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Schur functors for the symplecticgroupMihalis Maliakas aa School of Mathematics , University of Minnesota ,Minneapolis, MN, 55455Published online: 27 Jun 2007.

To cite this article: Mihalis Maliakas (1991) Schur functors for the symplectic group,Communications in Algebra, 19:1, 297-324, DOI: 10.1080/00927879108824141

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COMMUNICATIONS I N A L G E B R A , 1 9 ( 1 ) , 297-324 (1991)

SCHUR FUNCTORS FOR THE SYMPLECTIC GROUP

Mihalis Maliakas

School of Mathematics University of Minnesota Minneapolis, MN 55455

ABSTRACT

The purpose of this work is to describe some new connections between the characteristic-free representation theories of the symplectic group and the corresponding general linear group (Theorem 2.2 and Theorem 2.6) .

INTRODUCTION

The theory of Schur functors for the general linear

group has enjoyed a rapid growth over the past two

decades. However the corresponding theory for the other

classical groups is less developed. For some results in

this direction we refer to [4] and [5]. De Concini

defines in [2] a Z-form of the rational representations

of the symplectic group and proves for this a standard

basis theorem. We want to investigate the connections

between De Concini's Schur functor for the symplectic

group and the Schur functor for the corresponding

Copyright @ 1991 by Marcel Dekker, Inc.

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298 MALIAKAS

general linear group considered by Akin. Buchsbaum and

Weyman in [ I ] .

Let R be any commutative ring with 1 and let F

be a symplectic R-module of rank 2r . This means that

F is free of rank 2r and there exists an " F E A ~ F *

such that the induced map, F + F* , is an isomorphism.

Fix a partition A = (A > A . A ) with X1 < r. 1 - 2

By LAF (resp. VAF) we will denote the Schur module

corresponding to A for the general linear group GL(F)

(resp. for the symplectic group Sp(F)) . We refer to

section 1 and the beginning of section 2 for the various

definitions. Our first main result states that VAF is

the cokernel of an explicit Sp(F) - map between (skew)

Schur modules for GL(F). Namely. let w!') E A'F . where t E a+, be the t-th divided power of

wF. 1

(Informally one may think of wit) as - t! wFnwF". . . A W ~ )

Using the description of LAF in terms of generators

and relations ([I]) we show that the map

A -2t A2 Ak f

A n Fan Fa . . .an F - n l ~ a n Fa.. .en% .

where f = ~ ( ~ ) 8 1 8 . . .81 . induces a map between the F

corresponding GL(F)-modules, LA/(2t)

F - LAF, and thus we have an Sp(F)-map

- W~ : = LA/(2t) - L F , A

where the sum ranges from t = 1 to [h1/2] . Using

the work of DeConcini, 121. we show: -

coker wF = VAF . The dual statement holds for the Weyl

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SYMPLECTIC GROUP 299

modules (Proposition 2.5). Our other main result shows

that the Schur module VA(F@G) for Sp(F@G)

corresponding to the sum. F@G , of two symplectic

modules has a characteristic free decomposition into

Z V7F 8 LAI7G . i.e. VA(F@G) admits a natural O>LA

Sp(F) x Sp(G) filtration, which we construct

explicitly, whose associated graded object is isomorphic

to the previous sum (Theorem 2.6). This generalizes to

the symplectic case Theorem 11.4.11 of [I] for p=(O).

1.BACKGROUNG MATERIAL

In this section we will recall briefly some

fundamental facts concerning Schur and Weyl modules for

the general linear group that will be needed in :he next

section, thus establishing our notation. Our basic

reference here is [I] where the reader will find a

detailed and complete account of Schur functors and

Schur complexes.

Let R be a commutative ring with identity. F a

free R-module of finite rank, and A/p a skew-partition

i.e. h and p are partitions, A = (A1.A 2.....Ak) and

p = (pl,p 2,...,pk) , with hi-pi > 0 . By AF,SF and DF

we will denote the exterior algebra, the symmetric

algebra and the divided power algebra of F

respectively. We will use the following notation

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MALIAKAS

D F@D F@.. .8 D F . In general if - h2-v2 Ak-pk

a = (a l,a2,...,ak) is a sequence of integers we put

a 1 ak A(F;a) = A F8 . . . @A F and similarly for S(F;a) and

D(F;a) . If there is no danger of confusion we will

often write A(a) for A(F;a) and similarly we write

S(a) and D(a) . By we will denote the transpose

(or conjucate) of the partition h . The Schur module.

