RESOLUTIONS OF THREE-ROWED SKEW- AND ALMOST SKEW-SHAPES IN CHARACTERISTIC ZERO

Transcription

1 RESOLUTIONS OF THREE-ROWED SKEW- AND ALMOST SKEW-SHAPES IN CHARACTERISTIC ZERO MARIA ARTALE AND DAVID A. BUCHSBAUM Abstract. We find an exlicit descrition of the terms and boundary mas for the three-rowed skew-shae and almost skew-shae in characteristic zero. 1. Introduction In [2], a connection was made between the characteristic-free resolution of the three-rowed artition, 2, 2, 2, and its characteristic-zero resolution described by A. Lascoux. The method used there was to take the known general resolution, modify the boundary ma exloiting the fact that we can divide when we re over the rationals, and then reduce the large general resolution to the much slimmer one of Lascoux [4]. A similar rogram was carried out by [3] in his thesis, for the artition 3, 3, 3, but it was clear that the method had reached its limit there, since the exlicit descrition of the boundary mas in the characteristic-free case is not yet available. 1 Since it was still very temting to have an exlicit descrition of the boundary mas in the characteristic-zero situation, we decided to borrow another tool from the characteristic-free construction, namely the maing cone, and see if we couldn t arrive at the Lascoux resolutions in general, that is not just for artitions, but for all the three-rowed shaes that come u in the reresentation category: the skew-shaes and almost skew-shaes [1]. The combinatorial comlexity of a three-rowed skew- or almost skewshae is to a large extent tied u with the number of trile overlas that occur in the shae. If the shae is a skew-shae: t 2 t 1 q, r then the number of trile overlas is r t 1 t 2. To avoid drawing the diagram each time it s needed, we will often denote this shae as, q, r; t 1, t 2. 1 We do intend to return to this aroach after we make exlicit the boundary mas of the characteristic-free resolutions, but this will no doubt take some time. 1

2 2 MARIA ARTALE AND DAVID A. BUCHSBAUM If the shae is an almost skew-shae: s t q r, the number of trile overlas is r t + s. In this case, too, we will often denote the shae by, q, r; t, s. There is a theorem [1] that tells us that we have short exact sequences: 0 t 1 + t q + t t r t 2 1 t t 1 q r S t 2 t 1 q r 0, and, starting with an almost skew-shae: 0 s 1 s 1 + τ q r τ s 1 t q r A s t q r 0, where τ = t s+1. Here we have resorted to ictures rather than the more concise notation since the ictures give a clearer indication of the changes taking lace on the shaes as we move from right to left. In each of these exact sequences, the number of trile overlas in the middle and leftmost terms is one less than in the initial or rightmost term. By using induction on the number of trile overlas in the term whose resolution we are seeking, we may assume the resolutions known for the terms with fewer trile overlas. By roducing a ma between the known resolutions, we can take the maing cone of this ma, and it is wellknown that this maing cone will give us a resolution of the rightmost term. However, this resolution has more terms than are anticiated in the Lascoux resolution in the case of skew-shaes; Lascoux didn t consider almost skewshaes, and we would like to excise them. The first time this occurs in the course of our construction, we will rove a lemma which will rovide a rescrition for doing this excision. In the next section, we will give a quick review of some of the notation we will be using, and also easily disatch the resolutions of the two-rowed skew-shaes there are no almost skew-shaes with two rows. This case, while very secial, is essential in getting the induction rocess going.

