Characteristic-free representation theory of the general linear group

ADVANCES

IN MATHEMATICS

%,

149-200 (1985)

Characteristic- Free Representation Theory of the General Linear Group KAAN

AKIN*

Purdue University, West Lafayette, Indiana 47907, and Massachusetts Institute of Technology, Cambridge, Massachusetls

02139

AND DAVID Brana’eis

University,

A. BUCHSBAUM’ Waltham,

Massachusetts

02254

1. INTRODUCTION In attempting to apply the techniques of Schur functors and complexes as developed in [3] to extend the results on resolutions of determinantal ideals described in [2], we became increasingly aware of the need to study systematically Z-forms of rational representations of the general linear group. These Z-forms arise in a number of different contexts and, to make clear the kinds of things we are talking about, we shall lirst illustrate with some simple examples. If F is a free module over a commutative ring R, and p is a positive integer, we have the CL(F)-module A*FQF. Now for any integer k, conwhich sends A*+‘F into sider the map of A *+‘F into A*F@F@A*+‘F A*F@ F by diagonalization and A P+ ‘F into A*+ ‘F by multiplication by k. We will denote the cokernel of this map by H,Jp, 1). Clearly, when R = Q, HJp, 1) is isomorphic to A*F@ F as a CL(F)-module, but this is not true in general when R = Z. In fact, if we consider the exact sequence O+Ap+‘F+ApF@F~L,p,,~F+O,

(11

then the map A p + ‘F +k A*+ ‘F induces the exact sequence O--+Ap+lF-+~k(p,

l)+Lcp,IJF+O

(2)

* The author acknowledges the partial support of NSF Grant MCS U-02367. + The author acknowledges the partial support of NSF Grants MCS 80-03118 and MCS 8303348 and the support of the Science and Engineering Research Council of England.

149 OOOl-8708/85 $7.50 Copyright 0 1985 by Academic Press, Inc. Ail rights of reprcduction in any form reserved.

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under the Yoneda correspondence of Extl(Lo,l& .4p+1F) with the short exact sequences of the types (1) and (2). Thus we see that the Z-forms H,& 1) arise in the context of determining Ext groups. (In fact, it can be shown that when R=Z, Extl&,il, Ap+i) is a cyclic group of order p+ 1 and is generated by the extension (1) corresponding to fZl(p, 1 ).) Another context in which the Z-forms H,Jp, 1) (and, of course, many others) arise, and the one with which we will be mainly concerned in this paper, is that of realizing the Giambelli identities in the Grothendieck ring of CL(F)-modules. Without elaborating on the Giambelli identities in general at this point (see (7)) we will just look at some special cases. One instance of these identities is that (3) where [X] signifies the class of a CL(F)-module X in the Grothendieck ring. The exact sequence (1) is a realization of this identity as an acyclic complex over LCp,lJF whose Euler-Poincare characteristic is the right-hand side of (3). However, another instance of these identities is that [Lcp-~,21F] = LAP-‘F@A*Fl

- [A+‘F@F-l

(4)

and we would be led to believe that there should be an exact sequence O+ApF@F+Ap-‘F@A2F+L~p~l,2~F+0

(5)

since the Euler-Poincare characteristic of (5) would yield the identity (4). If we take ApF@ F+ Ap-‘F@ A*F to be the composition APF@ F +‘I@’ Ap-‘F@F@F +l@” App1F@A2F (where A is the diagonal and m is the multiplication in AF), then it is rather easy to show that the sequence (5) is exact over CD.It is also easy to show that for this or any other choice of a map ApF@F+ ApdlF@ A*F, the sequence (5) is not exact over Z. Nevertheless, the complex (5) can be enlarged to a sequence O+Ap+1F+ApF~F@Ap+1F-+Ap-‘F~A2F-+L~p~l,2~F-+0

(6j

which is always exact and which realizes the identity (4) because the terms Ap+ ‘F cancel each other in the Euler-Poincare characteristic of (6). The map Ap+‘F+ApF@F@A p+ ‘F is the map, described earlier, whose cokernel is H2(p, 1). The exactness of (6) is, then, equivalent to the statement that the kernel of the natural surjection ApblF@ A*F + L cpp l,2JF is the Z-form H2(p, 1). With these concrete examples in mind, we will now describe in more general terms the main thrust of this article and at the same time discuss

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the connections between polynomial representations of the general linear group and the classical symmetric functions. The Grothendieck ring of the category of polynomial representations of the general linear group GL,, over an intinite field is canonically isomorphic to the ring of symmetric polynomials in n variables, the correspondence being obtained by sending a representation to its formal character [4]. Let F denote the vector space k” of dimension n over an infinite field k. The fundamental representations of GLn = GL(F) are the exterior powers /I’(F),

A’(F),..., A”(F)

of the k-module F and their formal characters are the elementary symmetric polynomials

in n variables x = x1 ,..., x~. The formal character of the Schur functor LA(F) associated to a partition A = (,I , ,..., Am) is a Schur function which can be expressed in terms of the elementary symmetric polynomials by Giambelh’s determinantal expansion CL~Fl=det(e~,-~+j)~~i,j~m~

(7)

where it is understood that ek =0 unless 0~ k
k

whose formal character is ey. The symmetric group Sm acts on sequences of length m by coordinate permutation and this action can be used to rewrite (7) in the more convenient form

where 6 is the sequence (m- 1, m -2,..., 2, 1,O). Since e,, is the formal character of II? F, (8) can be rewritten as

in the Grothendieck

ring of polynomial

representations

of GL(F). It was

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asserted in [5 J that over a held of characteristic-zero, as a resolution

cl-a +)m

(9) may be realized

+ -.. +Bl(A)+&,(A)+LAF+O,

(10)

where Bi(L) is the direct sum &WjEi A,,,cA+hJp6F over all permutations w in S,,, of length i. Although formula (9) holds over any field or even the ring of integers Z, we have seen that there exist partitions 1 for which there cannot be a resolution of the form (10) over an arbitrary field, e.g., any partition 1 of the form (p - 1,2). In this case (10) is just the sequence (5) which is defined over any held, or Z, and is in fact exact as long as the held has characteristic different from two. Over a field of characteristic two, or over Z, the sequence (5) does have nonzero homology if the rank of F is large. However, we have seen that the enlarged sequence (6) is always exact. It is not unreasonable, therefore, to expect that the defects of the sequence (10) can be corrected by adding more terms, as illustrated in the above discussion. More precisely, we can ask the following question: does there exist, over an arbitrary field, a resolution . * * -+ Ci(l) + . . . + CI@.)+ G,~~~ + LB3

-, 0,

(11)

where each Ci(d) is a direct sum of tensor products of exterior powers of F. One can also ask that (11) be finite and universal (delined over the integers). In Section 4 we construct explicitly a universal resolution c(A) of LA F for a partition 1 = (Ar, &) with two rows. The terms of this resolution are

where the terms A” +’ F@ A’*- ‘F appear as summand of Ci(L) with multiplicity equal to the binomial coelhcient (i: i). In particular (i 1 i) is zero if i > r so that Ci(A) = 0 for i > AZ and the length of the resolution is &. For a discussion of the solution in the case of a general partition the reader is referred to Section 8. Before we continue our discussion it is necessary to bring the skew Schur functors into the picture. By a skew partition we mean a pair A, p of partitions A = (Ai ,..., A,,,) and ,u= (,~i ,..., ,u,,,). It is customary to denote a skew partition by the symbol J/p and to identify a partition L with the skew partition l/(O). The formal character of the skew Schur functor LA,JF) is a

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skew Schur function and can be expressed in terms of the elementary symmetric polynomials by

or equivalently by (13)

which yield (7) and (8) if we take p = (O,..., 0). Let us now go back to the formula [L2F] =det(e+i+j) and expand the determinant by the bottom row. The cofactor of the entry e2,..,,, +j is (-l)m+J times the character of the Schur functor LAci)F associated to the skew shape A(i) = (Al + l,..., AMP1 + l)/(lm-i,

O;-l),

where i = m -j + 1. If we expand by the bottom row we obtain the formula

or equivalently [LAF]=

f

(-l)i+l[A’m-i+‘F@LJ.F].

i=l

It follows from the structure of the resolution (10) (see, e.g., [7]) that over a field of characteristic zero the formula (14) can be realized as an exact sequence 0 -+ Em(A) + ... -+El(i)-+LAF-+O,

(151

where Ei( A) = A Amei+‘FQ LAcijF. As with the resolution (10) such a sequence can be defined over any held but will not be exact in general, and as with (10) there is a remedy. In Section 5 we show that there is a universal exact sequence Lice) I? O+LAcm,F-+

