, has the desired de- scription. The proof is concluded by the observation that the right vertical arrow is, up to sign, an isomorphism of d-modules (for @0 = id . . . ffij+1,j . . .id and j+1 = ffij+1,jO j). || Let us observe that Lemma 7.1 enables us to close (in the unique way) the diagram defining ~. Moreover, the proof of Lemma 7.1 provides an explicit description of ~ in terms of tableaux. A Y -standard tableau of shape ~ is sent (up to a nonzero scalar factor depending on cohomological degree and shapes of slices) to a Y -standard tableau of shape Fk(~) constructed in a 26 following way: if in a given box in ~ there is an element z, then in a slice corresponding to this box we put tableau (z). It follows immediately from this description that { ~} where ~ ranges over the set of all one-rowed dia- grams form a compatible family of transformations for any such a diagram. In fact, we proved in Lemma 7.1 the assertion which is slightly more general than Theorem 5.3 for one-rowed diagrams. We now turn to the general case. For an arbitrary diagram ~ we consider a diagram ~(1)[hk(~)] - ! S(1)~[hk(~)] # H*( Fk(~)) - ! H*(SFk(~)). The top arrow in this diagram is the structural arrow in the Schur complex, the left vertical arrow is the tensor product of transformations (~s)for one- rowed diagrams which we constructed in the previous paragraph. But the bottom arrows needs some explanation, for it is the first arrow which exists only on the level of cohomology. We recall that since ~ = (~1)|h . .|.h(~l), the structural arrow may be described as a composition of the map S(~1)(V ) . . .S(~l)(V ) -! S~(V . . .V ) obtained by l-times taking the smallest diagram (of appropriate weight) in the Decomposition Formula for S~(V . . .V ), with the map induced by addition. We perform an analogous construction in the bottom row of our diagram but on the level of cohomology. In order to construct a map H*(SFk((~1)))(V ) . . .H*(SFk((~l)))(V ) -! H*(Fk(S~))(V . . .V ) we observe that (we start with detaching the last row) the diagram Fk((~1, . .,.~l-1)) is the smallest subdiagram in Fk(~) among these which give in the Decomposition Formula for SFk(~)(V V ) a cohomologically nontrivial factor. As we noticed in Section 6, it enables to construct the map H*(SFk((~1,...,~l-1)))(V (l-1)) H*(SFk((~l)))(V ) -! H*(Fk(S~))(V l). Iterating this procedure we obtain the bottom arrow in the diagram. Now we would like to show that it is possible to close this diagram by the right vertical arrow (thanks to the epimorphicity of the top arrow it will be unique). This closure will be our transformation ~. Our task will be 27 completed if we show that the left arrow sends the kernel of the top arrow to the kernel of the bottom one. We start by recalling (and adapting to our context) a known description of the kernel of the structural arrow OE~ (cf. [ABW], p. 264). Let us introduce the temporary notation: we call a diagram (n + m - 1, m)=(m - 1) a zigzag and denote it by z(n, m). Since Sz(n,m)is very close to n m , the description of the kernel of the structural map is quite simple in this case. We just have an exact sequence ffiz(n,m) 0 -! n+m -! n m - ! Sz(n,m)-! 