3 RESOLUTIONS IN CHARACTERISTIC ZERO 3 In Section 3, we will get the descrition of the resolution for skew-shaes having at most one trile overla, and then tackle the case of an almost skew-shae with one trile overla there are no almost skew-shaes with fewer than one trile overla. These cases are rather secial; for the skewshaes, the Jacobi-Trudi determinant tells us we should exect fewer terms than in the general case, and that is what haens. In the subsequent section, we get the descrition of the resolutions in general, again working first on the skew-shaes, and then the almost skew. From the exact sequence A above, it is clear that the aroach to almost skew-shaes has to consider the case when s = 1 where the terms that occur to the left of it will be skew-shaes, and the case when s > Notation and the two-rowed case As usual, we will let D a stand for the a-fold divided ower of some underlying free module F, and we consider the n-fold tensor roduct, D a1 D an. Notation: We denote by t ji the comosition of the mas D a1 D ai D aj D an t Da1 D ai t D t D aj D an µ D a1 D ai t D aj +t D an, where t denotes the indicated diagonalization D ai D ai t D t, and µ stands for the multilication within the divided ower algebra, D t D aj D aj +t. We have obviously moved D t ast several laces to juxtaose it with D aj, but there are no sign changes to deal with when we do this since the divided ower algebra is commutative. Definition 2.1. The ma t ji olarization, ji. is called the t th divided ower of the lace The way we have written this definition, it looks as though i must be less than j. It is clear from the definition though that the relative size of the indices is of no consequence; it simly moves from the argument in lace i to the argument in lace j. However, in all that we do in this article, it will be the case that i < j. For a fuller discussion of letter-lace algebras and lace olarizations, we refer the reader to [1]. The main result we want to rove in this section is the following theorem. Theorem 2.1. Let M= t q be a two-rowed skew-shae, and let K, q; t denote the Weyl module corresonding to this shae. Then the following sequence: is exact. 0 D +t+1 D q t 1 t+1 D D q K, q; t 0

4 4 MARIA ARTALE AND DAVID A. BUCHSBAUM Proof. We could go through a ainstaking induction, using maing cone arguments, to show that this is so. However, we ll jum to a letter-lace roof of this easy fact, just to illustrate the use of that technique. It is well-known that, in characteristic zero, the sequence without the leftmost zero is exact. Therefore it will suffice to define a ma, s, from D D q to D +t+1 D q t 1 such that s t+1 = id. Now a basis element of D +t+1 D q t 1 can be written, in letter-lace double tableau notation, as w1 1 +t+1 2 l w 2 2 q t 1 l, and its image under t+1 is t l w1 1 2 t+1+l l w 2 2 q t 1 l To slit this ma, that is, to define our ma, s, we have to take a tyical basis element of D D q and say where it goes under s. A tyical such basis element is of the form w1 1 2 k w 2 2 q k Let s agree to define s on such a basis element by sending it to 0 if k < t + 1, and sending it to k 1 w1 1 +t+1 2 k t 1, otherwise. It is k t 1 trivial to check that s t+1 = id. w 2 2 q k. 3. At most one trile overla In this section, we start our maing cone aroach to constructing the resolutions of three-rowed shaes. The basic tool underlying this construction is the following easy lemma. Lemma 3.1. Let 0 A f B g C 0 be an exact sequence of modules, let X and Y be acyclic left comlexes over A and B resectively, and let F : X Y be a ma over f. Then the maing cone of F is an acyclic left comlex over C Skew-shaes. We ll first handle the case where r t 1 + t and r = t The short exact sequence S of Section 1 gives us two terms that we can resolve, and we have their resolutions and mas between them:. 0 D +t1 +t 2 +2 D q t1 1 D 0 t 1+t 2+2 D D q+t2 +1 D 0 F t 1, t 2 t D +t1 +1 D q t1 1 D r t 1+1 D D q D r,