‘.. +LAc,,F+LIF+O

(161

whose ith term is the Schur functor LA,iJF associated to the skew partition J(i)=(J,+l,...,

~~~~+l,~~-i+l)/(lm~i,Oi~‘)

(17)

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formed by adjoining a row of length A,,,+, + i - 1 to the skew partition l
where a(i)

(- l)i+‘[~~m-~m-i+lF~L~
(181

I

is the skew partition (21 + l,..., Am-* + l)/(/Ai,***,fim-i9..*9 )+(lm-i,oi-l)*

In Section 6 we construct a universal exact sequence Lac.,F

where a(i) is the shape

which is not in general a skew partition because p,,, - i may be greater than P !?I* Section 7 deals with the analogous results for the “coschur” functors. We show that there is a universal exact sequence Kat.)F

for any skew partition a = (A,,..., A,,J/(pi ,..., /A~). We also construct in the case m = 2 a universal resolution of KaF by direct sums of tensor products of divided powers of F. Finally in the last section we discuss how the sequences LatwjF, Kate,F may be used to construct resolutions of general Schur and coSchur functors, and we indicate the homological significance of these resolutions. The starting point for the proof of exactness of the sequences (16), (17), and (20) is a filtration decomposition of suitably chosen Schur complexes. The characteristic-free versions of the Pieri formulas discussed in Section 3 are special cases of this decomposition but are treated separately in some detail to elucidate the general decompositions which appear later.

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2. PRELIMINARIES Sequences, Partitions, Diagrams, and Shapes

A partition is a sequence 2 = (1, ,..., &) of nonnegative integers in nonincreasing order

The weight of a partition 1, or more generally any linite sequence 1 of nonnegative integers, is the sum of all the terms of J and is denoted by 111,

It is often convenient not to distinguish between (Jr,..., A,,) and (A 1,..., A,,, 0). For this purpose we let Na denote the set of all i&rite sequences of nonnegative integers containing only a fmite number of nonzero terms. Given any finite sequence 2 = (Ai ,..., &) we can think of it as a sequence (A, ,..., &, 0,O ,...) in N m by extension with zeroes. A relative sequence is a pair (1, p) of sequences in N a such that p s 1 meaning that pi < Ai for all i 2 I. We shall use the notation L/p to represent relative sequences. If both 2 and p are partitions then the relative sequences J/p will be called a skew partition. It is natural to think of a sequence 1 in N m as a relative sequence A/(O) by taking the zero sequence (0) = (0, O,...) as the second member of the pair. Suppose now that 1,‘~ = (Ai,..., ,$,)/(pr ,..., PJ is a skew partition. The diagram of A/p is dehned to be the set of all ordered pairs (i, j) of integers satisfying the inequalities

simultaneously. The shape matrix of A/p is an n x m matrix a = (av) defined by the rule LX&=

1 0

if pi
where we take m = L1. We will illustrate these delinitions with an example. If A/p = (5, 3, 3, 2, 1)/(2, 2, 1, 0, 0), then the shape matrix of A/p is

i 1 011000 01100 00111 1 0 01 0 0

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and the diagram of A/p is

where each dot represents an ordered pair (i, j). The diagram of A/p is ‘obtained from the shape matrix of A/p by erasing the zero entries and replacing the one entries by dots. For convenience the dots of a diagram are usually replaced by square boxes. With this convention the above diagram is represented by

In general we define a shape matrix to be a finite matrix with zeroes and ones as entries. Given any relative sequence A/p = (AI ,..., A,,)/(pr ,..., PJ we can associate to A/p a diagram and a shape matrix with the same rules (1) and (2) used for skew partitions, except that now, m = max(&). For example, if A//J = (4, 5, 2, 3)/( 1, 2,0,0) then the shape matrix for A/p is

i 1 000111 11100 11000 1 1 1 0

and the diagram for A/p is

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1:! :I

Just as it is convenient to think of a partition or sequence A= (A, ,..., A,,) as an intinite sequence (A i ,..., A,,, 0, 0 ,...) with Imite support it may be convenient to think of an n x m shape matrix CC= (au) as an iminite matrix 0

. ..

a11

. .. ...

elm 0

000

0

a nl

.‘.

G?tm

0

cl

0

00

0 . ..

...

. . . ..

of zeroes and ones with Iinite support. If a is the shape matrix of a relative sequence A/p then the support of IX is exactly the diagram of A./p. We will usually not distinguish between a relative sequence A/p, its shape matrix a, and its diagram. For example, we will often write IX= A/p to mean r~is the shape matrix of A/p. The weight of a shape matrix u = (CQ) is defined to be the sum of all the entries ati of IX and is denoted by ial. If u = A/p is the shape matrix associated to a relative sequence, then clearly 1ai = [A[ - 1~1. If 2 = (Al, A*,...) E Na is a partition then its conjugate or transpose is deIined to be the partition I= (11, I*,...), where Aj is the number of terms of A which are greater than or equal to j. Similarly if a = (cx~) is a shape matrix we debne the transpose & = (12~) of & in the usual way by taking ~2~= aji. We shall see below that these two dehnitions are compatible. Let LX= (Q) be an n x m shape matrix. We define the row sequence (al,..., a,,) of E by

and we deIine the column sequence (El,..., Em) of a to be the row sequence of its transpose 6. If tx is the shape matrix of a relative equence A/p then ai = ,$ - pi for all i. Moreover, if A/p happens to be a skew partition, then it is not hard to see that gj= lj- /ij for all j. In particular if IX is the shape matrix associated to a partition 1, then 1 is the row sequence of a and 1 is the column sequence of a. Finally, we make a note here of the convention used throughout Setions 4-7 of not distinguishing between a relative sequence (Al ,..., A,,)/ (P 1 ,..., p,,) and its horizontal translate (A, + l,..., An + l)/(pl + l,..., p,, + 1). Schur Functors Associated to General Shapes

Let R be a commutative ring and let 4: G + F be a map of Iinitely generated free R-modules. We let Ad and Q denote the exterior and sym-

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metric algebras on the map 4. A@ is the antisymmetric tensor product AF@ DG of the Hopf algebras LIF and DG, and @ is the usual tensor product SF@ AG of the Hopf algebras SF and AG. A4 and Sd are nonnegatively graded Hopf algebras Aq5= ZAk(i5,

@ = z& 4

(3)

meaning that the Hopf algebra structure maps are homogeneous with respect to the above gradings. Moreover, ,44 and Sd can be naturally made into chain complexes in manner compatible with their Hopf algebra structures, and the formulas in (3) are decompositions into direct sums of subcomplexes. The components in dimension i of the complexes Akd and Sk4 are given by (Ak&=

Ak-‘F@DiG,

(sk(b)i=skeiF?jjbiiG.

(41

For descriptions of the boundary maps and for details on the Hopf algebra structures of A4 and Sd see [3, Chap. V]. The reader should keep in mind that if 4 is the zero map 0 + F, then A+ is just the usual exterior algebra AF on the free R-module F and S’d is the symmetric algebra SF. Similarly if 4 is the map G -P 0, then Ad is the divided (symmetric) power algebra DG on G and S4 is AG. If IJ = (PI ,..., PJ is any sequence of nonnegative integers, we define complexes Ap 4 and S,,d Ap~=Ap’@l ... QApncj, (5) Sp4=Sp,4Q ... QSP”4 as tensor products of complexes over R. If IXis an n x m shape matrix, we set

where (a, ,..., a,,) is the row sequence of CL We can then define a map of chain complexes, called the Schur map, &(4): Ad -, M

(7)

to be the composition AmdA where X is the chain complex

XL

YA

S&,

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and Y is the chain complex

X and Y are actually the same complex because

A04 = R = So$,

where 4 denotes the complex 0 -+ G + @F + 0 living in dimensions 0 and 1 and R denotes the trivial complex 0 + R + 0 living in dimension 0. It should be observed that the complex R plays the role of the identity under the tensor product operation on complexes. So in fact X and Y are isomorphic to the tensor product of la1 copies of the complex 4, where [ai is the weight of a. As for the other maps in (8) the map u is the tensor product of diagonalizations A: ,la$+Aa~~q5Q

... QAai@

and b: Y + StZ,,,..,Emjdis n-fold multiplication AQ ... QA+A ” of the algebra A = Sq5@ . * * @ Sd which is the tensor product of m copies of sq5. The map &(q5) is a natural transformation and a morphism of complexes because each of the maps in (8) is so. The image of &(c$) is denoted by &(q+) and is called the Schur complex on 4 associated to the shape a. L.& - ) is a functor from maps to complexes over R. If we restrict our attention to the maps of the form 0 + F, then we recover the usual Schur functor La(F) on R-modules. Although traditionally the above terminology has been used in connection with shapes given by skew partitions there should be no harm in extending it to include general shapes. Similarly if we restrict attention to maps of the form G + 0 we obtain the coSchur functor Kzt(3. Since L=(F) is a functor on R-modules F we can view L=(F) as a GL(F)module in a natural way and the same observation holds for K=(F). When viewed as GL.( F)-modules LJ F) and K&F) will be referred to as Schur and Weyl modules, respectively. It should be observed that if A= p, then the shape matrix a of L/p is the zero matrix and the Schur complex LJq5) is the trivial complex R = (0 + R 40). This is because if a is the zero matrix, then A=(4)=

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R = SE(~) and the Schur map &(d) is the identity. It will be convenient for us to also have a shape a such that L=( -) is the zero functor. For this purpose we will use the empty matrix a = (4). If we have a pair A/p of sequences such that p ~6 A, then we will assign the empty matrix (#) as the shape matrix for A/p. Note that these conventions are consistent with the custom of taking A’4 and S,J to be zero for negative integers k. Two shape matrices a and p are said to be equivalent if one can be transformed into the other by permutations of its rows and columns, i.e., /& = agcijTc,jT where c and r are permutations. It is easy to show that if a and /3 are equivalent then the functors LJ - ) and L,J -) are naturally equivalent (see [ 31). Suppose now that 4: G + F is the direct sum 4, @& of two maps tii:Gi+Fi, i=l,2. By this we mean that G=G,@GZ, F=F,@F*, and d(gr,g*)= (d,(g,), &(g*)). For any nonnegative integer p we have the direct sum decomposition

of chain complexes. If P = (pl ,..., pn) is a sequence of nonnegative integers of length rr, then (11) immediately yields a natural direct sum decomposition of the chain complex APd = A4$ @ . . . @ Ap@ as follows:

where the sum is over sequences of nonnegative integers A, B of length rr. We then deline AP(4,, &; u, b) to the natural subcomplex of AP(#) given by the summation

over all sequences ,4, B of length n satisfying A + B = P, [Al = 0, and /Bi = b. It follows from (12) that there is a direct sum decomposition

where p = 1Pi. The above discussion and definitions may be repeated with S in place of A.

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If a is a shape matrix, then we can apply the above discussion to the row and column sequences of a to obtain natural direct sum decompositions

of chain complexes. It is immediate from homogeneity L&($) decomposes into a direct sum

that the Schur map

We can then define L.J4,, &; a, b) to be the image of the map dJ#, , & ; a, b) and obtain the direct sum decomposition

u+b=lal

of complexes. Schur Functors Associated to Skew Partitions Most of what is known about the functors LJ - ) is known in the case a = A/p is a skew partition. We will give a brief summary of those properties which will be needed in this paper, beginning with the presentation of Schur complexes by generators and relations. (See [3] for details and for references to treatments of related subjects.) If A is a Hopf-algebra we let C! A : A @ A + A @ A denote the composite map AQAS

AQAQA-

AQA,

where A,, and rnA are the diagonalization and multiplication Since A = A4 is a graded Hopf algebra we can let

maps of A.

denote the appropriate homogeneous component of q A4. It is clear that ( 16) is zero unless a + b = c + d and if the latter condition is satisiied we

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can describe (16) explicitly as follows. Let x@y~ Aad@ A’4 and let xj xj@x; denote the image of x under the diagonalization map A: A”4 + /P-=4@

AC4

(17)

which is the appropriate homogeneous component of the total diagonal A ,,d: A4 + A#@A#, The image of x@y under (16) is then equal to xixiQxry. Observe that (16) is zero unless CX>c because (17) is so. The relations for Schur complexes can be described in terms of the map 0. We begin with the two row case, meaning that a is a skew partition (AI, &)&A~, pZ) with two rows. The row sequence (a,, ~4 of a is (AI -pr, &-pJ We let IJa denote the map

which is the sum of maps

as described in (16), for ,~r-p~+l
which is defined as follows. If x, @ . . * 60 x,, is an element of A ~~I~-~~~l,Qj+LMj+ 1as in (19) we define Xl Q “’

f,U,+2,.-.,Un)

4

q Jx, @I * .. @Ix~) to be equal to QXj-IQ

•(xjQxj+~)Qxj+12Q

*”

QXns

is as defined in (18). More explicitly if we let where •bjQ~j+lW~a,,~j+,~ xi xji@xJi denote the image of xj~ A q+rd under the diagonalization map A: A?+*#-+ A=$@A’& then using the bar notation for tensors we can write

for (19). Observe that (18) is a special case of (19).

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It is proved as a part of the standard basis theorem for Schur complexes in [3] that the image of lJ= is the kernel of the Schur map &(4). We will identify the cokernel of q a with ,C,J#) in the canonical way. The universal freeness of&(d) is another important part of the standard basis theorem in [3]. More precisely this means that for a given map 4: G + F of free Rmodules of linite rank the component of ,!,Jd) in each dimension is a free R-module of finite rank which depends only on the ranks of F, G and not on the ground ring R. (In particular, the Schur modules LJF) and Weyl modules Ku(F) are universally free; see [3] for references to other treatments.) An important consequence of the standard basis theorem is the decomposition by a filtration of the Schur complex &(#, @ &). Let 4: G + F be the direct sum 4, @ & of two maps di: Gi + Fi and let a = A/p be a skew partition. From (12) we have the direct sum decomposition of the chain complex A ~C$ A=(~)=~A~,~(~~)QA~,~(~*), 0 where the sum is over all sequences CTsuch that p z u z 1. Now for any such sequence 7 we define subcomplexes M&J+,, &) and &Y&(#,, &) by the following formulas:

where da: A,& + sz# is the Schur map and the summations are restricted to partitions c satisfying p s 0 s 1. We shall observe below that one may allow the r~ to range over all sequences p s cr G ,l instead of just partitions. The inequalities 0 > 7 and ~7> 7 refer to the lexicographic order on sequences of natural numbers. Thus CJ> y means that e, = y1 ,..., pi = yi and cJi+I > yi+, for some i. We have a filtration { Mv LJtiI, &) 1p s y s A} of Laq5 by chain complexes, decreasing as y increases in the lexicographic order. Observe that if y is not a partition, then it is clear from (21) that yv&(#r, &) equals hv L,Jd,, &). If y is a partition it is equally clear that My L=(c,~~,&) equals 2y in the MfLJqjI, &) where j denotes the smallest partition lexicographic order. (22) THEOREM. Zf y is a partition, isomorphism of chain complexes ~y,J4,~Q~~,~~42~ -5

p G y s 1, then there is a natural

qL&,v

M~vuL

421

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This theorem, which is proved in [3] using the standard basis theorem, can be reformulated for convenience as follows. (23) COROLLARY. The associated graded object of {MYLz(q5,, $2)]Y is naturally isomorphic to ihe direct sum

the filtration

of complexes where the sum is over partitions 7 satisfying p s y s A.

As special cases of (23) we obtain natural tiltrations of LJF, @FJ and KJG, @ GZ) whose associated graded modules are naturally isomorphic to x7 L,,,PFI 8 LA,,,F2 and &, KY,PGI Q KA,,,G2, respectively, with y as in (23). It will be useful for us to know that the summations in (21) can be allowed to range over all sequences 0, p G 0 CE1, satisfying c > y and c B 7, respectively. This amounts to observing that if XE &J&) @ hA,O(#Z), then there exist partitions 82 cr and elements Xi in &,Jb,)@A&&) such that da(X) = z dJXJ. This is an immediate consequence of a result in [ ] used in the proof of (22), namely the analogue of 11.4.6 in [ ] for Schur complexes. One of the consequences of (22), (23) is a decomposition formula for the chains (Laq5)j in dimension j of the Schur complex L#qS, where again IX= A/p is a skew partition and 4: G + F is a map of free R-modules. The Rmodule (LXq5)j depends only on the modules F, G and not on the map 4 itself. Therefore if we take c$~: 0 + F and &: G + 0 to be the zero maps then we can write (,!,ad)j= (LJc$~ @&))j as R-modules. Let us now consider the filtration of LJ#, @&) defined in (21). Slightly altering the notation of (21) for this special case we have

where we can take e to be sequence satisfying /.Ls IJ s 1. Applying (23) we know that the associated graded object of this filtration is naturally isomorphic to the direct sum

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over partitions

y satisfying p s y G 2. Observing that

and that

we obtain the following result. (25) COROLLARY. There is a natural filtration by R-modules of (LEq5)j defined by (24) whose associated graded module is naturally isomorphic to

where the sum is over all partitions y satisfying p & y s 1 and IlJyi = j.

Although it is not necessary for the above corollary it will be useful for us to know that the filtration terms in (24) are actually subcomplexes of La(#). This amounts to showing that if yl @ --a @ yn E AY,PF, 2163 7-s Qzn~DAp.G,

ad

X=ylz,Q

... Q.Y,J,,EAJ~),

hen

aL4(dJX))

is in M?LJF, G). By detinition cYLJdJX)) = dJa,,+(X)), where ~3~~,a,,d denote the respective boundaries of Laq5 and &4. We have

denoting the boundary of P4, takes the submodule A,Pv,G into ,4vf+‘-PJF@ DA,-?,- ,G. Therefore the ith term of the summation in (26) is contained in the submodule where

iJib,

AVd-PiFQD

of Am& where yi denotes the sequence (y ,,..., yj+ l,..., yk). Since yi> j it follows that aL6(dm(X)) is in MYLE(F, G), or actually even in MY LJF, G), as desired. The decomposition results (22), (23) for a direct sum can easily be generalized to the case of a semi-direct sum or extension. Let di: Gi + Fi be maps of free R-modules and let G = GI @ G*, F= F, @ FI. We say that a map 4: G + F is a semi-direct sum or extension of & by #I if there exists a map c5:Gz+Fl such that #(g,+gz)=~I(g,)+&(gz)+~(g~). We let 4’: G + F denote the map defined by #(g, +gz) = (8(gz), 0). Observe that

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AKINANDBUCHSBAUM

the modules underlying the chain complexes L=(d), &(dr @&), &(c$‘) are all the same. Therefore the subcomplexes MY&(4,, &) of ,C.Jd, @&) delined in (21) can be viewed as submodules of &(4) and &(&). We propose to show that they are actually subcomplexes of the same. In fact since the boundary map of ,!,Jb) is the sum of the boundary maps of &(di @ &) and &(#‘) it is clearly suflicient to show that the M?&(c$, , &) are subcomplexes of &(&). By the discussion above we may assume that 4 = &, that is to say, 4 sends GZ into Fl and G, to zero. We proceed as in the discussion where we showed the tiltration terms in (24) to be subcomplexes, which was actually a special case of our present discussion. Let y1 @ . .. 8~” E A,,,J4,), z,@ **a @z~EA&&), and X=yrz,@ ... @JJ,,z”EAJ~). Again from the delinition ~9;~ takes the submodule AyZ-pib, @A’f-y$Z into ~~1+‘-~~~1~~~~-~~-1~2, where 8ib denotes the boundary of the complex AaJ 4. Therefore the ith term of

belongs to the submodule A,,,,J#r) C3,4&&) fi$&#r, &) as desired. We can now state

proving that ~‘J&(x))

E

(27) COROLLARY. The results (22) and (23) hold if the map q5= q5lCD& is replaced by a map I$ which is an extension of & by dl. The only thing that remains to be checked to prove (27) is the fact that the boundary map of the quotient complex

is the same whether it is viewed as a subquotient of the complex ,C.,J&,Cl3b2) or of the complex L=(d). But recall that the boundary maps of these two complexes differ exactly by the boundary map of the complex LJqS’), and we just showed that the boundary map of &J&‘) takes ~yLA41, d21 ha3 QyLthv d21.

3. PIERI FORMULAS In this section we will prove the Pieri formulas in a form that is valid over any commutative ring and which yields the classical formulas over a field of characteristic zero. Before doing this, however, we will review the classical situation.

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Over a field of characteristic zero the tensor product LAFQ SP F decomposes into the direct sum z0 LOF where cr runs over all partitions of weight /Ai +p containing 1 subject to the restriction that the diagram of D/I contain at most one square in each row. A typical such cr is pictured below:

p= 10.

Similarly, the tensor product LA F@ APF decomposes into the direct sum x0 LOF where ~7again ranges over those partitions of weight 121+p containing ,I, but now subject to the condition that TV/,?contain at most one square in each solumn, e.g.,

p= 10.

These two decompositions, which were originally stated in terms of Schur functions and Schubert cycles, will now be restated formally. Ouer u field of characteristic zero one ( 1) THEOREM (Pieri formulas). has the two direct sum decompositions:

(a) (b)

LAF@SPFzx,,

L,,F, where 101= 1114-p und li<~i<,li-t-

LAF@ApF~xO

LCF, where ICI= IAl +p and lj
1; 1.

There is a closely related pair of formulas for skew Schur functors which are actually adjoint to the above formulas with respect to the usual inner product on symmetric functions or polynomial functors [6]. Over afield of characteristic zero (2) THEOREM (Skew Pieri formulas). one has the two direct sum decompositions: (4

@I

JL,~,~,F%zO LOF, where 1~1= IAl -p and Ai- 1 < oi< Ji; La/m F%xO LOF, where lcr[ = IAl -p and lj- 1
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Without using the adjoint property mentioned above, we will show how Theorem (1) can be recovered as a special case of Theorem (2). If 2 = (Al ,..., &), & # 0, then LAF@ SrF= LA,,CIrjF, where A’ is the partition 1) formed by adjoining p ones to 1. If we now apply formula ~~~‘o~%rt&ern (2) we immediately recover formula (a) of Theorem (1). For the other Pieri iormula, we take 1” to be the partition (A, +p, AZ,..., &) and note that LAF@ ApF= LAC,,Cpj F. We then apply (b) of Theorem (2) to F and obtain formula (b) of Theorem ( 1). LiXP, To prove theorems like (1) and (2) over lields of arbitrary characteristic or, more generally, over orbitrary commutative rings, we must replace the statements of existence of direct sum decompositions by statements asserting the existence of natural tiltrations whose associated graded modules are those specilied in the formulas of Theorems (1) and (2). To be precise, we state now the following two theorems. (3) THEOREM. Let R be any commutative ring, F a free R-module of finite type, and 2 any partition. Then

(a) LAFQ SpF admits a natural filtration whose associated graded module is xg LCF where the sum is taken over all partitions a such that ~a~=~A~+pandA.i
It is sufficient to prove Theorem (4) because, as we showed in the characteristic zero case, Theorem (3) can be recovered as a special case of Theorem (4). Proof of Theorem (4). (a) Using (2.23) we know that LA(R@F), which is isomorphic to LA(F@ R), has two natural tiltrations which yield the following two decompositions:

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LINEAR GROUP

WWF&~

k,RQ Lj.,J

LA(F@R)~~LgFQLA,cR

up to filtration, up to filtration.

Recall that Ll(R @ F) is the direct sum x0+ bZ ,j., Li(R, F; u, b), where LJR, F; u, b) is the submodule of LJR@ F) of degree u in R and degree b in F. Each component LA(R,P, u, b) admits a natural filtration whose factors are Lo R @ LA,0F, where c can be any partition of weight u. Since R is a free module of rank one, Lo R = 0 unless CJis the partition (1 p’ for some p, and in this case L c,pjR~R. Therefore we have Lj.(R,~p, 111 -p)= L A,clpjF.On the other hand, Li(F, R; 111-p, p) admits a natural filtration whose associated graded module is E L,,F@ LA,oR, where the sum is taken over all partitions 0 of weight jJ.1--p which are contained in I. Since L?.,,,R= 0 unless A/cr contains at most one box in each row, in which case LA,oRs R, we get exactly the statement or (a). (b) The decomposition of the Schur function Li(F@G) used to prove (a) does not yield, at least in any straightforward manner, the result asserted in (b). Fortunately, the decomposition of Schur complexes provides the appropriate machinery. Let us take 4: R -+ F to be the zero map. Using (2.25), we know that (LAq5)pdecomposes, up to a natural filtration, into the summation z LcFQ KA,gR over all partitions 0 of weight 121-p which are contained in A. Since R has rank one, KA,gR= 0 unless 1/g has at most one square in each column, and in this case Kl,oR = R. On the other hand, it follows from (2.23) and (2.25) that (LAq5)p = & =~ KgR Q LA,nF.Again using the fact that R has rank one, we observe that KoR = 0 unless rs is the partition (p), in which case Kcp,R= R. Therefore (LAr+h)p =Ll,cpj F and formula (b) now follows from the above discussion.

4. RESOLUTIONSOF TWO-ROWED SCHUR FUNCTORS Let r~= (AI, &)/(pr, pZ) be the two-rowed skew partition. that there exists a short exact sequence of Schur modules

O+LyF+LpF+LxF+O,

We will show

(11

where /I= (A, + 1, AZ)/@, + 1, &) and y=(AI-p2+ 1,&-p,-1). We will then utilize (1) to construct a resolution of La F by direct sums of tensor products of exterior powers of F. Since the R-module F will be fixed, we will usually omit it and write La or L(a) in place of L=F. Similarly we shall write A@ or A(a) for AEF and S* or S(a) for SmF.

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Observing that CQ= Ai- pi= pi (i = 1,2) we have the equalities A(a) = A AipplFQ AAzep2F=A(fl). We will show that the identity map by No -+ = A(y) on the generators induces a map n(a): L(b) +L(a) examining the relations for I,(/?) and L(a). We recall from (2.18) that the relations for L(a) are given by the summation

and those for L(b) are given by

where the terms in (2) and (3) are sent into A(a)=A(fl) by the map Cl. Therefore every relation for L(/3) is also a relation for L(a) and so there is an induced natural surjection II(a): Lb + La as promised. Next we observe that only the relation corresponding to the index t = pI - /A~+ 1 in (2) is missing from (3), so that the kernel of n(a) is exactly the image in LB of this missing term. When t = /A, - ,uS+ 1 we have a~+t=~,-~*+l=~~anda*-t=~~-~,-l=y*,sothemissingtermis A(y) = ,4”’ @Ay2F.Consequently the kernel of n(a) is the image of the composite map

We claim that the image of (4) is exactly the Schur module L(y). To see this, it is sufticient to detine a map G: S(T) + S(j?) which makes the diagram

commutative.

Observing tat the shape matrix for fi is 0 . . . . . . 0 I...... L 1 . . . . . . Il......

-e02 Y2

1 I...... 1 1 () . . . . . . () I%- Y2

Y2

and that the one for y is 1 . ..I 1 . ..I

-Y2

I...... 1 (-J. . . . . . (-J

1

Y1-Y2

1

LINEAR

GROUP

171

we see thats(fl)=s( 1(~2-y2))~~(2y2)~~(l(~J-y2)) and s(v)=S(2y2)@ ~(l(~2-y2))~~(l(~1-y2)). Therefore we can take cx s(T) + S(p) to be the isomorphism which switches the tensor factors s(2Y2) and ,S(l(p2-Y2)). The commutativity of (5) now can be verified by a direct computation or can be deduced from the Lemmas 11.2.5, 6, and 7 of [3]. Having linished the discussion of the short exact sequence (1) what we propose to do now is to write down an explicit exact sequence ... + C;(g) + ... -+ C,(a) -+ Co(a) + L(a) + 0, where each module powers. We lirst ~?,(a)= A(a) so that 0 + C,,(a) +,5(a) + have a sequence

(61

Ci(a) is a direct sum of tensor products of exterior dispose of the trivial case p, > AZ when we have we can take C,,(a) = A(a) and obtain an exact sequence 0 of the desired form (6). In the case p1 < & we will

0 + Ck(a) + . . . + C,(a) -+ Co(a) + L(a) -+ 0, where k = AZ- p,, Co(a) = A(a), and

for i > 0. Suppose, for example, that a has one overlap: a=

that is to say AZ= pi + 1. As with (1.1) we have a short exact sequence (see C31)

so that we do have C,,(a) = A(al, az) and C,(a) = A(a, + a2, 0) as desired. Befor we describe the general case we introduce an arithmetic Koszul complex detined as follows. Let u and a be nonnegative integers, with u 2 u. Let R” = R @ ... @R be the free R-module of rank V with the canonical basis {&i=(o ,..., 0, li, 0 ,..., 0) 1 i= l,..., v} and let R[;] denote the complex O+A”R”+A”-lF’+

... +A’Rv+AoR”+O

whose boundary maps 8” + , : A”+ ‘R” + AnRv are delined by

(71

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The symbol (;) denotes the binomial coellicient and kG indicates that q, is to be omitted. The only terms that are not yet delined in the above sum are those for which j= n + 1 and j= 1. To take care of these two cases, we arbitrarily set in+ 2 = ~4and iO= 0. In other words, the lirst and last terms f the summation in (8) are (z) cjZA *.* A ain+, and (- l)n (iO;;jin &i, A *. . A &i”. Just as the 1ordinary Koszul complex may be obtained as the mapping cone of a map between Koszul complexes of smaller size, the complex delined above can also be described as a mapping cone. (9) LEMMA. Let u and v be nonnegative integers with u> v, and let {v~: AnRv + A”R’} be the sequence of maps defined by

Then q = {v~} is a map of !6[“;

l ] + K[ E], and the mapping cone of q is

KL”Y,l. Proof In the above statement, the only part that may be unclear is the definition of q,,. However, if we use the convention employed above of setting iO= 0, we see that q,,: AoRu + AoR” is simply multiplication by (“:,I To see that q is a map of complexes, one makes use of the identity (g)(t) = (;:;:)(;) for binomial coeflicients. In particular, this identity is applied to prove that

which establishes that q is a map of complexes. To establish the fact that K[ “y ,] is the mapping cone of q, one uses the canonical identilication of A’+‘R”+’ with the sum Afl+‘Ru@ A”R”. Under this identification, the basis element .si, A ... A siV in AnR”, considered as with the basis element an element of K[“Tl] is identified The rest of the proof is now trivial. &i,A .** A&inA&u+,. With this auxiliary construction accomplished, we can now deline, for a skew partition a = (Lr, &)/(pr, pZ), the modules

173

LINEAR GROUP

where ~=P~--/J~+ 1. The module K[rT’]+I denotes the chains of K[;] ofdimension~-l.Notethat~~~~-~~-~=~~-~~-l,sothatC~(~)=O for n>&--p,. Next, we detine the map &,(a): C,,(a) -+ L(a) to be the Schur map da. We also observe that Cl(a) is canonically isomorphic to the sum zra ~A(a, + t, a2 - t) so that we can define the boundary map Jl(a): Cl(a) + C,,(a) to be the map q z of (2.18). To detine the maps

it suffkes to detine, for each 1, the map

To this end, let si, A +. * A 8:” be a basis element of [rT ‘In, x an element of Act-r-l . Then, Aal+r+l and y an element of bn+ ,(a) (4, /Y ..: A e@x@ y) is detined to be the sum

A

...

j-4 A &in-i,

@xj@x;y,

(11)

where 8’ is the boundary map in K [r 7’1, xj xj @ xi is the image of x under the diagonal map

and xj y is the exterior product x,!

A y.

(12) THEOREM. Let a = (A,, A*)/@,, ,u2) be a two-rowed skew partition and k = A2- p, 2 0. Then the sequence

is exact. ProojI We shall proceed by induction on k, having already seen that the sequence 0 + C,,(a) -+ L(a) -P 0 is exact in the trivial case k = 0. Assuming the theorem is true for k - 1, let us prove it for k. From (1) we have the short exact sequence 0 + L(y) + L(p) + L(a) + where

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BUCHSBAUM

(13)

We have to show that the augmented complex c(a) + I,(a) + 0 is exact, knowing by our induction hypothesis that the augmented complexes c(y)+L(y)+O and @Jj?)+,5@)+0 are exact. We shall lift the map L(y) + L(p) to a map $: c(y) + c(j?) of complexes. We know from (4) that the map 15(y) * I,(/?) is induced on the generators by the map Cl: A(y) +A(/?). Since C,(~)=A(JJ) and C&?) = A(b), we can take $,,: Cc,(y) + C&?) to be the map Cl. Next we observe

