a note on polynomial functors and ext 1 groups

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This article was downloaded by: [University of Chicago Library] On: 12 November 2014, At: 06:06 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Communications in Algebra Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lagb20 A note on polynomial functors and ext 1 Groups Geoffrey M. L. Powell a a L'institut galilee, universite de parisnord , Villetaneuse, France Published online: 27 Jun 2007. To cite this article: Geoffrey M. L. Powell (1997) A note on polynomial functors and ext 1 Groups, Communications in Algebra, 25:3, 979-987, DOI: 10.1080/00927879708825903 To link to this article: http://dx.doi.org/10.1080/00927879708825903 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

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Page 1: A note on polynomial functors and ext               1               Groups

This article was downloaded by: [University of Chicago Library]On: 12 November 2014, At: 06:06Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20

A note on polynomial functors and ext1 GroupsGeoffrey M. L. Powell aa L'institut galilee, universite de paris‐nord , Villetaneuse, FrancePublished online: 27 Jun 2007.

To cite this article: Geoffrey M. L. Powell (1997) A note on polynomial functors and ext1 Groups, Communications inAlgebra, 25:3, 979-987, DOI: 10.1080/00927879708825903

To link to this article: http://dx.doi.org/10.1080/00927879708825903

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”)contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensorsmake no representations or warranties whatsoever as to the accuracy, completeness, or suitabilityfor any purpose of the Content. Any opinions and views expressed in this publication are the opinionsand views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy ofthe Content should not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings,demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arisingdirectly or indirectly in connection with, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial orsystematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distributionin any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found athttp://www.tandfonline.com/page/terms-and-conditions

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COMMUNICATIONS IN ALGEBRA, 25(3), 979-987 (1997)

A Note on Polynomial Functors and ~ x t ' Groups

Geoffrey M.L. Powell

L'INSTITUT GALILEE, UNIVERSITE DE PARIS-NORD, VILLETANEUSE, FRANCE

1 Introduction

Let F be a finite field of characteristic p and let F = F ( F ) denote the category of functors Ef + E , where E is the category of IF-vector spaces and Ef is the full subcategory of finite dimensional vector spaces.

3 is an abelian category; an object is szmple if it has no non-trivial subobjects and is finite if it has a finite composition series. It is known that the isomorphism classes of simple objects in F are indexed by the q-regular partitions; however, the details of this are not required for this paper. References for the study of the category 3 are [Kl, K2, K3, S].

No ta t ion 1.1 Suppose that F E 3 takes values in E f . Put dF(n) := d i m F ( F n ) .

Kuhn has shown that, for a functor F E F which takes finite-dimensional values, there exist uniquely-determined integral coefficients dk = dk(F) , such that:

for all natural numbers n (see [Kl] ) .

Definit ion 1.2 A functor F E 3 is polynomial if it takes finite-dimensional values and there exists n such that d k ( F ) = 0 for k > n. If d,(F) # 0, then F is said to have polynomial degree exactly n. 0

The category 3 comes equipped with the dzfterence functor A : F -+ F; this may be defined on objects as the kernel A F ( V ) = ker {F(V@IF) + F ( V ) ) , of the map induced by the natural projection: V @IF + V .

It is known (see [Kl]) that a functor is polynomial if and only if it is finite; moreover F is polynomial of degree exactly n if and only if An+'(F) = 0 but An(F) # 0. The full subcategory of functors of degree < d is denoted by Fd.

Copyr~gh t C 1997 by Marcel Dekker, Inc

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980 POWELL

This defines a filtration F , C Fl C . . . c Fd C . . .3, by subcategories of the category 3, where the category F , is defined as the colimit of this filtration by the subcategories Fd and is called the category of analytic functors. 3 is a natural object of study in algebraic K-theory; it is of particular importance since the calculation of Ezt groups in F is equivalent to the calculation of MacLane cohomology [FLS]. For F = F , the prime field, it is of interest to topologists due to the connection established in [HLS] between F and the category U of unstable modules over the mod-p Steenrod algebra. Namely there are functors 1 : U + F and r : 3 + U, with 1 left adjoint to T ,

which induce an equivalence of categories U I N i l Z F , ( F ) , where N i l is the localizing subcategory of nilpotent unstable modules.