5 RESOLUTIONS IN CHARACTERISTIC ZERO 5 where F t 1, t 2 = t 2 +1 β=0 t β t β β t 2+1 β 31. This definition of F t 1, t 2 is retty much forced by the need to make the diagram commute. To get the resolution of the original shae,, q, r; t 1, t 2, we take the maing cone and get: 0 D + t +2 D q t1 1 D 0 D +t1+1 D q t1 1 D r D D q+t2+1 D 0 D D q D r, which has the terms that the Lascoux resolution says we should have in this case. We have written t for the sum t 1 + t 2. The mas from the center summands are the obvious lace olarizations, the ma to the uer central summand is what we ve called F t 1, t 2, and the ma to the lower central summand is again the obvious lace olarization. To go further, we ll now still insist that r t 1 + t 2 + 1, but let r = t This case, while again being very secial, is what indicates what the inductive rocedure is in general. Using the same short exact sequence: 0, q + t 2 + 1, 1; t 1 + t 2 + 1, 0, q, r; t 1, t 2 + 1, q, r; t 1, t 2 0, we get a ma of the resolution of the left-most into that of the center one: 0 D + t +3 D q t1 1 D 0 D + t +2 D q t1 1 D 1 D D q+t2 +2 D 0 D D q+t2 +1 D 1 t F t 1, t 2 t t D + t +3 D q t1 1 D 0 D +t1 +1 D q t1 1 D r D D q D r, D D q+t2 +2 D 0 where the sum on the center arrow indicates that F t 1, t 2 alies to the uer term, and multilication by t alies to the lower summand. The roof that this diagram does really commute is trivial in the right square, but involves some calculation in the second square. It boils down to roving the identity: F t 1, t 2 F t + 1, 0 = t 2 + 2F t 1, t We defer the roof of this fact until later; for now, we continue with the rest of the rocedure. First, though, we introduce some shorthand notation for the tensor roduct of divided owers, namely, instead of writing D a1 D a2 D an, we will write this as a 1 a 2 a n.

6 6 MARIA ARTALE AND DAVID A. BUCHSBAUM Returning to the ma of comlexes above, we now form its maing cone, and we get the exact comlex: 0 + t + 3 q t t + 2 q t q + t t + 3 q t q + t t q t 1 1 r q + t q r. We note that we have the subcomlex: 0 + t +3 q t and the quotient comlex: 0 + t + 2 q t q + t q+t t + 3 q t t q t 1 1 r q + t q r. This last, of course, is the one that we want for our resolution, so we have to show that the subcomlex we re dividing out by is exact. But the exactness of that comlex in characteristic zero is trivial to show. Just to be exlicit, the mas from the sum to q r are the usual lace olarizations. The ma from the tail to the sum is exactly what it was in the revious case: F t 1, t 2 t +2. Proceeding in this way, we obtain the result for skew-shaes having no more than one trile overla. Theorem 3.2. Under the hyothesis that r t + 1, the resolution in characteristic zero of, q, r; t 1, t 2 is 0 + t + 2 q t 1 1 r t 2 1 Or, more grahically, 0 + t + 2 q t 1 1 r t 2 1 F t 1,t 2 t +2 We now rovide the roof of the fact that + t q t 1 1 r q + t r t 2 1 q r. + t q t 1 1 r q r. q + t r t 2 1 F t 1, t 2 F t + 1, 0 = t 2 + 2F t 1, t A heuristic roof might go like this.

7 We can use the basic fact that RESOLUTIONS IN CHARACTERISTIC ZERO 7 t 1+1 F t 1, t 2 = t 2+1 t 1+t 2 +2, and aly t 1+1 to both sides of the equation. This equality is then very easy to see. But unfortunately, since t 1+1 isn t a monomorhism, this can t serve as a roof, and we have to go through the calculations. First, let and A = t 2 + 2F t 1, t = B = F t 1, t 2 F t + 1, 0 = 1. t 2+1 β 0 t 2+2 β 0 k β β β t2+2 β 31, k β β 1 β t2+1 β t 1 + t 2 + 3, where k β = t 1 +1+β t 1 +1 Taking into account the different ranges of summation of the index β, we see that A B = t 2+2 β 1 βk β β β t2+2 β 31 1 k γ γ t 1 + t γ t2+1 γ 31. γ 0 We distinguish between the indices β and γ because they run over different limits. We now want to bring the ast the γ in the last exression so that we can collect our terms. That is, we use the fact that γ = γ + γ If we do that, and then collect the terms that modify a fixed ower of 31, we see that they add u to zero, and we re done The almost skew case. Recall that we use the notation, q, r; t, s for an almost skew-shae, with 0 < s t. To say that the almost skew-shae has only one trile overla is simly to say that r = t s+1. If that is the case, then we have the exact sequence 0 + t s + 1, q, 0; s 1, 0, q, r; t, 0, q, r; t, s 0. The left-most shae is a two-rowed skew-shae, and the middle term is a three-rowed skew shae with no trile overlas. If we write down their