L F , of the general linear group , GL(F) ,

corresponding to the skew-partition A/p is defined in

[l.chapter 111 as the image of a natural GL(F) - map

d F : A(A/p) - s(~/G) . Similarly the Weyl module, KAIpF , of GL(F) is defined

as the image of a natural GL(F) - map

' F : D(A/p) - A ( X / C ) . dA/p

(In [I] the term "coschur" is used for KAIpF instead

of "Weyl"). There is a useful description of LAIvF in

terms of generators and relations which we now outline

We will use this description in Section 2. For each

pair of adjacent rows, (Ai.Ai+l)/(pi.pi+l) . of A/v

define the map

as the sum of compositions of multiplication and

diagonalization

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SYMPLECTIC GROUP

Now put

where k is the number of rows of the partition

h . This defines a map from

THEOREM 1.1. ([I] Thm 11.2.11). The composition

dh/pF Onh/p is zero.

In fact more is true.

THEOREM 1.2 ([I] Thm 11.2.16). (i) L F is a free

R-module with a basis in one to one correspondence with

the standard tableaux of shape h/p with entries from

the set (1.2 , . . . , n) , where n = rank F . (ii) There

is a canonical isomorphism LAlUF cz coker h/p '

Now let . G be a second free R-module of finite

rank. In section 2 we will also need the followrng.

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302 MALIAKAS

THEOREM 1.3 ([I] Thm 11. 4.11). LhIp (FQG) admits a

natural filtration whose associated graded module is

isomorphic to H L F 8 LhI,G . ,A29

Note that for partitions a = (a . a k ) and

p = (P l,...,Pk) we write a C /3 if ai < Pi for all

i = 1 , . . . . k . The explicit description of the

filtration on L (FOG) that was constructed in [ I ]

in order to prove the above theorem will be recalled and

utilized in the proof of Theorem 2.6.

If p = (0) we write LhF for LXIpF . Over a

field of characteristic zero the modules LhF , as

h = (A l,...,hk) ranges over all partitions with

hl rank F , form a complete set of inequivalent

irreducible polynomial representations of GL(F) .

partitions h,p and v we denote the multiplicty

LAF in LVF8L F by ( v ) . The integers (v,p P

have a well-known combinatorial description. The

Littlewood - Richardson rule for skew-partitions gi

the decomposition of LhlpF which will be needed 1

For

0 f

A

ve s

ater

THEOREM 1.4. If R contains the rationals, then there

is a canonical isomorphism Lh,pF 2 H (v,p;h)LvF . where the sum ranges over all partitions v .

We have concentrated our attention mainly on Schur

modules. For the corresponding results on Weyl modules

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SYMPLECTIC GROUP 303

we once again refer to [ I ] . For further use we recall

here a duality statement.

PROPOSITION 1.5. (i) There is a canonical isomorphism

* (LAlpF)* - K-,_(F ) . where F* = HomR(F.R) . (ii) If

A l.l

R contains the rationals then there is a canonical

isomorphism L F 1 K- _F . Alp

We end this section by remarking that the Sthur

functor (i.e. the functor L ( -1 defined for every

pair (R.F) . where R is a commutative ring with

identity and F is a finitely generated free R-nodule)

is, in light of Theorem 1.2, a universally free functor

that is. LAlyF is free and commutes with change of the

base ring R .

2. SCHUR AND WEYL MODULES FOR THE

SYMPLECTIC GROUP

In this section we show in a characteristic - free

setting that the Schur module for the symplectic group.

Sp(F) , is the cokernel of an explicit Sp(F) - map

between (skew) Schur modules for GL(F) (Theorem 2.2).