No ta t ion 1.3 Let M, denote the category of modules over M,(F) , the semi-group of n x n matrices with entries in F. 0

This note studies polynomial functors; the main results are the following:

T h e o r e m 1 If F zs a functor of polynomial degree 5 d and G is any functor, then the natural map Hornr(F, G ) 4 H o ~ ~ , + , ( F ( F ~ + ' ) , G ( F ~ + ' ) ) is an isomorphism.

Definit ion 1.4 Suppose that F E F is a functor; the connectivity of F is the smallest integer c such that F(IFC) # 0. If FA is a simple functor, indexed by a partition A, write c(X) for the connectivity of FA. 0

The next result is an immediate corollary of the preceding theorem:

T h e o r e m 2 Suppose that G zs a polynomial functor of degree d and that F is a functor with connectivity c ( F ) > d + 1; then Ext;(F, G ) = 0 = E&G, F ) .

The methods used to prove these results also permit the proof of the following

T h e o r e m 3 If H zs a functor which takes finite-dzmensional values and d n ( H ) = 0, for some n , then H splzts as H % G @ F, where d,(F) = 0 for s 5 n and dt (G) = 0 for t 2 n.

R e m a r k 1.5 The method of proof relies on constructing a 'model' for the zntermediate extension of F ( I F ~ ) , as explained in the next section; the results may become clearer to the reader when viewed in terms of the embedding theorem of [Kl], together with the understanding of the polynomial filtration of the symmetric powers given in [KK]. 0

2 Polynomial functors

This section provides an approach to the study of polynomial functors; the results will probably be clear to the experts and many are implicit in [Kl].

There is a natural functor en : F -+ M,, given by evaluation: F - F(IFn). The functor en admits left and right adjoints: r,, 1, : M , -+ T (see [K2]) which are exten- sions: the unit M 2 e,l,M and the co-unit e,r,M 2 M of the respective adjunctions are isomorphisms for all M E M,.

The zntennedzate extenszon is the functor c, : M , -, 3 which is defined on objects, M E M,, as the image h M of the map y~ : 1,M + rnM which is adjoint to

: enlnM + M . This is an extension, since e,c,,M M and is the smallest such extension, in that c,M is a sub-quot,ient of any functor F such that e,F Z M . The following holds:

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POLYNOMIAL FUNCTORS 98 1

Propos i t i on 2.1 If F is a polynomzal functor of degree d and n 2 d, write N := F(Fn) There is an zsomorphzsm F c,,N.

Proof : By [K2], c,,N appears as a subquotient of F ; it is therefore polynomial of degree 5 d. However, since c,,(N) is an extension, c,,N(IFk) F(IFk) for k 5 n. The dimen- sions of these spaces 0 5 k 5 d determine d ~ ( * ) and d % ~ ( * ) , which are equal, since these are polynomial functions of degree 5 d which agree on d + 1 points. Hence, we may conclude that the functors are equal. 0

This paper essentially gives an explicit model for the intermediate extension in the situation of the proposition. The aim of this section is to give a description of the coefficients dk and a choice of basis of the vector spaces F ( F ~ ) , when F E Fd and k > d. The methods require the choice of a standard skeleton for Ef, together with a choice of basis for each functor.

No ta t ion 2.2 Fix a basis {xl , . . . , xn, . . .) for IFm and embed IFn -+ IFm as the subspace spanned by the first n vectors. Let I ( n , k ) denote the set of 'standard' embeddings F' -+ .Fk, which are induced by monomorphisms a : [n] -3 [k], via x, I-+ x,,. (Here, [N] denotes the set { I , . . . , N)). Thus; if C,v denotes the symmetric group acting on the left of [N] as the group of permutations, then I ( n , k) admits a right &,-action and a left Ck-action.