so that in particular C,,+,(y) and Cn+,(/3) are isomorphic modules for all Let n>O. We detine I/I,,+, : CH+ ,(y) + Cn+ r(p) as follows. ***@si~~K[‘~‘],,,x@~~~(aI+r+l+1,a2-r-l-~),andset &I = &i,Q

Lt is an easy computation, similar to (9), that $: c(y) + c(p) defmes a map of complexes and that the mapping cone of @ is the complex c(a). It follows that C(a) is a complex whose homology is zero in positive dimensions and whose zeroth homology is L(a). This establishes the exactness of the augmented complex C(a) + L(a) +O and hence the truth of the theorem. As an illustration of the use to which the above complexes may be put, we shall calculate A2(A2F) over an arbitrary field of characteristic p. (See also [8].)

LINEAR

Let A be the partition

175

GROUP

(2,2). We have the commutative

diagram

0

0 ~

/l4F

h(d) -A4F@

A3F@ F-=+

A2FQ A’F-

b

??I

A4F

o-

----f-+

S2(A2F) -

I

0

LIF-+O

I 0

where A is the diagonal map, m is multiplication, and u is the map sending to Xl AX2 A x3 A x4 the alternating sum txl /Y x2)(x3 * x4)(x1 A x3)(x2 A x4)+ (x, A x4)(x2 A x3). The map b is the transpose of d2(A). Since the two bottom rows and the two right-hand columns are exact, it is clear that A2(A2F) is the l-dimensional homology of the following small double complex: 0

0-

0

I

I

A4FA

A3F@F

2

In I

O-

(14)

A4FL

I

A4F

where 2 denotes multiplication by 2, and A and m are the diagonal and multiplication maps of AF. If the characteristic p of our ground lield is different from 2, then 2 is invertible and the l-dimensional homology of the complex (14) is clearly seen to be isomorphic to the cokernel of A. Therefore if p # 2, then A’( A2F) is isomorphic to the cokernel of A, which is LC3.,, F. If, however, p = 2, then the l-dimensional homology of (14) is easily seen to be the direct sum of A4F and ker(m)/A4F. But ker(m)= Kc2, l, ljF, so A2(A2F) is F A4F. Thus, A2(A2F) is an example of a Z-form of Lc3,1jF ~4F@J&,~,~, I which is not indecomposable over every field. It is known that Lc3 ,)F is an irreducible CL(F)-module over any infinite lield of characteristic p # 2.

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Therefore, over any infinite field, we have that A*(,42F) is irreducible p # 2 and is decomposable if p = 2.

5. THE SEQUENCE L&

if

FOR A PARTITION

In this section we will construct the fundamental exact sequence LAC.)F as a subcomplex of a suitably chosen Schur complex. We let 4 be the map

which maps R identically onto R in F@l R, i.e., 4(r) = (0, r). To avoid confusion, we shall denote the domain of 4 by R2, and the summand R of F@ R by R,. Then 4 is the direct sum 41 @ 42 of the maps 4,: 0 + F and qb2:R2 += R,. We will also, in the context of Young tableaux, denote the identity elements of RI and R2 by the symbols 1 and 2. Let A be the partition (A ,,..., &) with k rows and denote by 1’ the partition (A1 + l,..., & + 1). The Schur complex LAP(#) decomposes, as in (2.15), into the direct sum of subcomplexes x LA, (#,, qS2;a, b) where the sum is over all pairs a, b of nonnegative integers such that a+ b= 11’1= A+ k. From [3] we know that the complex L,J4) is exact in every dimension except zero, and that the homology in dimension zero is exactly LA,(F). It follows that each subcomplex LA,(#l, 42; a, b) is exact, with the exception of (a, b) = (/Aft, 0). (1) DEFINITION. L-%4&h

X’(A)

is the subcomplex

LIT(qhl, qh2; [A’[ - s, s) of

From what has been said, then, we have that X’(A) is an exact complex for 0~s~ iA’1= 111+k. We know from (2.22)-(2.25) that, up to filtration,

where X’(A.)i denotes, as usual, the i-dimensional chains of X’(A), and the summation is over all pairs of partitions (0,~) such that r~< y < A’, jyl= IA’i-i, and ~c~~=s-i. Since R,=R2=R, we have L,,Rl=O unless c = ( 1’ ~ ‘) and KA,,?R2 = 0 unless Al/y has at most one box in each column. In terms of Young tableaux, the picture to keep in mind is

LINEAR

I

177

GROUP

mi

where the overall shape is that of I’, the boxes indicated by ones and twos are filled by the respective identities of R, and RI, and the space denoted by F is filled in with elements of F. Consider now the special case that A. is a rectangle, i.e., that A = (pk). From the above observations, we have XS(A)i = L, RI Q L,,, F@ KAs,,R2 where 0 = (lSei) and y = (A, + l,..., I,-,+1,&+1-i). Whens=k, y/o= (A, + l,..., Ak-, + 1 -i)/(l”-j) is just the shape A(i) defined in (1.17). But then, since L,,a-,,R = R and KAslyR= D,R = R for 0 6 i< k, we have Xk(A)i= L,,,,F. Therefore, when I is a rectangle, the complex J?(A) is the fundamental exact sequence II,(.,F. When I is not a rectangle, but an arbitrary partition, the exact complex P(A) contains ILic*,F as a subcomplex, but is generally larger. To see this, and to understand the complex Xk(A) better, we will introduce additional filtrations. However, before making the formal constructins, we will give an informal sketch of the ideas they involve. Recall that Xk(A)i is generated by standard tableaux that look like 1 121 .*. 12F

x

1.

(4)

121 . . . 12 121 . . . 12

where the shape is A’, the number of twos is i, and the number of ones is k-i. If we restrict attention to those tableaux in which twos appear in the bottom row only, then the only tableaux in dimension i are of the form

I 1 F l!---c

/I1

k-i

I

i

2 . ..

2

I

i i

(5)

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corresponding to the shape n(i)=(L, + l,..., &-i + 1, A,+ 1 -i)/(l’-j), explaining the assertion that LnCijF~ Xk(J)i. From the discussion following (2.25) and the above observations it is easy to see that J?(L) contains a subcomplex of the form

which we will take to be II ,(.,F. To prove the exactness of L1(.) we shall introduce a filtration on X“(n) of which L,(., forms the top level. Essentially, the rth level of the filtration, denoted by J?(J), is the subcomplex generated by the tableaux of type (4) in which the twos can occur only up to the (I + 1)-st row of A’: f t

1

11,

(6)

Thus J?‘(n) = J?(n) as no restriction is placed on the rows when r =O, and Xkak- ‘(A) = [II(*) as discussed earlier. An explicit description of the boundary maps in the complex I,(., will be given in Section 6. The remainder of this section is devoted to showing the exactness of I-,,*,. To proceed we must generalize the foregoing discussion a bit, and consider the complexes X’(A) for all s, together with their filtrations ocx-‘(A)&

... GP’(n)~~“(n)=xyn),

where P’(A) is the subcomplex of J!?(n) generated by tableaux of the form (6) in which the twos can appear only up to the (r + 1)th row of 1’. This formulation allows us to interpret the quotients Xs*‘-‘(n)/P’(L). If then clearly P- ‘(L)/P’(d) must be zero. However, if 4=&+1, Jr>&+,, then P-’ (n)i/X’~‘(n)j is generated by tableaux of the form

LINEAR

GROUP

179

where there is at least one 2 in the rth row. Observing that s - i= (s- 1) - (i- l), we see that the tableaux in (7) describe the generators of the complex .Y- I*‘- ‘(p) in dimension i- 1, where p is the partition &). In fact we have a short exact sequence of (A I,..., k-1, A,1, &+1,..., complexes

where u is a map of degree - 1 of complexes. With these observations we can prove by induction on r that if 0 < r < s, and A is any partition, then P’(A) is exact, with the case r = 0 being the exactness of the complex P(A) for s > 0. We can therefore conclude that the complex [L,(,, = J?-‘(A) is exact as desired. Now that the idea of the proof is clear, we merely have to show that the formal machinery of Schur complexes allows us to make the preceding discussion rigorous. Consider, then, any map of the form $: R + F’ and any partition y = (yi,..., yk). In our situation F will be F@ R and y will be 1’. The complex A,(+) consists of the terms LY”‘F@Di,R@

..a QAYk-“‘F@D,R

(9)

and contains, for each r, the subcomplex whose terms are those in (9) subject to the condition that m, = ... = m, = 0. We define the subcomplex MY of L,($) to be the image of this subcomplex under the Schur map d,$, for O< r
(10) LEMMA- (a) If Yr’Yr+l,

then there is a short exact sequence -+“M’-l-+0 where v denotes the partition a*\, u is ‘the natural inclusion, and v is a map of , ,..., a, , , ar l,..., (a degree - 1. O-FM;

(b)

-PMj,-’

If~~=y,+~,

then Mt=Mj-‘.

Proof: (a) We let E denote the canonical generator of the module R corresponding to the identity element of R. The complex A,$ is generated by elements of the form

X,Q&(m’)Q

... QXkQ&(mk),

(11)

where xi E AY1-mlF. We define a map A, + w A,$ by sending (11) to the tensor

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AKIN AND BUCHSBAUM

where (12) is understood to be zero if m, = 0. It is a straightforward computation to check that w sends relations for M’-’ of type (2.20) into relations for M’- ’ so that w induces a map u: M;- ’ + M;- l. Moreover since w maps the generators of M;- ’ onto the generators of A4- ’ the map u is surjective. It is also easy to check that Y is a map of complexes and that u sends MY to zero. To see that the kernel of u is MY it is sufficient to show that M;- i/M; and M’ are free R-modules of the same rank. This is an easy consequence of (2.22). (b) This is again an easy consequence of (2.22). As an immediate proposition.

application

(13) PROPOSITION.

of this lemma

we obtain

the following

ZfO < r < s < II) + k, then A?‘(l) is exact.

Proof. We use induction on r. When r =O, P”(1) = F(J) which we observed to be exact for 0 < s < 121+ k. Suppose r > 0. If I, = A, + 1, then by part (b) of Lemma (lo), P’(J) = P-‘(A) and we are done by induction. then it is an immediate consequence of part (a) of Lemma (10) Ifk>4+, that we have a short exact sequence of complexes

where p = (A,,..., A,- 1, A,- 1, A,+ 1,..., A,). Since J?-‘(n) and J?‘,‘-‘(P) are exact by induction, it follows that P’(J) is also exact. (14) COROLLARY.

L,,.,F=X

k-k-l(A.) is exact.

6. THE SEQUENCE [I,,,, FOR A SKEW-PARTITION The Construction of L,,.,F Let a = (Al )...) n,)/(pL, )...) pk) be a skew-partition. seen that there is a short exact sequence

When k= 2 we have

0 + L(y) + L(p) + L(a) + 0

(1)

of Schur modules where /I=(~1,&-1)/(~1,~L21) and y=(1,-p,+ 1, 1, - p1 - 1). An analogous sequence does exist in the case k > 2 if one takes p =

tn,

,...,

I,

- 1) Izk -

1 )/b,

y = (4 ,*-*,~k-1,~k-l)/(~,,...,rUk-2,rUk-l,~k-1).

,...,

pk

- 1, pk

-

1 1,

(2)