The dual statement holds for the Weyl modules. We also

describe a characteristic-free decomposition of the

Schur module for the symplectic group corresponding to

the sum of two free modules (Theorem 2 . 6 . )

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MALIAKAS

Unless otherwise stated our ground ring, R , will

be any commutative ring (with 1). Let F be a free

R-module of finite rank. Recall that AF* can be made

a left AF-module in a natural way: Since F is free of

finite rank we have a natural Hopf-algebra isomorphism

AF* - (AF)* . ( 1 )

Now let a € ASF* . b € AtF* , s 1 t and A(a)=Pai(t) @ i

ai(s-t)' . where A(a) is the image of a under the

s-t * diagonalization h:ASF*+ AtF*@ A F. The action of AF

on AF* is defined by b(a) = Z<ai(t) ,b>ai(s-t) ' . i

where <ai(t),b> is the value of b of the image of

ai(t) under (1).

DEFINITION 2.1. Let F have even rank, say 2r. Then

F is called a symplectic module if there exists an

e 1 ement wF of A ~ F * such that the map F + F* . defined by x + x(wF) . is an R-ismorphism

From this point on F will be a symp lectic module.

There exists a basis of F , {ei) . i = 1.2. . . . , 2r such

that under the identification F F* we have

wF = elAe +e Ae 2r 2 2r-1 + . . . + e Ae

r r+l (2 1 (see [ 4 ] ) . We will sometimes drop the subscript F and

r - write w = 2 e.Ae- , where i = 2r+l-i . Using w we

1 i = 1 i

define an alternating bilinear form, < > , on F by

setting <x,y> = xAy(w) . If we use (2) we have

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SYMPLECTIC GROUP

1. if i j = 2 and i<j

<ei,ej> = {-I. if i+j = 2r+l and i>j

1 0 , otherwise.

The symplectic group Sp(F) is defined as the

group of all R-automorphisms, f, of F . such that <f(x),f(y)> = <x.y> for all x,y E F .

Now let A'= (A l,....Ak) be a partition with

Al < r . We use the terminology "Schur module for

Sp(F) " and the notation VAF for the representation of

Sp(F) that in [2 , Definition 5.11 is called

"symplectic Specht module" and is denoted by Ah . As

an R-module VAF is free with a basis given by the

(right canonical) symplectic standard tableaux of shape

A with entries out of 1,2. . . . , 2r . ([2,Thm. 4.81). If

R is a field of characteristic zero, then the modules

VAF , as A ranges over all partitions with A I I r ,

form a complete set of inequivalent irreducible

representations of Sp(F) ([2,Thm 4.111).

Let X 5 LAF be the R-submodule of L F generated A

by all elements of the form

hi-2t where x E A i

F and w(~) E A ~ ~ F stands for the

t-th divided power of w . 1.e.

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MALIAKAS

',( t), H e Ae- Aei Ae- A...Aei Ae- ,

l<il<. . . < i <r '1 i 1 2 i2 t i t t -

where ?=2r+l-i .

(Recall that the map d F was considered in section 1) A

It is clear that X is an Sp(F)-submodule of LhF

Our starting point is the following observation

which follows immediately from the proof of

Theorem 2.4 of [2]. (Notice that all the relations

described by Proposition [2, 1.7(2)]-and hence by

Proposition [2. 1.81-corresponding to A are precisely

those given by X above. As for the other relations

used in the proof of Theorem [2. 2.41 we note that these

are given by the image of the map OA of section 1 and

we recall that L F = coker II ) . h A

In view of (4) we will usually write

LhF V F - 0 where a is the obvious map. By A

[A1/2] we will denote the largest integer less than or

A 1 equal to - 2 -

THEOREM 2.2. Let F be a symplectic module of rank 2r

and let A = (A l.....Ak) be a partition with Al r .

Then: (i) For every integer t < [A1/2] the map

induces an Sp(F)-map

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SYMPLECTIC GROUP

Lh/(2t)F + LhF '

(ii) There is an exact sequence of Sp(F)-modules

- where w is the indicated sum of the maps given by part

(i).