Define S ( n , k) c T(n , k) to be the set of standard embeddings induced by increuzng functions; S ( n , k) gives a set of coset representatives of I ( n , k) under the right En-action. Thus, for any a' E T ( n , k) , there are unique elements a E S(n. k ) and p E C, such that a' = ap . 0

For a fixed k, let {q,(1 5 z 5 k) be the set of standard projections IFk -+ pk onto k - 1 dimensional subspaces, which factor through elements of S ( k - 1, k). Choose the numbering of these elements so that ker(q,) = ( s t ) .

Definit ion 2.3 Suppose that F is a functor in F; define the following subspaces of

F ( F k )

1. C k ( F ) := { r E F(IF~)IF(I~ , ) (Z) = 0 for q,, 1 5 i 5 k)

The following lemma presents no difficulty:

L e m m a 2.4

1. x E Ck(F) zf and only zf x represents a non-tnvzal element zn Ck-l(AF), where AF(pk- ' ) zs the cokernel of the standard embedding F (pk - ' ) -+ F ( F ~ ) .

2. Ck zs a complement of Vk, so that F(IF~) 2 Vk $ Ck

The first part of the lemma shows that d imCk(F) = dimCk-'(AF); it follows that d imCk(F) = dim Akp(0 ) ; hence, the work of [Kl ] shows that:

P ropos i t i on 2.5 Suppose that F E FU 2s an analytzc functor, whzch takes valves of finite dimenszon, then dk = dim Ck.

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R e m a r k 2.6 Definition 2.3 provides another characterization of functors of polynomial degree 5 d. 0

This allows for the choice of a basis for F ( F n ) when n > d , the polynomial degree of F ; the result may be proved by an induction using the functor A and the first part of Lemma 2.4:

Proposi t ion 2.7 Suppose that F E Fd zs a polynomzal functor of degree d and that n 2 d; then the map

d

$ @ (a), - F ( F " ) > k = O a € S ( k , n )

defined as the sum of the maps (Ck) , 2 Ck - F ( I F ~ ) F2) F(IFn), is a n isomorphism. (Here, by conventton, S(0, n) zs taken to be the szngleton set and Co --t F ( F n ) is the incluszon of the constant part of the functor F ) .

This result has the following important, but straightforward, corollary

Corollary 2.8 A functor F whzch takes finzte-dzmenszonal values zs polynomzal of degree 5 d z f and only z f , V n > d , the maps F ( a ) . for a E S ( d , n ) , znduce a suqectzon

Corollary 2.8 may be refined as follows, by a method analogous to simplicia1 realization. The construction below may be thought of a s a suitable colimit in the category of vector spaces.

Definition 2.9 Suppose that F E F is a functor; define a vector space Gd(k) as follows. For k 2 d , Gd(k) := F ( I F ~ ) and, for k > d

where 5 is the quotient relation given as follows. If

is a commuting diagram, with y, 6 E S(a, d) and a,,G E S ( d , k ) , then, for all z E F ( F a ) , set ( F ( ~ ) x ) , = ( F ( ~ ) X ) ~ .

By inspection, the map @ f a : $ n E S ( d , k , F ( I F ~ ) , -+ F ( F ~ ) passes to a map f : &(k) -+

F ( P k ) . In fact, the following holds:

Proposi t ion 2.10 If F E Fd zs a functor of degree d , then the n a p f : Gd(k) --, F ( F ~ ) is a n isomorphzsm of vector spaces.

Proof: That f is surject~ve is clear. since the conditions imposed by the equivalence relation are necessary.

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POLYNOMIAL FUNCTORS 983

Proposition 2.7 implies that F ( F d ) has a decomposition as @$o @aES(n,d) (Cn)a,

whereas F(F? hhas a decomposition: @$o @bES(n,k) (Cn)S, for k > d.