0, where the first arrow is a comultiplication in the Hopf algebra * (cf. [ABW], sect. V.1). For a general diagram ~ we say that ff ~ generates zigzag if there exists the diagram ff0 ~ (it is unique, if exists) such that ff0=ff = z is a zigzag _______________ | | | | | ff ___| | | |_|___|_______|____z_oe |______|_ | | | | 0 _____| |~=ff __|_ | |______|_ To any ff generating zigzag we shall associate a space Kff contained in ker( ~ -! S~). To do this we observe that both ff and ff0 are the smallest subdiagrams of their weights. Hence we have the commutative diagram 0 ff n m ~=ff0 ff n+m ~=ff - ! - ! Sff Sw S~=ff0 | | | | |? |? ~ ____________________- S~, in which the composition of the top arrows is trivial. Therefore the space 0 ~ Kff= im ( ff n+m ~=ff) - ! ) is contained in the kernel of the structural arrow. The main point of ([ABW], Th. V.1.10) is that the spaces Kfffor all ff generating zigzags span this kernel. Thus it suffices to show that 28 (~1) . . . (~l)(Kff) is contained in the kernel of a homological analogue of the structural arrow, for all ff generating zigzag. To see this, let us consider the commutative diagram 0(1) _____________-~(1) ff(1) n+m(1) ~=ff [hk(~)] [hk(~)] | | | | |? |? 0) * F (~) * H*( Fk(ff)) H*( Fk((n+m))) H*( Fk(~=ff) -! H ( k ) - ! H (S~), in which the bottom arrows exist only on the level of cohomology, by lex- icographic properties of ff and ff0. It will be sufficient if we show that the composition of the bottom arrows is trivial. Moreover, it suffices to show it for ~ being a zigzag, because tensoring with a trivial map remains trivial. But in this special case, the considered sequence of cohomology groups H*(SFk((n+m)))- !H*(SFk((n)) SFk((m)))- !H*(SFk(W(n,m))) is induced by the "real" sequence of complexes SFk((n+m))-!SFk((m)) SFk((n)))- !SFk(W(n,m)), because Fk((n)) is the smallest (not only among nonacyclic!) subdiagram of a given weight in Fk(w(n, m)). Now it is easy to check on tableaux that the composition of the arrows in this sequence of complexes is trivial. This finishes the construction of ~. What remains is to show that the family ~ satisfies the conditions of Defi- nition 6.2. Since our construction was uniform for all diagrams, it suffices to check the conditions for ff=ff0= ~. In this situation, to shorten notation, we will denote the diagram ff00just by ff. We take an arbitrary ff ~ and consider the diagram L fi(1) ~=fi(1) fi ~=fi L * Fk(fi) * Fk(~* *=fi) fi ff (V ) (W )[hk(~)]____________- fi ffH ( )(V ) H ( * * )(W ) | | | | |? |? ~ (V W ) ~(1)(V W )[hk(~)] _________________________- H*( Fk(~))(V W ) | | | | |? |? ~(V W ) S(1)~(V W )[hk(~)] _________________________- H*(SFk(~))(V W ), 29 in which fi: fi[hk(~)] -! H*( Fk(fi)) stands for the product (fi1) * * . . . (fil). By the definition of filtration yielding the Decomposition For* *mula ([ABW], Def. V.1.11), M(1)ffis the image of the composition of the arro* *ws in the first column, while M(1)Fk(ff)_ in the second. Hence to show th* *at ~ preserves the filtration it suffices to show that this diagram commutes* *. But the commutativity of the upper square follows from the compatibility of* * the family for one-rowed diagrams, and the commutativity of the lower one, from the diagram defining ~. It remains to identify the quotient map. We consider a diagram which* * is best seen as a cube (in order not to make it even larger I neglect shif* *ts of grading) L * Fk(fi) * Fk(~=fi) __________- * F (ff) * F (~=ff) fi ffH ( )(V ) H ( )(W ) H ( k )(V ) H ( k )* *(W ) | Z" æ> | | Z æ | | Z fi ~=fi ff ~=ff æ | | Z æ | | Z Z æ æ | ||L fi(1) ~=fi(1) _____-ff(1) ~=ff(1) || || fi ff (V ) (W ) (V ) (W ) || | | | | | | | æ| | | | |? |? | | | | ___________- | || M*(1)ff(V W ) Sff(1)(V ) S~=ff(1)(W ) || | | | æ Z | | æ |C*(1) Z | | æ ~ ff ? ff ~=ff Z | |? æææ= Z ZZ~ |? H*(MFk(ff))(V W ) ____________________- H*(SFk(ff))(V ) H*(SFk(~=* *ff))(W ) We have to show the commutativity of a square with question mark insert* *ed. I claim that the commutativity of all other squares follows from the ea* *r- lier considerations. Indeed: the commutativity of the external and inte* *rnal squares