8 8 MARIA ARTALE AND DAVID A. BUCHSBAUM resolutions and ma the left one into the other, we get the following: t + 2 q t 1 r 1 + t + 1 q s 0 1 t s+1 t s+1 Gs,t F t, 0 t+2 + t + 1 q t 1 r q + 1 r 1 s t+1 + t s + 1 q 0 t s+1 31 q r, where we define Gs, t = β>0 1 β β 1 β 1 s+β t s+1 β 31. Note that if t = s. then Gs, s = s+1. The definition of Gs, t is insired by the Caelli identity as are the definitions of most of the other mas that have aeared or will aear in this article, which in this case is: t+1 τ = 1 α τ α t+1 α α 31. α 0 A discussion of the Caelli identities can be found in [1], ages All of this discussion then gives us the following theorem. Theorem 3.3. Let,q,r;t,s be a three-rowed almost skew-shae, with r = t s + 1. The resolution of the Weyl module corresonding to this shae is the maing cone of the above ma, namely, the comlex: 0 + t + 1 q s 0 + t + 2 q t 1 r 1 δ 2 + t + 1 q t 1 r + τ q 0 q + 1 r 1 δ 1 q r. where the mas δ 1 and δ 2 are defined as follows:

9 RESOLUTIONS IN CHARACTERISTIC ZERO 9 δ 2 = δ 1 = 1 τ 1 τ s Gs, t F t, 0 0 t+2 t+1 τ 31 and where we have set τ = t s + 1., We should exlain our matrix notation and conventions. If A = m i=1 A i and B = n j=1 B j are two modules, we describe a ma α : A B as an m n-matrix α ij where α ij : A i B j. The rows, therefore, reresent the image of each A i in B. This convention will be used throughout. 4. The general case In the revious section, the resolutions we obtained were both of length 2. For the skew-shaes, this was anticiated by the Jacobi-Trudi determinantal formula for Weyl modules associated to skew-shaes. Of course, we have no classical formula for the almost skew-shaes, so it is fortunate that even here the length of the resolution is, if not what we could anticiate, at least what we hoed would be the case. As we would exect, once the number of trile overlas in a skew-shae is at least two, we get a full-length resolution, namely one of length 3. The way we aroach the general case is to use the same sort of induction rocedure that we used before: we start with our skew-shae with a certain number, n, of trile overlas, we look at the fundamental exact sequence S of Section 2, and notice that the skew-shae and almost skew-shae that occur there both have fewer trile overlas n 1, to be exact. Assuming we know the resolutions for all shaes both skew and almost skew having fewer than n, we take those resolutions, find a ma between them, take the maing cone, see that it s larger than the one we had in mind, and see if some reduction can be carried out to arrive at the conjectured form. This reduction rocess is not simly finding an acyclic subcomlex of the maing cone as we ve done in the revious section; for skew-shaes, it requires more work than that. For almost skew-shaes we start out following the same attern, using the exact sequence A of Section 2. But there, we run into the roblem that if s = 1, the shaes that aear in the exact sequence are skew, not almost skew, so that their resolutions are not the same as the ones we would use for almost skew-shaes. As a result, when we aly our inductive ste, we must consider the two cases, s = 1 and s > 1 searately. Since the discussions of the skew- and almost skew-shaes require that we hyothesize the forms of their resolutions, we will state here the results. In each of the following subsections, we will indicate in more detail how we arrive at them.