Consider a map F ( I F ~ ) F J ) F ( P " ) . for y E S ( d , k ) . Under this map, (C,), has image (C,),,, with ?a E S ( n , k ) . Thus, the map @ E s ( d , k ) ( ~ ( ~ d ) ) , -' F ( F k ) acts sending

{(Cn),), (Cn),a. In particular, to prove that the map on the quotient is injective, it suffices to show

that the map to each component of the decomposition of F(IFk) is injective. This follows since {(C,),), and {(C,,),I),I map to the same component if and only if y a = 7'a1, if and only if the quotient relation identifies the two summands in the decomposition.

Remark 2 . 1 1 The result of Corollary 2 8 may be compared with [FLS, Definition 1.21: A functor F E 3 zs generated zn finzte dzmenszon zf there exzsts a finzte dzmenszonal vector space E such that the canonzcal morphzsm F ( E ) @ PE -+ F (evaluation) is surjective, where PE denotes the standard projectwe in 3, P E ( V ) = IF[Homt(E, V ) ] . I t is easy to see that, if F is analytic, then F is generated in finite dimension if and only if F is polynomial. Thus the space F ( E ) should be compared with the use of F(IFd) above in Corollary 2.8. 0

Definition 2.12 Suppose that F E F is a functor and that x E F(IFk)),

1. x is strictly generated in degree d if there exists y E F ( F d ) and a E S ( d , k ) such that x = F ( a ) ( y ) .

2. x is generated in degree d if x = C x,, where each x , is strictly generated in degree d. 0

Remark 2.13 The argument of Proposition 2.10 shows that , for any functor F , Gd(k) is a sub-vector space of F(IFk), namely the subspace of elements generated in degree d; it is natural to ask when this may be given the structure of an object of F. 0

The following lemma depends only on the coset decomposition of I ( d , k ) with respect to the Ed action.

Lemma 2.14 Suppose that F E 3 and that x zs strictly generated i n degree d , say x = F ( a ) y , as above. If u E Ck represents a permutation matrix zn Mk(IF), then F ( o ) x = F ( O ) { F ( t ) y } , where P E S(d. k ) , t E Cd are uniquely determined by ucy = Pt i n T ( d , k ) .

I n partzcular, the element F ( u ) x zs stnctly generated i n dimension d .

Recall that a transuectzon of IFk, with respect to a given basis, is a linear transfor- mation in G L k ( F ) defined by a pair ( 2 , j) with i # j , so that x, +-+ x , for a # i and x , H x, + x j . The matrix semi-group n4k(p) is generated by the diagonal matrices d iagk (P) , the permutation matrices and the transvections.

Suppose that F E 3 and that z E F ( I F ~ ) is strictly generated in degree d , say x = F ( a ) y . with a E S ( d , k ) and y E F ( F ~ ) . Consider the action of these matrices on the element x.

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984 POWELL

Actzon of diagonal matnces: Suppose that 6 E diagk(p) , then there exists a unique 8 E diagd(F) making the following diagram commute:

Then F (6 ) s = F'(a){F(&)y}, which is strictly generated in degree d. Actzon of transvectzons: Suppose that 7 E M k ( F ) is a transvection; then, either

the composite Fd 5 F k TFk is cr. or there exist a transvection i E Md+l(F) and i E 7 ( d + 1, k) such that the following diagram commutes:

where j is the standard embedding. z, i are uniquely defined given j and the composite along the top row is a .

In the first case, F(T)x = x; in the second F ( r ) ( x ) = F(i ){F( i )F( j )y} . In partic- ular, F(T)(x) is generated in degree d if and only if either we are in the first case or { F ( i ) F ( j ) y } is generated in degree d .

These remarks suffice to prove the following result.

P ropos i t i on 2.15 Suppose that F 6 F takes finite-dzmensional values, that d 2 0 and that every element of F(F*+') zs generated in degree d. Then the gnzded vector space Gd(k) (fork > 0) may be gzven the structure of an element in F.