Proof. (i) Recall (Theorem 1.2) that the relations of

LA/(2t)F are indexed by i . i = 1.2, . . . . k-1 . Let

i 2 2 . For each t , where 0 J t < . consider the diagram:

f A(A1-2t.h 2...hi...hk) + A(Al...Ai.Ai+l...hk)

f A(hl-2t.h 2...hi+hi+l-e,e...A,) + A(hl...Ai+hi+l-l,

e.. .Ak)

where we have written Oi for the appropriate component

of the map Oi that was defined in section 1 and

f = w(t)@l@. . .81 . This diagram clearly commutes.

Now let i=l . We may assume that A is a two

rowed partition, A = (Al.Ag) . For each 8 , where

0 ( e < h2-2t . consider the diagram

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MALIAKAS

where the vertical maps, O1 . are the appropriate components of the maps Dl defined in section 1. We

want to define Sp(F) maps Po.Pl....,P2t such that

diagram (*) commutes. For each i = 0.1, . . . , 2t let

be the image of w (t) under the diagonalization

A(2t) - A(2t-i,i) .

Define a map

7 : ~ ( ~ ~ - 2 t + ~ ~ - e , e ) - ~ ( h ~ - i + h ~ - e , e + i ) i

by T~(x@Y) = Ex~u~~)(2t-i)@w~~)(i) ' ~ y . Put a

i Pi = (-1) -ri . i = 0.1. . . . , 2t . We prove now that

diagram (*) commutes. Let x E A(A1-2t+h2-P) and

y E A(-!) . For one direction we have, since deg w = 2:

where Zxb(A1-2t)@xb(h2-e)' is the image of x under b

the diagonalization A(Al-2t+A2-@) - A(A1-2t,h2-e) .

We will from this point on use this type of notation

without special mention sinLe i t is self explanatory.

For the other direction using that deg GI = 2 we have:

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SYMPLECTIC GROUP

where Wac 2 - s ) ( s - 1 is the image of

~ ( ~ ) ( 2 t - 1 ) € A(2t-1) under the diagonalization a

A(2t-1) + A(2t-s,s-1) . In general for any

i = 0.1, . . . . 2t ,

where Oat 2 - s ) ( s - i ) is the image of

~ ( ~ ) ( 2 t - i ) € A(2t-i) under the diagonalization a

A(2t-i) + A(2t-s,s-i) . Therefore

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310 MALIAKAS

By rearranging (according to s ) the terms of (5) we

have

ol(Po+P1+. . .+P2t) = Ao+A1+. . .+A2t . where for each s = 0.1. . . . , 2t i t is

and it is understood that o!:)(s-i)' = 0 if s-i < 0 .

We claim that As = 0 for s > 1 . Indeed.

This follows from the co-associativity of

diagonalization and the fact that the composition of

multiplication and diagonalization

A(s) + A(s-i,i) + A(s)

is multiplication by the binomial coefficient

( ) . Now since (-l)i() = 0 , (6) and (7) give i =O

As = 0 . Therefore U ~ ( P ~ + P ~ + . . . + P ~ ~ ) = A. , but

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SYMPLECTIC GROUP

proving the commutativity of (*) . Since

is an Sp(F) - map, each 7i is an Sp(F) map and this

concludes the proof of part (i).

(ii) In view of (4) we will show that X = im(;)

Towards this end let Xi be the R-submodule of LhF

generated by all elements of the form (3). where t i > 0

- and ti+l - ti+2 = . . . = tk = 0 . Then obviously

X = B Xi (not a direct sum). i

Claim: We have a chain of submodules

Xk 5 Xk-l 5 . . . 5 X1 . Firstly we note that the claim

follows from the two-rowed case. Indeed, suppose the

claim is true for any two-rowed partition. Take a

generator, a. of Xi given by (3). where i > 2 , and

apply the claim to the partition formed by the i-1 and

i-th rows of a . thus obtaining an element in X i-1 '

Secondly let h = (hl.Xg) . x E A(hl) . Y E A(X2-2t) . and z = d ~(x@u(~)dy) E X2. We will show that i E X I .

h

Towards this end consider the map

O1 : A(h1+2t,h2-2t) + A(hl,X2)

and the element

U(~)AX@~ E A(hl+2t.h2-2t).