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First, observe that any skew-partition c1can be represented by a pair A/p where pi B 1 for all i so that we need not concern ourselves with negative entries. Second, the shape y is not a skew-partition although it may be equivalent to one. The latter is always the case when k = 2: the shape (A, -p2 + 1, (43 A,- lMP21, P,) is equivalent to the skew-partition A2 - pL1- 1). Third, when pLkP, 2 &, then Ak - 1 < pLkP, and y is therefore the empty matrix (4) which means that L(y) = 0. We will now proceed to describe the construction and exactness of the above sequence, beginning with the surjective map L(b) + L(U). It is clear that A(a)=A(/?) b ecause ai = bi for all i. We claim that the identity map A(/?) -A(a) on the generators induces a map L(p) -+L(cc). This can be seen from an examination of the relations for L(B) and L(E). Recall that the relations for L(a) are given by the double summation i

j=2

[

1

A(a,,...,

OLj-l

+t,

aj-t,...,

‘>+-A+

OZk)]

(3)

and analogously for L(p). It is clear that L(p) has exactly the same relations for i < k, and when j = k the inner summation is

Consequently the only difference between the two sets of relations is the term A(y,,..., yk) with indices j = k and t = pLk_ i - pLk+ 1 in the summation (3). Since every relation for L(b) is also a relation for L(B) the identity map 4B) + = A(a) induces a natural surjection L(p) + L(U) and thus the kernel of this surjection must be the image in L(b) of the one missing term A(y,,..., yk). More precisely the kernel of L(p) -+ L(a) is the image of the composite map A(Y) --=+

A(/?)

dM) b L(P)

(4)

and we claim that the image of d(P)0 0 in (4) is L(y). To see this one must first observe that S(y’) = ,S(p) and then check that the diagram

(5) is commutative. The verilication of the commutativity of (5) is a straightforward computation which actually reduces to the two-row case

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discussed in Section 4. This completes the description sequence O-+L,-+L,+L,-+O,

of the short exact (6)

where y and B are as in (2). We observed earlier that although the shape y in (2) is not a skew-partition it may sometimes be equivalent to one. When the number k of rows is greater than two this is usually not the case as the following minimal example illustrates. If we take a = (3,2,2), then y = (4, 3,2)/( LO, 1). The matrix representation for y is 0 1 1 1 1 1 1 0 0 1 0 0 and it is easy to see that this matrix is not equivalent to a skew-partition. To utilize the exact sequences (6) in a recursive manner one has to enlarge the family {a} of skew-shapes to a family of shapes which stay closed under short exact sequences of type (6). Let a = l/p = (A, ,..., A,)/@, ,..., pk) be a skew-partition with k rows. The shape a can be represented by the diagram

I

I

I

s,-1 Is2-1 , ,

I

al a2 a3

ak-l ak

with rows of length a I ,..., ak and protuberances si - l,..., Sk-, - 1 on the right, where ai = lj - pi and si = Izi - li+ i + 1. For any nonnegative integer p we let p = zi30pi2’ be its dyadic expansion and define a skew-partition associated to a and p by the rule

dpl = (A,..., ~k-lI,IZk-(P))/(~LI-Pk-l,...,~k-PO),

(7)

where (p ) = C pi, provided that the relative sequence in (7) is a skew-partition and that p < 2k-1. Otherwise we take a[p] to be the empty matrix (4). In formula (7) we will allow ourselves to replace A/,u by the columnequivalent shape A’/,’ = (A1 + l&k + l)/(pi + l,..., pk + 1) to guarantee that the sequence (cl1 -pk -,,..., pk -PO) have nonnegative entries. If p < 2k - ’ we define a sequence ?(P)=

bk-lmPl,h)

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whose weight is (p) = Cpi so that we may write (7) in the convenient form

a1 = (A- (P> V(l)MP-V(P)) provided a[~] is nonempty. In the non-empty case the shape C&J] can be represented by the diagram a1 +Pk-I s, - I /

/

/ /

I

/

/

/

a2

+Pkp2

I

Sk-z-

1

elk-

Is,-,-n-l

1 +Pl

ak+pO-n

whose rows have length a, +p k _ I )...) ak-,+pi,ak+po-n and whose protuberances on the right are s1 - l,..., skp2-1, Sk-,-n-l, where n= (p). In other words, if we write ~=2~-‘1+ .a. +2k-in with l
bkpf,...,

pk,

(8)

pk-th

where the symbol j&-, is to be omitted and it is understood that t # a is the empty matrix if t > k. In the case t < k the shape t # a can be represented by the diagram

I _’

whose

have

J

a-+

1 (9)

length a ,,..., ak--I--l, ak-t+l+ik --I)..., ak+jk-,, + Sk_ ,) and whose protuberances on the right are the same as those of a, where ii denotes Si - 1. We are now ready to define the maps from which the fundamental ak-t-(ik-t+l

rows

r’

I

+ “’

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sequences will be constructed. Let /I = (a ,,..., CJ~)/(P,,..., tk) be a relative sequence where r~ is a partition and set b[ 1 ] = (a, ,..., bk _, , e‘k - 1)/ (5 I ,..., Tk ~ r, rk - 1). We claim that the identity map Apcr7 + = A, on the generators induces a surjection

To see this it suffices to construct a map S, -+ Sg, where y = p[ 11, which makes the diagram

(11)

commutative. We first observe that the shape matrices p and y differ only in their pth and qth columns, where p = pk - 1 and q = &, so that bi = Ti for i #p, q. Moreover it is easy to see that jr, = g, - 1 and 7, = BP + 1. The map S, + S, can then be defined by sending a tensor x1 Q * * * 8 x, in S,=S,,Q . * . Q SYm(m = cl) to the tensor cx,Q where the summation map

... QxpjQ

... Qx’,~x,Q

... Qx,,

xi xPj Q xbj is the image of x, under the diagonal

It is routine computation, formally similar to (5) and (4), to check that diagram (11) is commutative. Let us now return to our skew partition 01= A/p and show that there is a natural inclusion

L((r+ 1) #a) -

L(r#a)

(12)

for any t 2 0. For the sake of convenience we set y = (r + 1) #a and p = ?#a. To show (12) it is sufficient to construct a map A, --* A, on the generators which makes the diagram

(13)

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commutative. It is easy to see from (8) or (9) that the /I differ only in their kth and pth rows, where p= kfor i #p, k. Observing also that yP = flk + 1, - 1, and can then define a map A, + A, by sending a x,0 ... Ox, in /i Y =/iy’@ . . . @ nrk to the tensor

shape matrices y and t - 1, so that yi=Bi yk = p, - I, + 1,, we tensor of the form

xx, Q... QXkXpjQ ... QXLj, where the summation map

cjxPj@xI,

is the image of xP under the diagonal

Once again it is a straightforward computation to verify the commutativity of diagram (13). With the above maps at our disposal we turn to the task of constructing the fundamental sequences. If t and m are a pair of nonnegative integers we assign a shape a(t; m) to t, m, and LY= (2, ,..., A,)/@, ,..., pk) by the rule a(t;m)=t#(a[m2r+2’-

l]),

(14)

where it is understood that the shape t #/I is empty if /I is empty. More explicitly if m=C m,2j, then a(t; m) is the relative sequence (1 I?“‘? Ak--l, ~k-(m)-t)/(CL1-mk-r-l,..., Ilk-r-l-ml? pk-r+l-19-? pk - 1, pk _ r - mo) or possibly empty. Using this notation the short exact sequence (6) can be written

(15) We will construct similar sequences O-+L a(t+ I;O)+ L(,;,, + &;O) + 0

(16)

for t >O. We can take the map L,(,;,, --f ~5,~~~~)to be the surjection I7(a(t; 0)) because if /I = a( t; 0) it is easy to see that /I[ 1 ] = a( t; 1). Also if we take /I = a[2’+ ’ -13, then (t+l)#/I=a(t+l;O)and t#j?=a(t;l)so that from (12) we have a natural inclusion L,(,+ ,:oj t L,c,;l,. It is not hard to check that Lorc,+,;o, is contained in the kernel of Z7(a(t; 0)) and thus the sequence (16) is a complex. By construction the sequence (16) is

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exact at both ends and the remainder of this section is devoted to proving exactness in the middle. The fundamental sequence II,,,, is constructed by splicing together the sequence ( 16) as follows. For i > 1 we have a(i) = a(i - 1; 1) and so we can define the boundary map J&+ ,) + LEciJ of te complex II,,,, to be the composition L,(i;i) + L*(i;O) + L,,i- iii). Finally since a(0) = a we can take the boundary L,,,) -, JL~) to be the surjection n(a). (17) THEOREM.

The sequenceO_,,.,is exact.

It is clear from the construction of L,,., that the theorem the exactness of the sequences (16) for all t. The proof involves a spectral sequence argument and in the analysis sequence it will be convenient to have available sequences O+L

a(t+

1;m) +

L(r;zm

+ 1) +

La(,;Zm)

+

0.

is equivalent to of the theorem of the spectral of the form (18)

To define such sequences we observe that if /I = a[m2’+ ‘1, then a(~; 2m) = P(t; O), cr(t; 2m + 1) = JI(t; l), and a(t + 1; m) = /3(t + 1; 0). Thus, we may simply take (18) to be the sequence 0-L

B(r+1;O)+ Lm1, + &?(r:O)+ 0,

(19)

where /? = a[m2’+ ‘1. The Spectral Sequenceand the Exactness of II,,,,

We begin by reformulating the constructions in Section 5 in terms of the skew-partition a = (J., ,..., &)/(pi ,..., pk). As we did there we let tj: R2 + (‘J) F@ R, denote the direct sum d1 @ 42 of the maps 4i: 0 + F and e2 : R2 + ’ R, , where R, = R2 = R. We also let a’ be the skew partition n’/p = (2, + l,..., & + l)/(pi ,..., &) whose rows have length ai + l,,,., ak + 1 and whose protrusions on the right are the same as those of a. The Schur complex L,.(4) decomposes into the direct sum C L,.(d,, 42; a, b) of complexes, the summation being taken over all pairs a, b of nonnegative integers whose sum a+ b equals Ia’1 = 11’1- IpI = 121+k- 1~1. If we let X”(a) denote the complex t,(#i, d2; Ia’1-s, s), then we can write L,.(d) = C, X’(a). From [3, V.1.163 we know that the complex L,.(d) is exact in every dimension except zero and that its homology in dimension zero is L,.(F). Consequently the complexes .-Y(a) are exact for s#O. It follows from (2.22-5) that the component A?(a), in dimension i decomposes by a natural filtration into C L,,,R, 81L,,,F@ KnrlyR2, where the sum is taken over all pairs 0, y of partitions satisfying PC Amy 5 A’,

187

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[(TI = s - i, and In’/yl= i. We note here that because R, is a free R-module of rank one we know L,,,,R, vanishes unless a/p has at most one box in each row (a E p’). Similarly KAslyRz = 0 unless n’/y has at most one box in each column. As in Section 5 we obtain a filtration 0 G J?(a) E . . . E J?(a) E . . . E P’(a) G Pa(a) = F(a) by taking P’(a)

= Y(a)n

M,~,) L,.(F@ R,, R2),

(20)

where r{l} denotes the partition (A, + l,..., 1, + 1, pLr+, ,..., pk). We have the following results analogous to (5.10) and (5.13). (21) LEMMA. (b)

(a)

V &>&+I,

~s~‘-l(a)--rXS-l,r-‘(~)~O,

If 1, = Iz,, ,, then JY(a) = X’2r-‘(a). then there is a short exact sequence 0 --f Pr(a) + where P=(A, ,..., Jr-,, A,- 1, Iz,+1 ,..., 2,)/p

and the surjection has degree - 1 as a map of complexes. (22) PROPOSITION. lf 0 c r < s, then JP(a) is exact. the exact complex Xkgk- ‘(a). Zf a is a partition 1, as in is the fundamental exact sequence [L,,(.). In the general are related through a spectral sequence arising from {A,(a)} which we describe below.

We let A(a) denote Section 5, then A(1) case A(a) and II,,,, a certain filtration