3 Proof of the Main Results

Using the above details, it is now possible to prove the main results of this note. If F is a functor of degree d and G E F, then the functor en : 3 -+ Mn(B?) induces

a map Homr(F, G) -+ H~rn,~, ,(e,F, e,G). This is an embedding for n > d, since the definition of c, implies that, for any IM E M,, there is a surjection l n M --+ c,M, where 1, is the left adjoint of en. Proposition 2.1 shows that F c,,enF, so that the surjection Z,e,F -+ c ,enF induces the following injection:

T h e o r e m 3.1 Suppose that F zs a functor of polynomzal degree 5 d and that G E 3, then the natural ensbeddzng HomF(F, G) - HornM,+, (ed+lF, ed+lG) zs an isomorphism.

P roo f : It suffices to show that any f E HornM, (ed+1F, ed+lG) extends to a map f : F --+

G in F. By Proposition 2.10, to define f on F'(Fk), it suffices to define f on elements strictly generated in degree d (namely on F(IF~),, for a E S(d, k)) and to check that the resulting map is well-defined with respect to the equivalence relation defining Bd(k).

If x = F ( a ) y , for some y E F ( I F ~ ) , then define f ( z ) = G(a)y. It is not difficult to check that this defines a linear map f : F(lFk) 2 Gd(k) -+ G ( F ~ ) .

Moreover, by a combination of the lemmas above and Proposition 2.15, this is a map of Mk(F)-modules, noting that the above definition implies that f takes values in the space of elements of G(lFk) generated in degree d. 0

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POLYNOMIAL FUNCTORS 985

Example 3.2 The following example shows that it is necessary to consider M d + l ( F ) - modules as opposed to the Md(F)-nlodules. Let q = p = 2 and consider the nth sym- metric power functor S n . The Ml(1F)-modules, Sn(lF), are all isomorphic, whereas Homr(S1, Sn) is zero unless n is a power of 2 . 0

This example, together with Theorem 1, should be compared with the following:

Proof : Given a map g : e d F -+ edG, take the intermediate extension cdg, which is a map F + G, since cdedF % F and cdedG "= G by Proposition 2.1. Moreover edcdg g, so that the map Homr(F, G) -- H ~ m , ~ , ( e d F , edG) is surjective, as required. 0

Definit ion 3.4 The duality functor D : F + Fop is the functor defined by (DF) (V) = {F(V*)}*; this corresponds to transpose duality in M n . It induces an isomorphism Hom3(F, G) r Hom3(DG, D F ) , for any functors F , G E 3, and is exact. 0

The second theorem follows as a corollary of Theorem 1

P r o o f of T h e o r e m 2: Suppose that G is a polynomial functor of degree d and F is a functor with connectivity c (F ) > d + 1; consider a short exact sequence F + H 5 G, representing a class in EX~$(G, F). To show that this class is zero, it suffices to find a section to the map p.

By the connectivity hypothesis on F, G(IF*+') r H(F~+ ' ) , SO that there is a map s : ed+ lF -- ed+lH in Md+lr providing a section to ed+lp. Theorem 1 then implies that this extends to a map s : G + H in F. By construction, ed+l(ps) = l(,,+lc); this implies that ps is an isomorphism of G, since G has polynomial degree d. (To see this, consider the cokernel and kernel of the map; these both have degree < d, since Fd is a thick sub-category, and thus have connectivity 5 d). Thus, s provides the required section.

For the statement concerning ~ x t i ( ~ , G ) ; one may apply the duality functor, to deduce the result from the above case, since D restricts to a contravariant functor D :

Fd -+ Fd, which is necessarily exact. Thus, an extension G -* H -- F is split if and only if the dual extension D F -t D H -, DG is split. Now, the functor D preserves the polynomial degree and the connectivity of functors, so that the result follows.

The theorem has the following special case; if F, is a simple functor indexed by a partition p, write Ip1 for the polynomial degree of F,.

Corollary 3.5 Suppose that A , p are q-regular partitions zndexing szmple functors FA, F, If c(X) > Ip/ + 1, then E x t & ( ~ x , F,) = 0 = E&(F,, FA).