Using Theorem 1 . 1 we have

0 = d A = dh~(x@u(t)~y) +

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MALI AKAS

euLt)(2t-s) .axb(s) lay) .

Thus in order to show that z E X1 i t is enough to show

that, for all s = 1.2. . . . , 2t ,

We prove this by a descending induction argument on s .

For s = 2t the sum in question is

I dA~(u(t)~xb(h,-2t)@xb(2t) 'Ay) , b

which clearly belongs to X1 . Suppose the statement is

true for s = 2t . 2t-1, . . . . 2t-i+l . Consider the map

01: A(hl+i,A2-i) + h(hl.A2)

and the element

z' = E u(t)~xb(hl-2t+i)@xb(2t-i)'hy E A(hi+i .A2-i) . b

Again we compute:

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SYMPLECTIC GROUP

.But notice that the sum

P dh~(w(t)(s)~xbc(Al-s)0w~t)(2t-~) a 'A a.b.c

Axbc(s-2t+i)'Axb(2t-i)'Ay)

is an integral multiple of

P dh~(w(t)(s)~xb(hl-s)0w~t)(2t-s) a qxb(~) O A Y ) . a,b

a

This last sum is, by induction, in Xi for s > 21:-i+l . Hence from (8)

P dh~(u!t)(2t-i)~xb(hl-2t+i)0u(t)(i) 'A a.b

a

Axb(2t-i)'Ay E XI .

This concludes the proof of the claim. The claim

implies that X1 = X , and since im(;) = X1 , we have

finished the proof of the theorem.

COROLLARY 2.3 There is an exact sequence of

Domain Oh+ HA(hl-2t.A2.. . . .hk)a,h

where a = a + ~w(~)010. . .O1 and the h

t=l to t = [h1/2] .

(Al. . . . . hk)+l'hF4 ,

sums range f rorn

Proof. From Theorem 2.2 (and its proof) we have the

following commutative diagram with exact columns and top

row.

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Z Domain 0 A/(2t) - Domain

A '

where g = ~w(~)@l@. . .81 and the sums are taken from

t = 1 to t = [A1/2] . The mapping cone theorem gives

the result.

C O R O L L A R Y 2.4 VA is a universally free functor on

symplectic modules.

Proof. VA is the cokernel of map between two

universally free functions and VAF is free over R .

REMARK. If R contains the rationals then all - but

one - of the relations of Theorem 2.2(ii) are redundant

Namely there is an exact sequence of Sp(F) - modules

This follows from the fact w(~) = - wA. Aw t !

t times

Next we wish to introduce the Weyl module for the

symplectic group. A s usual F is a symplectic module

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SYMPLECTIC GROUP 315

of rank 2r . If 2t j s < r , by u(t)V we denote the

dual map

w (t)v : A(s) + A(s-2t) ,

of the multiplication map

where for x E A(s-2t) . w(~)(x) = W(~)AX . An easy

computation shows that

t where T = 2 (i j - 1 and the sum ranges over all

u u u=l

possible choices of il,...,it,jl,...,jt,al,...,:i s-2 t

out of 1.2, . . . . s such that il<j l.....it<jt and

Given a partition h = (A l.....hk) with k < r

put V F = HomR(V_F.R) . This has a natural (lelt) h

h

Sp(F)-module structure in the usual way. We call VAF

the Weyl module for Sp(F) corresponding to h . From

Theorem 2.2 we have immediately the following result.

PROPOSITION 2.5. Let A = (A l.....hk) be a partition

with k j r . Then we have the following.

(a) VA is a universally free functor on

symplectic modules.

h "here is an exact sequence of Sp(F)- nodules

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-v where w is the restriction to KhF of the map

2 w(t)veie.. .el : A(X) - 2 ~(~/(2t)), all the sums are taken from t = 1 to t = [k/2] and

It times

Proof. (a) Let R 4 S be a ring homomorphism. Since

V-F is free over R of finite rank we have h

SBHomR(V-F,R) % HomS(S8VhF,S) . By Corollary 2.4 h

S8V-F 2 V-F(SBF) . Thus S B V ~ F = Vh(s@~) . h h

(b) Dualize the exact sequence of Theorem 2.2 that

corresponds to the partition X . apply Proposition 1.5(i) and recall F Z F* .