If we let 0, : 0 + R, and 0,: R, + F denote the zero maps, then we can think of the map 4 as an extension of O2 by 8,. Therefore we have from (2.27) a filtration {M, L,.( 8 i, 0,) of the complex L,.( 4) whose associated graded object is the sum C L,,,@, @LA.,,@, over all partitions (T satisfying p ECJ ~2’. Actually we need only sum over /J E CTE$ because if G YLp’ then some row of a/p has more than one box so that L,,@, = L,, R, = 0. Observing that any (T satisfying PG (Tc p’ can be written in the form (T= p’ -n(i) and that i >j if and only if CL’-q(i) Q p’- q(j), we have an increasing filtration {M,, _ 4Ci)L,.( 8,) S,)} indexed by i = O,..., 2k - 1. We can then define

A,(a)=A(a)nM,,-,(,,+,,L,.(Q,,

69

(23)

for p = O,..., 2k- ’ - 1 and obtain an increasing filtration OsAo(a)sA,(a)s

**. sA,,-t-,(a)=A(a)

with the property that A,(a) = A,- 1(a) if $ - n(2p + 1) is not a partition.

607%/?-6

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We let {E;Jcr)} denote the spectral sequence associated to the filtration {A,(a)} of the complex A(a). It will be more convenient for us to use the total degree n instead of the complementary index q. So we shall let,!&(a) denote the E’ term with filtration degree p and total degree n:

We begin the analysis of E’(a) which is by definition just the {A,(a)}. We know from (2.25) the complex A,(a) in dimension the sum

over all pairs 0, y of partitions (a)

~Lfrdyyc’,

(b) (c) (d) (e)

REP’,

(0

IdA + IJ’hl =k.

fJ2$-1(2p+

by describing explicitly the term E”(a) associated graded object of the filtration and (2.27) that the component A,(a), of n decomposes by a natural filtration into

satisfying

11,

(25)

Il’l~l =n, r>(&,..., L,,n),

Conditions (e) and (f) are due to the fact that we are inside the subcomplex A(a) = Xk*k-l(a) of L,.(4). Recalling the fact that any u satisfying ~1E CJE $ can be written uniquely in the form p’ -q(m) with 0 j the condition (c) on (T= $ - q(m) becomes m < 2p + 1. Also the condition (d) on y = Iz’ - (m) . q( 1) is just the equality (m ) = n. Combining all of the above observations we can rewrite the summation (24) subject to (25) in the more convenient form 2p+ c m=O (m>=n

where a[m]

I Lwr,rl(m,,,pR~

(26)

~‘L,c,,FOK~,(~,-<~>,*(,~R~,

is the skew shape (J.‘-(m)

q(l))/(p’--q(m))

if &-q(m)

is a

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partition and is empty otherwise. Using the fact that R,, Rz are free Rmodules of rank one the summation (26) can be rewritten simply as 2p+

I

1 ??I=0

(27)

L,rn,F.

(m)=n

Since q(a) is the quotient complex A,(a)/A,-,(a) it follows from the above discussion that the component ,!$,(a) in dimension n is isomorphic by a natural filtration to the sum 2p+

I

c

LabIF

m = 2p
which clearly has at most one term. Therefore we have proved the following statement: L @C2Pl F

q;,(a)

=

LaCb+ UF

I

0

if (p)=n, if (p)=n-1, otherwise.

(28)

Setting h = k - 1 for convenience, in the case (p) = PI we can write p=2h-‘1+ . . . +2h-in with l
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can be written completely

as

AND

BUCHSBAUM

aI a,+ 1 a3- 1 a,+ 1

a2 a3- 1

a,+1 a,+ 1 a,-2

The

only possibly

nonzero

boundary maps in the E” term + 1 and it is not hard to check that is up to a sign equal to the map n(a[2p]): LaC2p+ ,, + L,,,, induced the identity map Azc2P+, , + = A,,,, on the generators. It follows from exactness of the sequence (6) or (15) that the E’ term is

a;,,:qin--t-qn- 1, where n = (p)

are a;,, by the

otherwise if

n=(p)+l, 3

(29)

where a( 1;~) = 1 #a[2p + l] is the shape whose rows have length if p is even and aI ,..., a,+1 ,..., a,-,+1 ,..., a,,+l+S,,,ah-Sb-n+l al ,..., c(i, + 19...,ai,_, + I,..., ah + 1 + sh, a,, - sh - n + 2 if p iS odd. In either case the protuberances are the same as those of a[2p + 11, namely Sl - l)...) She1 - 1, s,i-n1. Before proceeding with the analysis of the spectral sequence in the general case we will examine the case k = 3 (h = 2) in greater detail. Let us assume that the fundamental sequence is exact for all shapes whose third row is shorter than a3 (the case a3 =0 is trivial and the case a3 = 1 is

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191

actually the exactness of the sequence (24)). It follows from the above discussion that the E’ term 0 0 t E&l t EL

0 0

t

t

0

G;*

t

t

0

G;3

0

0

t

can be written as 0 t L,(W) t Ld,:,, t 0

and the shapes involved can be pictured as

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It is a routine but tedious computation to check that the differentials Lcl;lj -, Lcl;oj and Laclt3) -+ L,cl;2j are up to a sign equal to the surjections U(U( 1; 0)) and Z7(cr(1; l)), respectively, induced by the identity maps on the appropriate generators. Consequently their respective kernels form the only possibly nonvanishing terms E:,2 and E:,, of E’: 0

0

EL 0 2 0 0

0 0 G;3 0

We will show that ETi2 = LacztO,, E& = Lmc,;,,, where a(2; 1) = ~~;~~~-s2

a(2; 0) = v;;;z;ms2

and the differential Lac2;,) + Lac2;0j is an isomorphism. Before doing so we first observe that the equality Efi2 = L,czzoj implies the exactness of the sequence

and hence the exactness of L,,.,. We recall from (19) that the sequence (30) is just the sequence

where /I is the shape

Since fi3 = aj - 1 < a3 we know that LB(.) is exact by our induction hypothesis. We can then conclude that the above sequence (30) is exact and hence that 2*. = L . For r>2 we k:iw t$r?‘the only possibly nonvanishing terms of E’ are Esi3 and E;;*. It follows that the differentials 8 of E’ must be trivial for r 2 3 and we can conclude that E3 = I? = ... = E”. Since our spectral

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sequence arises from an exact complex we see that E3 = E” = 0 proving that the differential c&: E& + Ef.2 is an isomorphism. It can be checked directly that ai;, is up to sign equal to the map Z7(cr(2; 0)): L(,;,) + L(m) E Ef.2 so that we have the other desired equality ’ E:;z = La(w). We now return to the analysis of the spectral sequence in the general case k> 3. As in the case k= 3 we will assume that the fundamental sequence [LBC * ) is exact for all shapes /I whose last row has length flk less than elk (the case flk =0 is trivial and the case Pk = 1 is the exactness of (15)). Let us recall that EL;” equals L,(l;P1 if n = (p) + 1 and vanishes otherwise. It is clear from the gradings on E’ that the only possibly nonzero differentials a;.,: EL.,, + EA- i.+, are those where p is odd and n = (p) + 1. It is a routine but tedious exercise in diagram chasing that the map a:;,,: JL(~;~) + Lcr(l;p- 1) is up to a sign equal to the surjection Z7(a(l;p-1)) induced by the identity ~olCIP) += ,4Eo;P-l) on the generators. This is the kind of analysis we make of the higher terms of the spectral sequence in the following statements: E$, vanishes unless p - 2’ - 1 mod 2’ and n = ( p ) + 1. Ifp=m2’+2t-1

and n=(p)+l,

then E~~,=L,cI+,i,,.

(31) (32)

The differential a’& : E$, + Ei’- 2,.n_ I is zero unless p = 2’+ ’ - 1 mod 2’+’ and n=‘(p)‘+ 1. ’ (33) If p=m2’+‘+2’+’ - 1 and n = (p) + 1, then the differential a;;“: L m(t+ 1:2m+ I) + Gx(t+l;Zm) is equal up to sign to the surjection lir(a(t + 1; 2m)). (34) We will proceed to prove these statements by induction on t, the case t = 0 already having been taken care of. So let us assume (3 l )-(34) to be true for t and prove them for s = t + 1. By (31) and (34) the only nonvanishing terms of E2’+’ are the kernels of the differentials a&, where p= -1 mod2”’ and n = (p) + 1. It follows immediately that EEn vanishes unless p and n are as above, proving (3 1) for s. Statement (33) follows from (3 1) because of grading considerations on EZs. Moreover the differentials of E’ are trivial for 2’ + 1 < t- < 2” - 1 since E; = 0 for p&2’+‘-lmod2’+‘. Therefore we have the chain of equalities of the terms E2’+ ’ = . . . = E2’ as modules. Let us now recall from (18) the sequences 0 + J&x+ I;m) -+

&;2m

+ I) +

L(s;Zm)

+ 0

(35)

which we know to be exact at the ends. Setting p = m2” + 2” - 1 and n=(p)+1 we have ET,, = Ej$ ’ = ker c?& = ker(Z7(a(s; 2m))

(36)

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from the above discussion. It follows from (35) and (36) that I,,(,+ ,CZmjE E,$. Before showing equality we make a note of the following fact. The differential of E*’ takes L,(,+ ,;2m+ ,,into I,,(,+ 1;2mj and this map is up to sign equal to the surjection I7(a(s + 1; 2m)) in (35). (37) The verification of (37) is a straightforward but tedious computation which we shall omit. Observing that once (32) is proved for s the statement (3) becomes equivalent to (34) for s, we return to the proof of (32). For any m > 1 the sequence (35) can be rewritten as (38) where /I is the shape a[m2s+‘]. S’mce fik
(39)

form>l,p=m2”+2”-l,n=(p)+l. To finish the proof of (32) it remains only to show that (39) holds for m = 0 as well. For this purpose we will examine the sequences E;z+,-,;,+,A

E;s-,a~E;s-,-,;,-l

(40)

arising in the E’ term for r 2 2”. We first observe that the term on the right is zero because its filtration index 2” - I - 1 is negative. When r = 2” we know from (39) that the first term in (40) is L,t,+,;lj and we know from is mapped onto L,(,+ l;oJ by C. Therefore to show (37) that L(s+lil) L ah+ 1;O)-- E2”*s- 1;s we need to prove that (40) is exact for r= 2” or equivalently that Es,- I;r = 0 for I = 2” + 1. We claim that the first term in (40) is zero for all r > 2” + 1. To check this it is sufficient to consider those values of r which satisfy the two equations 2”+r-1=2”-lmod2”, s+l=(2”+r-l)+l

(41)

simultaneously because Ei:, = 0 unless p 3 2” - 1 mod 2” and n = (p) + 1. The first equation in (41)’ implies that r = m2” for some m, where m > 2 because r 2 2” + 1. But then the second equation in (41) becomes (m ) + s - 1 = s which says that m must itself be a power of 2. Now setting p = 2” + m2” - 1 we use (37) and (39) to observe that a*’ maps

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195

ET+2s;s+ z = L,(, + iirn + , ) onto Egl+ i = L,(, + , zm,. Therefore the term E2S PJ++' 1

vanishes and the claim is proved. We have shown that both end terms of the sequence (40) are zero for all r > 2” + 1. Consequently we must have E;,- l;s = Eg- i . Since our spectral sequence arises from a bounded filtration of a bounded exact complex, E" = 0 and we have shown the vanishing of Es.,-I;s as desired. This completes the proof of the statements (31)-(34) concerning our spectral sequence and it follows immediately that the sequences

O+L cdr+I;m)--) La(t;Zm+ are exact for all 1 and m, establishing

I) + Lcr:zm,+ 0 the exactness of IL,(,,.

7. TRANSCRIPTION FOR COSCHLJR FUNCTORS We introduce this section by recalling from [6] the Jacobi-Trudi sion of a Schur function sA ~,=det(h+;+j)

expan(1)