R e m a r k 3.6 This is a considerable improvement on the result which is yielded by the methods of [FLS, Section 101, which suffice to yield a similar result for c(X) > $(/p\), where 6 is not a linear function. 0

3.1 An application to the coefficients d k ( F )

Theorem 3 may be wewed as a straightforward corollary to either Theorem 1 or Theo- rem 2 the fol~owirig lemma concerning the structure coefficients for slmple functors is reassuring (It appears, w ~ t h d different proof as [CK, Proposition 4 6 part 1))

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Lemma 3.7 Suppose that FA zs a szmple functor wzth connectzuzty c (X) and polynomzal degree IXI Then, zf c(X) 5 n 5 / X I , d,(Fx) # 0 Proof: Observe that , by the very definition of the constants c(X) and / A / , one has d,(,,)(Fx) # 0 and d l x l ( F x ) # 0. Thus. one considers the following case: suppose tha t there exists an integer n such that c(X) < n < / A ( and d,(Fx) = 0 . We show that the space of elements generated in degree n forms a sub-functor of FA.

The hypothesis implies that every element in Fx(lFn) is generated in degree n - 1. Consider the construction & l ( k ) for k > n ; Proposition 2.15 shows that G := G n - l ( k ) has the structure of a functor in F. Moreover, by this construction, G is a non-trivial functor of polynomial degree 5 n - 1. which is a subfunctor of FA. Since G has polyno- mial degree strictly less than \ A / , this implies G # FA, contradicting the fact that FA is simple. Therefore we conclude that no such n exists. 0

Theorem 3.8 If H is a functor whlch ta,kes finzte-dzmensional values, then H splits as H G @ F , where d , ( F ) = 0 Jor s 5 n a n d d , ( G ) = 0 for t 2 n.

Proof: The proof is a generalization of that of the lemma. G is defined to be the space of elements generated in degree 5 71 - 1. By the argument of Proposition 2.15, G has the structure of a sub-functor of H. which is polynomial of degree 5 n - 1. Form the short exact sequence: G - H + F, to define F . By the construction of H, d , (F ) = 0 for s < n and d t ( G ) = 0 for t > n. Thus. G is polynomial of degree 5 n - 1 and F has connectivity 2 n + 1. Theorem 2 implies that the short exact sequence splits, proving the theorem. 0

Acknowledgement: The author would like to thank Lionel Schwartz and Nick Kuhn for their comments on an earlier version of this paper'.

References

[CK] D. CARLISLE and N.J. KUHN. Smash products of summands of B(Z /p ) ; , Con- temporary Math. , A M S 96 (1989), 87-102.

[FLS] V. FRANJOU, J . LANNES and L. SCHWARTZ, Autour de la cohomologie de MacLane des corps finis, Invent. Math. 115 (1994), 513-538.

[HLS] H.-W. HENN, J . LANNES and L. SCHWARTZ, The categories of unstable mod- ules and unstable algebras over the Steenrod algebra modulo nilpotent objects, Amer. J . Math. 115 (1993), 1053.1106.

[KK] P. KRASON and N.J . KUHN. On embedding polynomial functors in symmetric powers, J. Algebra 163, (1994): 281-294.

[K] L. KROP, On comparison of 31-, G-, and S - representations, J. Algebra 146 (1992), 497-513.

[Kl] N.J. KUHN, Generic representations of the finite general linear groups and the Steenrod algebra: I , Amer J. Math. 116 (1993), 327-360.

'This earlier version appeared as 96-02, P~ipublrcations Mathimattpues d e I'Universiti d e Pam-Nod

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POLYNOMIAL FUNCTORS 987

[K2] N.J. KUHN, Generic representations of the finite general linear groups and the Steenrod algebra: 11: K-Theory 8 (1994), 395-426.

[K3] N.J. KUHN, Generic representations of the finite general linear groups and the Steenrod algebra: 111. K- Theory 9 (1995), 273-303.

[S] L. SCHWARTZ. Unstable Modules over the Steenrod Algebra and Sullivan's Fixed Point Set Conjecture, Chzcago Lecture Notes in Mathematics, Univ. Chzcago Press, Chicago and London (1994).

Received: June 1996

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