Remark. If R contains the rationals we have a natural

isomorphism V F u VAF , where X is as in the previous x proposition. However no such isomorphism exists over

- Z. Thus the modules VhF form a second Z-form of

rational representations of Sp(F) .

Next we deal with the following problem. Let F

and G be two symplectic modules. Then F@G is a

symplectic module and the question is whether Vh(F@G)

has a characteristic-free decomposition analogous to

Theorem 1.3. The answer is given by the next result.

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SYMPLECTIC GROUP 31 7

THEOREM 2.6. Let F and G be symplectic modules and

let h = (A l.....hk) be a partition with Al J r ,

where rkF = 2r . Then F@G is a symplectic module and

VA(F@G) admits a natural filtration whose associated

graded object is isomorphic to

For the proof-of this we will need a classical

character formula of D.E. Littlewood. Let R = C the

complex numbers. If H is a 2r-dimensional sy:nplectic

vector space over C and if h = (A l,....hk) is a

partition with hl < r we will denote the (forlcal)

character of LhH by xH(A) . while by qH(h) we will

denote the (formal) character of VhH . If

al>a >. . . 2 >as > 0 , where the ai are integers, let

(al.a 2.....aslal-l.a2-l,...,a -1) be the partition

pictured below

The diagonal consists of s boxes; the i-th rolw

consists of a.+i boxes and the j-th column co~nsists of

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318 MALIAKAS

a.+j-1 boxes. Put lhl = hl+. . .+Ak . Ihl is called J

the wieght of h . From [3,Prop. 1.5.3(2)] and Theorem

1.4 we have the following.

where p runs over all partitions and a runs over all

partitions (al.. . . .as lal-1.. . . .a -1). a >a > . . . >a >O . 1 2

REMARK: In [6] we construct an exact sequence over

Sp(F) (in characteristic zero) whose Euler - Poincare

characteristic is given by Prop. 2.7 above. However our

construction is valid only under an extra condition on

A , namely r+l > Ihl .

LEMMA 2.8. Using the notation of Theorem 2.6 we have:

rank Vh(FQG) = rank 2 V F@Lhl7G OC-rCh

Proof. By the universal freeness of the functors Lh/?

and V7 we may assume R = C . Now we have

~ ~ ( 7 ) = I(-l) 1u1/2xF(7/u) (u as in Prop. 2.7) u

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SYMPLECTIC GROUP

xFOG(h/u) = X xF(-r/u)x (h1-r) (Thm 1.3) . 0579 G

Hence X ~~(-r)~~(h/-r)= 8 Z(-l)'a"2x (T/U)%~(X/T)= O c _ 7 9 0 9 9 a

F

= 8(-1) I f f 112 YFBG("u) = OIFBG(X). a

Now the desired equality of ranks follows.

We order the subpartitions of h

lexicographically. For example the subpartitions of

X = (2,l.l) are ordered as follows:

( 0 ( 1 ( I ) < ( l l ) < ( 2 (21)< ( 2 1 ) . Recall

that there is a natural isomorphism

Define a filtration. {M7}7Ch . on LX(F@G) as f~llows.

Consider for each a 5 h the map

which is obtained by tensoring the maps

A(F;ui) 8 A(G;hi-ai) A(F@G;hi) .

(This last map sends f8g to fAg) .

Now put

M7 = dA(F@G)(im( 2 A(F;a) 8 A(G;h/a) - A(F@(:;X)) , ai7 09

where a i 7 means o is lexicographically greater

than or equal to 7 . Also let

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320 MALIAKAS

THEOREM 2.9 (11,Thm. II.4.11]). There is an isomorphism

induced by the map \L, of (9)

For further use we note that for x @...@x E 1 k

A(F:-r) and yl@ . . . 8yk € h(G;h/~) we have

Now we are ready for the:

Proof of Theorem 2.6. Note that, since

W A((F@G)*;~) 2 A(F ;~)@F*@G*@A(G*;~),

F@G is a symplectic module because we may take

"FBG = "F + W G . ( 1 1)

Now we define the following total ordering among

subpartitions of h . For U.T h put a < T if

1 0 1 < I T I or i f 1 0 1 = I T I and a ) T .