10 10 MARIA ARTALE AND DAVID A. BUCHSBAUM Theorem 4.1. Let, q, r; t 1, t 2 be a three-rowed skew-shae. The terms of the resolution of the Weyl module associated to this shae are: 0 + t + 2 q r t t + 2 q t 1 1 r t t q + t r t 2 2 q + t r t t q t 1 1 r 1 q r, where we have set t = t 1 + t 2. The boundary mas, which we describe in matrix form, are the following: 3 = 1 t1 t1+1 t2+1 2 = 1 = t +2 F t 1, t 2 Lt 1, t 2 1 t1 t +2 t2+1 t1+1 where F t 1, t 2 has already been defined, and we define Lt 1, t 2 = 1 β t β 1 β t β t 1+1 β 31. β 0 Note: When r < t + 2, we recover the resolution for skew-shaes with at most one trile overla. Theorem 4.2. Let, q, r; t, s be a three-rowed almost skew-shae. The terms of the resolution of the Weyl module associated to this shae are: 0 + t + 2 q s r τ 1 δ3 + t + 2 q t 1 r 1 + t + 1 q s r τ + τ q + 1 r τ 1 δ 2 + t + 1 q t 1 r q + 1 r 1 + τ q r τ δ 1 q r, where, as usual, we have set τ = t s + 1.

11 RESOLUTIONS IN CHARACTERISTIC ZERO 11 The boundary mas, which we will again describe in matrix form, are the following: δ 3 = δ 2 = δ 1 = 1 τ t+2 s+1 τ F s 1, 0 s+1 F t, 0 t τ 1 τ Gs, t s 0 τ 31 t+1 τ 31 Note: Here again we see easily that when r τ, we recover our old resolution. Before we get into the details to rove the above results, let us describe the simlification rocess that we will have to use. Simlification Procedure Suose that you have a comlex, X which, in each dimension, i, breaks u into a direct sum of modules, X i = A i B i. We will write the boundary ma of X as {δ i }, and the boundary ma on its A and B comonents as δ Ai A i 1, δ Ai B i 1, δ Bi,B i 1, δ Bi,A i 1. We are assuming that X is a comlex, but we make no assumtions at this oint about either A or B. Assume now that B 0 = 0, and that we have mas σ i : B i A i satisfying the following identities:. δ A1 A 0 σ 1 = δ 1 σ 1 = δ B1 A 0 ; i σ i = δ Bi A i 1 + σ i 1 δ Bi B i 1 for i 2, where the newly-defined boundary mas on A, = { i }, are defined by setting i = δ Ai A i 1 + σ i 1 δ Ai B i 1. With this boundary, it is easy to see that A is turned into a comlex the identities were chosen to ensure this. Now define the boundary in B in a similar way, namely, i = δ Bi B i 1 δ Ai B i 1 σ i, and you get an exact sequence of comlexes: 2 0 B σi A B I+σ A 0, 2 The same set of identities, *, serves to make both of the mas bona fide mas of comlexes.

12 12 MARIA ARTALE AND DAVID A. BUCHSBAUM where I is the identity ma. If you know that the comlexes B and X are exact, you then have the exactness of A Skew-shaes. Here we describe the stes used to rove Theorem 4.1. We use induction on the number of trile overlas, that is, on the number ω = r t. We have the result for ω 1, so we assume ω = 2. The exact sequence S of Section 2 shows us that the middle term is a skew-shae with ω = 1, and the leftmost shae is an almost skew-shae also with the number of trile overlas equal to 1. If we take the resolutions of these shaes, we can define a ma, Ω, from the one into the other in terms of matrices as follows: t2 + 2 Ω 2 = 0 Ω 1 = Ω 0 = 0 F t 1, t 2 t Ht 1, t 2 β>0 1 t1+1 t +2 t2+1 where Ht 1, t 2 = 1 t 1 1 β t β 1 β 1 β t t 1+1 β 31. The ma Ht 1, t 2 suggests itself in order to satisfy the following identity: t 2+1 t = 1 t 1+1 t 1+1 t +2 + t 2+2 Ht 1, t 2.. When we take the maing cone of the ma, Ω, we obtain a comlex which, while a resolution of our skew-shae, has too many terms in it, that is, it isn t of the form described in Theorem 4.1. In fact, we see that this maing cone, which we will denote by X, is as a module the direct sum of two submodules, A and B as in the simlification rocedure, with the B art reresenting the excess that we want to get rid of. However, neither the A art nor the B art is a subcomlex, so here is where we have to aly our simlification rocedure to make them into comlexes by modifying the boundary mas by finding mas σ i : B i A i. To be recise, the B 1 iece consists of the term q+t 2 +2 r t 2 2 and we set the ma σ 1 = 1 t The iece B 2 is the direct sum of two ieces: q + t r t t + 3 q t 1 1 r t 2 2. We set σ 2 to be zero on the first iece, and equal to 1 the second iece. t 2 +2 F t + 1, 0 on