in terms of the complete symmetric polynomials h,, h 1,... . The expansion above and the expansion in (1.7) of the introduction are related through the involution on symmetric functions which exchanges ei with hi and sA with sl. Since hi is the formal character of the divided power Di(F) of the module F and sA is the formal character of the Weyl module K,(F), we can rewrite ( 1) in the form

analogous to (1.9) in the Introduction. In fact most of the discussion in the Introduction may be repeated with K, F, D,F and hi in place of L, F, niF, and ei. We will indicate, in this section, how the results of Section 4-6 can be reformulated in terms of coSchur functors and divided powers. We first note that every statement in 4 through 6 involving LF, AF, and SF remains true if these are replaced with KF, DF, and AF, respectively. However, the statements dealing with the auxiliary constructions involving filtrations on Schur complexes require a little more care. Therefore, in what follows, we will simply restate the main results in these sections and at the same time point out the correct reformulations of the auxiliary constructions. If a = (A,, A,)/@,, pz) is a two-rowed skew-partition, then there exists a short exact sequence of Weyl modules

O+K,F-+KBF+KaF-,O,

(3)

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where b and y are as in (4.1). Also, there is an exact sequence O+D,(ap+

... -+D,(a+

Do(a)2

K,F+O

(4)

where k = A2 - p, , Do(a) = D,, F@ D2 F,

and the maps 6, are defined as in Section 4, but replacing the diagonal and multiplication maps of AF by those of DF. Now let a = (A,,..., &)/(p i,..., pLk) be a general skew partition with k rows. We have short exact sequences

analogous to (6.16) and a long exact sequence H,,.,F, analogous to II,,.,F, formed by splicing together the short exact sequences (5). These are delined by replacing the Hopf algebras AF and SF by DF and AF, respectively, in the constructions of Section 6. For the proofs of exactness, we need to reformulate the auxiliary constructions X’*‘(a) and A,(a) in a suitable manner. To do this, we start with a map 4: F@ R2 + R, which is the direct sum of the two maps til : F+ 0 and &: R2 + RI. The Schur complex L,(4) decomposes into the direct sum C L,.(#,, &; t, s) is of subcomplexes. We know from [3, V] that the complex L,.(4) is exact in every dimension except [a’( = i + k - p and that the homology in this dimension is K,, F. Consequently, the complexes L,.(#,, &; t, s) are exact for s different from 0. We define X’(a) to be the complex L,.(#,, &; t, s), where t = Ia’1 -3, with a shift in dimensions so that the component X”(a), in dimension i equals the component L,.(b,, &; t, s)~+, in dimension i+ t. It follows from (2.23) that Xs(a)i decomposes by a natural filtration into 1 L,,, R, @ K,,.,, F@ Kifly RS, where (rrJ= s - i and J’fy = i. If we let $,: F+ R, and rj2: R2 --t 0 be the zero maps, then we can think of 4 as an extension of tiZ by til. So we have, from (2.27), a filtration {M, L,.(+, , tiZ)} of the complex L,.(4), and we obtain a filtration {P’(a)] of Y(a) by taking

where y(r) = (A, + l,..., 1, + 1, fir+, ,..., pLk). We analogous to (6.21) and (6.22).

then

have the results

As in Section 6, we let A(a) denote the exact complex Xk3k-1(a).

If a is a

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partition 1, then A(1) is the fundamental general case, we define a filtration {A,(U)} A,(a)=

exact sequence lK,(.,F. In the of A(a) by

A(a)nM,,-,,,,+,,L,,(Q,,

8,)

(7)

as in (6.23) but where 0, and 0, now denote the zero maps 0, : 0 + R, and O2 : F@ R2 --f 0. Finally, we let {E’(a)} denote the spectral sequence associated to the filtration { AJa)}. Then, the remainder of the proof of exactness of K a(,) now goes through by formally substituting KF and DF for LF and AF. We close this section by making an observation that may have become apparent to the reader: it is possible to reformulate the results in Sections 3-6 in terms of Schur complexes and then obtain the results for Schur and coSchur functors as special cases. For example, there exists an exact sequence IL,,.,4 of Schur complexes for an arbitrary map 4, and the sequences L,, )F, Dd,,.,F can be obtained from [la(*)4 by making suitable choices of 4. l

8. A SKETCH OF SOME FORTHCOMING

RESULTS

For a skew-partition a= (A,, L,)/(p,, p2) with two rows, we constructed in Section 4 a resolution @(a) of the Schur module L,F using direct sums of AyF, and in Section 7 we constructed a parallel resolution D(a) of the Weyl module K,F using direct sums of D,F. In this section we will indicate how the fundamental exact sequence II,,,,F and &,,,,F may be used to construct analogous resolutions C(a) and D(a) in the case of a general nrowed skew-partition a, and discuss some applications of these resolutions. Details wil be given in [ 11. For the sake of convenience our discussion will take place in the stable category of polynomial representations of GL, as n + cc or, equivalently, if the reader prefers, in the category of polynomial functors. It follows easily from the theory of Schur algebras [4] that the tensor products D,F= D,,,F@ ... @ D,,F of divided powers are projective in the above category, so that the desired D(a) will be a projective resolution of the Weyl module K, F. We let Y denote the family of all shapes /I that appear in the fundamental exact sequences II or(.JF and I&,., F, as a ranges over all skew partitions. Y consists (with but few exceptions) of relative sequences n/p, where ’ ’ and ,u = (pl ,..., p,, ) c ;1 is a sequence such that, 1 = (A, ,..., 1,) i s a partition for some k = l,..., m- 1, we have p, > .*. >~~>/p,,,>p~+~ > .** a~,,-,. On this family F we define the (nonnegative) integer-valued function j(/?) by setting j(b) to be the sum of the number of overlaps in each pair of rows

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of the shape /I. Thus j has the property that j(p) =0 if and only if L, Fr AP F (or, equivalently, if K, Fz D, F). Using the short exact sequences (6.18) and (7.3), it is easy to show that for any shape /I in Y there exist short exact sequences

O+L,F+L,F-tL,F-tO (2) O-+K,F-+K,F-+KpF-+O,

(1)

and

where 6 nd y are also shapes in Y such that j(S) 0 we can use the short exact sequences (1) and (2): by induction K6F and K,,F have resolutions D(6) and D(y) whose terms are sums of tensor products of divided powers. Since these are projective resolutions, the map K&F+ K,,F can be lifted to a map $: D(6) + 119(y)of resolutions, and we can take D(p) to be the mapping cone of $, obtaining a resolution of KBF as desired. For the Schur module, LpF, the procedure is the same except for the complication of not being able to use the comparison theorem to lift the map L,F+ L,F to a map C(6) -+ C(r) of resolutions, because the terms of C(6) are not projective. However, using the connection between the representation theories of the symmetric groups and Schur algebras, it is not hard to see that there is a natural equivalence of functors Hom(D,,

Dy) 2

Hom(/i,,

A,)

(3)

when the ground ring is Z. We also know from the origin of the sequences (1) and (2) that the map Da + D, which lifts K6 + K,, corresponds under (3) to the map A, + A, which lifts L, + L,. Since the map Ds --t D, can be extended to D(6) + D(y), it follows that the map ,4, + A, can be extended to a map C(6) + C(y) as desired. A comparison of the parallel constructions of these resolutions can be used to extend (3) to a natural equivalence of functors Ext’(K,,

KB)4

Ext’(L,,

Ls)

for all i and all shapes 8, y in Y for any commutative ground ring. It is also an immediate consequence of the construction of the projective resolution D(a) that the Weyl module K, has homological dimension less than or equal to j(cr). Using this it is possible to prove that the Schur algebras, over any field or the ring of integers, have finite global dimension. The function j(a) turns out to be a rather poor estimate of the homological dimension of K,. It would actually be more eficient to carry

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out the above procedure with a different family, Y’, of shapes in place of .Y, and a different function j’. In order to introduce the new family Y, we will take another look at the determinantal expansion (1.12). Instead of expanding by the bottom row to obtain (1.18), we will this time expand by the top row. This yields an analogous formula [L,F]

= f i=

( -l)‘+‘[n”l-“+i-‘FOL,,,i>F], I

where a’(i) = (A*,..., LMP, + lY.9 PLi- 1 + 1, Pi, I,-., AJ We define a’(i) to be the shape (A, + i, &,...,

nm)/(Pj

+

l,

PC, +

l,...,

Pi-

19 Pi+

1 Y...F Pm)

obtained by adjoining a top row of length 1, - pi + i - 1 to the skew partition cr’(i) in the indicated manner. Using the fundamental exact sequences II z(.,I; and K,,.,F, it may be possible to build analogous exact sequences L,.,., F and I&,.,. The shapes that arise in the construction of these exact sequences define the new family Y. If one now takes j’(b) to be the weight of /I minus the number of nonzero columns of fl, then the procedure described earlier can be repeated with 9’ andj’ in place of Y and j to construct resolutions D’(p) and C(p) of K# and L,F for all shapes /3 in Y’. The length of D’(p) (or C’(p)) is exactly j’(p), and is generally less than the length of D(p). We have seen how not only skew shapes, but more general shapes arise naturally in the context of constructing resolutions of the modules K,F and L,F. They arise, too, in the study of extensions of these modules, For example, let 1= (2, ,..., A,) be a partition such that 1, _ 2 > ;1, _, , and take p to be the partition (Ai,..., &,--2, J,,-i + 1, I,-- 1). It can be shown that over h the group Ext’(L,, LP) is cyclic of order L, _ , - A, + 2 and is generated by the extension

where /I is the shape (1, ,..., ;lmP2, Am-i + 1, &,)/(O’+*, 1,O). As we have said, detailed proofs of the results sketched in this section, as well as further applications, will be given in [ 11.

REFERENCES AKIN AND D. A. BUCHSBAUM, Characteristic-free representation theory of the general linear group II: Homological considerations, in preparation. K. AKIN, D. A. BUCHSBAUM, AND J. WEYMAN, Resolutions of determinantal ideals: The submaximal minors, Advan. in Math. 39 (1981), I-10.

1. K. 2.

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3. K. AKIN, D. A. BUCHSBALJM,AND J. WEYMAN, Schur functors and Schur complexes, Advan. in Math. 44 (1982), 207-278. 4. J. A. GREEN, “Polynomial Representations of GL,,” Lecture Notes in Mathematics, No. 380, Springer-Verlag, Berlin, 1980. 5. A. LASCOUX, These, Paris, 1977. 6. I. G. MACDONALD, “Symmetric Functions and Hall Polynomials,” Oxford Univ. Press (Clarendon), Oxford, 1979. 7. H. A. NIELSEN, Tensor functors of complexes, Aarhus University Preprint Series, No. 15, 1978. 8. P. PRAGACZ,Characteristic free resolution of (n - 2)-order Pfaflians of n x n antisymmetric matrix, J. Algebra 78 (1982), 386-396.