follows from the definition of the filtration. The commutativi* *ty of the side squares follows from the second condition in the definition of* * a com- patible family. The commutativity of the top square is obvious. Now, * *to obtain the commutativity of the square we are interested in, it suffice* *s to ob- serve that the arrow æ is epic and perform a standard diagram chasing. * *This 30 completes the proof of compatibility of the family ~ and hence, of Theorem 5.3. || 8 Remarks on multiple quotients A situation when a diagram has a trivial p-core but its p-quotient consists of a few diagrams is much more complicated and I have not succeed yet in understanding the cohomology of the Schur-De-Rham complex in this case. In this section I would like to discuss some conjectures suggested by numerical computations and support them by a simple example where a complete calculation of the cohomology is possible. Both numerical computations and common sense suggest the existence of some relation between the cohomology of S~ and the tensor product S(1)q0(~) . . .S(1)qp-1(~)for diagrams appearing in the p-quotient of ~. Alas, the simple* *st possible example shows that one cannot hope for an isomorphism Fact 8.1 ( 2(1) for n = 2p - 4, 2p - 3, 2p - 1, 2p Hn(S(p,p)) = 0 otherwise. Proof: Applying the Littlewood-Richardson rule to the skew diagram (2p - 1,Lp)=(p) we get the following decomposition (up to filtration) S(2p-1,p)=(p-1)= p a=1S(2p - a, a). But among the diagrams at the right-hand side only (2p-1, 1) and (p, p) have trivial p-cores. Hence in the spectral sequence of fi* *l- tration converging to H*(S(2p-1,p)=(p)) there are only two nontrivial columns. Thus this spectral sequence degenerates to the long exact sequence . .-.! H*(S(2p-1,1)) -! H*(S(2p-1,p)=(p-1)) -! H*(S(p,p)) -! H*+1(S(2p-1,1)) -!* * . . . We recall that according to Lemma 7.1 (and its Kuhn dual), we have H*(S(2p-1,p)* *=(p)) = S2(1)[2p - 2], and H*(S(2p-1,1)) = 2(1)[2p - 3]. Therefore, a nontrivial part of this sequence looks like this 0 -! H2p-4(S(p,p)) -! 2(1)-! 0 -! H2p-3(S(p,p)) -! I2(1)-! - ! S2(1)-! H2p-2(S(p,p)) -! D2(1)-! I2(1)-! H2p-1(S(p,p)) -! 0 -! - ! 2(1)-! H2p(S(p,p)) -! 0. 31 From this we immediately get that Hn(S(p,p)) = 0 for n < 2p - 4, and for n > 2p, and also that H2p-4(S(p,p)) = 2(1)= H2p(S(p,p)). Moreover, we observe that since H2p-3(S(p,p)) is a subobject in I2(1), it must be equal to 2(1)or to D2(1). To rule out the second possibility it suffices to notice that H*(S(p,p)) evaluated on a one-dimensional space is trivial since the underly- ing complex is trivial. Thus we conclude that H2p-3(S(p,p)) = 2(1). By a similar reasoning H2p-1(S(p,p)) = 2(1). Now, the exactness of the sequence forces that H2p-2(S(p,p)) = 0 which completes the proof. || Thus we may suppose that H*(S(p,p)) comes from (suitably shifted) 2(1) S2(1)divided thorough some differential acting between the factors. To con- nect this observation with remarks made at the beginning of this section we recall that qp-2((p, p)) = qp-1((p, p)) = (1). Therefore, the Littlewood- Richardson rule applied to the tensor product of diagrams from p-quotient gives S(1) S(1)= S(2)+ S(12), that is, exactly these Schur complexes whose remains we discovered in H*(S(p,p)). Also some other partial calculations which I have performed with the aid of Mathematica support the following conjecture. It seems, there exists a spectral sequence converging to H*(S~), whose columns in E1-term are Schur complexes for diagrams appearing in the Littlewood-Richardson decomposition of a tensor product of diagrams from the p-quotient. 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