13 RESOLUTIONS IN CHARACTERISTIC ZERO 13 Finally, on B 3, which is + t + 3 q t 1 1 r t 2 2, we define σ 3 to be zero. It is as a result of the modifications on the boundary mas to make A and B into comlexes that we arrive at the announced form of the resolution. It should be ointed out that the comlex B roduced in this way is clearly homologically trivial. Since the maing cone is known to be acyclic, we get the fact that the comlex A is acyclic as well, and thus is a resolution. As the reader can see, the inductive rocedure is reetitive, but yields the stated result. That is, when the number of trile overlas is greater than 2, we roceed in the same way, emloying the fundamental short exact sequence to reduce the number of trile overlas, we find the ma corresonding to Ω it is essentially the same ma as in the described case, we take the maing cone, and get rid of the excess The almost skew case. As we indicated, the almost skew case is treated inductively in the same way as the skew, but as a careful look at the exact sequence A in section 2 will convince you, we have to treat the case s = 1 searately always using the induction hyothesis on skew- and almost skew-shaes on the number of trile overlas, since the shaes that occur in that case are skew- not almost skew-shaes. If we write down the resolutions of the two shaes that occur in A using Theorem 4.1, we must define a ma between them. We shall call this ma Θ, and describe Θ 3, Θ 2, Θ 1 and Θ 0 as matrices. Θ 3 = Θ 2 = Θ 1 = Θ 0 = t+2 2 1t t+2 2 t 0 t 0 t G1, t 1 t 1 t t 31 In this case, we see that the suerfluous terms of the maing cone, X, actually form a subcomlex of X, namely: 0 + t + 2 q r t 2 α2 + t + 2 q r t 2 + t + 1 q + 1 r t 2 α 1 + t + 1 q + 1 r t 2 0 where each of α 1 and α 2 have a comonent which involves simle multilication by t+2 2. Therefore this comlex is homologically trivial, and the factor comlex is our desired resolution.

14 14 MARIA ARTALE AND DAVID A. BUCHSBAUM When s > 1, we again resort to the exact sequence, A, and this time we use the resolutions of Theorem 4.2 to resolve the central and leftmost terms. The ma between those resolutions, which we will call Ψ, is defined to be Ψ 3 = τ + 1 Ψ 2 = Ψ 1 = 1 τ t+2 s+1 τ τ τ τ 1 τ Gt, s 0 0 τ τ + 1 Ψ 0 = τ 31 In this case, too, the maing cone has suerfluous terms, these terms form a homologically trivial subcomlex of it, and the quotient comlex is the desired resolution. This concludes the discussion of the results Theorem 4.1 and Theorem References [1] G. Boffi and D.A. Buchsbaum, Threading Homology Through Algebra: Selected Patterns. Oxford University Press Clarendon, Oxford [2] D. A. Buchsbaum, A Characteristic-free examle of a Lascoux resolution, and letterlace methods for intertwining numbers. Euroean Journal of Combinatorics, Vol 25, Issue 8, November, 2004, ages [3] H. Hassan, Alication of the characteristic-free resolution of the Weyl module to the Lascoux resolution in the case 3, 3, 3. Ph.D. Dissertation, Università di Roma Tor Vergata, [4] A. Lascoux. Syzygies des variétés déterminantales. Advan. in Math , Diartimento di Matematica, Università di Roma, Tor Vergata, Roma, Italia address: artale@ax.mat.uniroma2.it Deartment of Mathematics, Brandeis University, Waltham, MA, USA address: buchsbau@brandeis.edu