Invariant Subspaces of the Ring of Functions on a Vector Space over a Finite Field

191, 212]227 Ž1997. JA966922

JOURNAL OF ALGEBRA ARTICLE NO.

Invariant Subspaces of the Ring of Functions on a Vector Space over a Finite Field Nicholas J. KuhnU Department of Mathematics, Uni¨ ersity of Virginia, Charlottes¨ ille, Virginia 22903 Communicated by Wilberd ¨ an der Kallen Received August 20, 1996

If Fq is the finite field of characteristic p and order q s p s, let F Ž q . be the category whose objects are functors from finite dimensional Fq-vector spaces to Fq-vector spaces, and with morphisms the natural transformations between such functors. A fundamental object in F Ž q . is the injective I Fq defined by U

I FqŽ V . s FqV s SU Ž V . r Ž x q y x . . We determine the lattice of subobjects of I Fq. It is the distributive lattice associated to a certain combinatorially defined poset I Ž p, s . whose q connected components are all infinite Žwith one trivial exception.. An analysis of I Ž p, s . reveals that every proper subobject of an indecomposable summand of I Fq is finite. Thus I Fq is Artinian. Filtering I Fq and I Ž p, s . in various ways yields various finite posets, and we recover the main results of papers by Doty, Kovacs, ´ and Krop on the structure of SU Ž V .rŽ x q . over Fq , and SU Ž V . over Fp . Q 1997 Academic Press

1. INTRODUCTION If Fq is the finite field of characteristic p and order q s p s, let F Ž q . be the category with objects the functors F: finite dimensional Fq-vector spaces ª Fq-vector spaces, and with morphisms the natural transformations. This is an abelian category in the obvious way, e.g., G is a subobject of F means that GŽ V . : F Ž V . for all vector spaces V. We like to view an object F g F Ž q . as a * Partially supported by the N.S.F. and the C.N.R.S. 212 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

INVARIANT SUBSPACES

213

‘‘generic representation’’ of the general linear groups over Fq , as F Ž V . becomes an Fq w GLŽ V .x-module for all Fq-vector spaces V. The tight relationship between F Ž q . and the categories of Fq w GLnŽFq .x-modules, for all n, makes the study of F Ž q . of great representation theoretic interest. ŽFor an overview, see our series of papers wK:I, K:II, K:IIIx.. U Let I Fq g F Ž q . be defined by I FqŽ V . s FqV . Thus I FqŽ V . is the ring of Fq-valued functions on the dual space of V. This is a fundamental object in F Ž q .: I Fq is injective, and, indeed, the collection  I Fmqk < k G 04 is a set of injective cogenerators for F Ž q . wK:I, Sect. 3x. In wK:Ix, it is noted that there is a decomposition into indecomposable summands I Fq , I FqŽ 0 . [ I FqŽ 1 . [ ??? [ I FqŽ q y 1 . . Furthermore, there is a revealing alternate description of I Fq: I Fq Ž V . , S U Ž V . r Ž x q y x . .

Here SU Ž V . is the polynomial algebra on V, with dth homogeneous component S d Ž V . s Ž V m d . S d , and Ž x q y x . denotes the Žnonhomogeneous. ideal generated by elements of the form x q y x, x g V. Then I FqŽ0. s S 0 is the constant functor, and, for 1 F d F q y 1, I FqŽ d . is the ` image in I Fq of [rs0 S dq r Ž qy1.. Since the functors S d are known to have only a finite number of simple composition factors Žsee wK:Ix or Section 5., one concludes that each I FqŽ d . with 1 F d F q y 1 is a locally finite object 1 with an infinite number of composition factors. The main result of this paper is a complete determination of the lattice L Ž I Fq . of subobjects of I Fq and, individually, the lattices L Ž I FqŽ d .. of subobjects of each of the I FqŽ d .. Before defining our lattice of subobjects, we comment briefly on other interpretations of our work, and its relation to previous results. Let P Fq g F Ž q . be defined by P FqŽ V . s Fq w V x. Thus P FqŽ V . is the Fq-vector space with the set V as basis. Then P Fq is the projective in F Ž q . dual to I Fq under the duality D: F Ž q . op ª F Ž q . defined by Ž DF .Ž V . s F Ž V U .U . Determining L Ž I Fq . is equivalent to determining the lattice of quotient objects of P Fq. With Vn s ŽFq . n, I FqŽ Vn . is a module for the multiplicative semigroup MnŽFq .. As will be explained in Section 5, determining L Ž I Fq . allows us to immediately determine the lattice of MnŽFq .-submodules of I FqŽ Vn . for all n. ŽWe confess to not completely determining the GLnŽFq .-submodule lattices.. 1

An object in an abelian category is finite if it admits a finite composition series with simple subquotients, and is locally finite if it is the sum of its finite subobjects.