For example the subpartitions of h = (2.1.1) are

ordered as follows(O)< (I)< (2)< (1,1)< (2.1)< (1.1,1)<

< (21.1) . Define a filtration. {N7ITLh . on Lh(FBG)

as follows. Put

N7 = dh(F@G)(im( H A(F;a)@A(G;A/o) + A(F@G;h))) 6<7

U ~ A

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SYMPLECTIC GROUP

and

So N (0

= LAG and NA = LA(F@G) . Consider the

natural projection

LA(F@G) a VA(F@G) + 0 .

We thus have a natural filtration, { U ( N ~ ) ) ~ ~ ~ . 0 x 1 -

VA(F@G) . The claim is that there is a natural

Sp(F) x Sp(G) surjection

Indeed, by Theorem 2.9 we have a natural GL(F) x GL(G)

surjection -

M L7F O LAI7G A 2 - 0 . (12 )

hi 7

Now from the defintiion of M7 we have

M7 A 7 - - - - ( 1 3 )

M7

where A7 = dh(F@G)(im(A(F;-r)OA(G;h/7) + A(F@G;h))) .

Since the map dA(F@G) restricted to the image of

A(F;u) 8 A(G:A/u) + A(F@G;A) preserves the weighr of

the partition, u . corresponding to F we have:

N7 - As in (13) - - A 7 . Thus from (12),(13) and (14)

7 N7

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3 22 MALIAKAS

and the definition of the ordering " 5 " there exists a

natural surjection

and hence there is a natural Sp(F) x Sp(G) surjection

which we call a . By Theorem 2.2 our claim will 7

follow if we show that a is zero on the image of 7

where 7 = ( 7 1,...,7k) . Let x = x @...@xk E 1

A ( F : ~ ~ - 2 t . 7 ~ , . . . , 7 ~ ) and y = yl@ . . . @yk E

A(G;Al--r1.A2-72....,A -7 ) . We have k k

Notice that

because deg w = deg uG = 2 . Now from (11) and the

relations in VA(F@G) we have

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SYMPLECTIC GROUP

Each summand of the right hand side of (15) lies in

a(Nq) . because in each case the F-content (i.e. the weight of the partition corresponding to F ) is

strictly less than 171 . This proves our claim. We

thus have an (R-module) surjection

B VqF8LA,7G + VA(F@G) + 0 , 7 9

which by Lemma 2.8 is an isomorphism. This ends the

proof of the theorem.

ACKNOWLEDGEMENTS. The results of this paper are

contained in the authors Ph.D. dissertation (Brandeis

University). I would like to express my deep gratitude

to Professor David Buchsbaum for most generously sharing

his time and insights. Also many thanks are due to

Professor Jerzy Weyman for some enlightening

discussions.

REFERENCES

K. Akin, D. Buchsbaum, J. Weyman: Schur functors and Schur complexes. Adv. in Math. vol. 44 No. 3, 207-278 (1982).

C. DeConcini: Symplectic standard tableaux. Adv. in Math. 34, 1-27 (1979)

K. Koike, I. Terada: Young diagramatic methods for the representation theory of the classical groups of type Bn.Cn.Dn . J. Alg. 107, 466-511 (1987).

G. Lancaster, J. Towber: Representation functors and flag algebras for the classical groups I. J. Alg. 5 9 , 16-38 (1979).

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5. G. Lancaster. J. Towber: Representation functors and flag algebras for the classical groups 1 1 . J . Alg. 94. 265-316 (1985).

6. M. Maliakas: Representation-theoretic realizations of two classical character formulas of D.E. Littlewood. Communications in Algebra g, 2 7 1 - 2 9 6 ( 1 9 9 1 )

Received: December 1 9 8 9

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