214

NICHOLAS J. KUHN

We relate our results to others in the literature. Filtering I Fq by polynomial degree and taking the associated graded object, we recover the main results of wKox: the determination of the submodules of S d Ž Vn ., viewed as either an MnŽFq ., GLnŽFq ., or SL nŽFq .-module, where SU Ž V . s SU Ž V .rŽ x q .. Letting q ‘‘go to infinity,’’ we recover the main results of wDx and wKr1x: the determination of the submodules of S d Ž Vn ., where Vn s ŽFp . n, viewed as either an MnŽFp ., GLnŽFp ., or SL nŽFp .-module. As in all these previous papers Žand also wK1x., subobjects are the ‘‘obvious’’ ones defined using polynomial multiplication and pth powers, and the subobject lattice is isomorphic to the distributive one associated to a certain combinatorially defined poset. By ‘‘polynomial multiplication,’’ we mean the maps S i m S j ª S iqj, and by ‘‘ pth powers,’’ we mean the inclusions Sji ¨ S p i, where Gj denotes the functor G twisted by the Frobenius Žas in wK:IIx.. Let N s be the additive monoid of s-tuples I s Ž i 0 , . . . , i sy1 . of nonnegasy 1 tive integers. Given I s Ž i 0 , . . . , i sy1 . g N s, let dŽ I . s Ý rs0 i r p r , and deI I ˜I d Ž . fine S , F , S g F q as follows. First defining F to be the degree d component of F U Ž V . s SU Ž V .rŽ x p ., we let i

1 sy 1 S I s S i 0 m Sji1 m ??? m Sj sy

and i

sy 1 1. F I s F i 0 m Fji1 m ??? m Fj sy

Now let F I : S I ª S dŽ I . be the composite S i 0 m S p i1 m ??? m S p

sy 1

i sy 1

multiply

S dŽ I . .

6

pth powers

6

sy11 S i 0 m Sji1 m ??? m Sji sy

FI

p

Then S˜I ; I Fq is defined to be the image of the composite S I ª S dŽ I . ª I Fq, where p is the inclusion S dŽ I . ¨ SU followed by the projection SU ª I Fq. Let R 0 , . . . , R sy1 g Z s be the following vectors: if s s 1, R 0 s p y 1, and, if s ) 1, R 0 s Žy1, 0, . . . , 0, p ., R1 s Ž p, y1, 0, . . . , 0., R 2 s Ž0, p, y1, 0, . . . , 0., . . . , and R sy 1 s Ž0, . . . , 0, p, y1..Let I Ž p, s . be the poset ŽN s, F., where ‘‘F ’’ is the partial ordering generated by the inequalities J - I, if I s J q R r , for some r. PROPOSITION 1.1. Ž1. I Fq s Ý I S˜I. Ž2. S˜IrRadŽ S˜I . , F I, and is simple. Ž3. RadŽ S˜I . s Ý J - I S˜J. Ž4. F I , F J if and only if I s J.

INVARIANT SUBSPACES

215

To state our main theorem, we need a standard construction from lattice theory Žas in, e.g., wG, p. 72x or wS, p. 100x.. If I is a poset, call K : I an order ideal if K g K and J F K implies J g K. Let L Ž I . s  K : I ¬ K is an order ideal4 . This becomes a distributive lattice using union and intersection for the join and meet lattice operations. THEOREM 1.2. The assignment I ¬ S˜I induces an isomorphism of lattices L Ž I Ž p, s . . , L Ž I Fq . . Thus the finite join-irreducible subobjects 2 of I Fq are precisely the S˜I, I g I Ž p, s ., and every subobject G : I Fq has a representation Gs

Ý S˜I Ig K

for a unique order ideal K : I Ž p, s .. Remark 1.3. Our lattice isomorphism carries two little bits of extra structure. Ž1. I Ž p, s . has a monoid structure compatible with the partial ordering, and this induces a product on L Ž J Ž p, s ..: if K1 and K2 are two order ideals of I Ž p, s ., then K1 ? K2 is defined to be the smallest order ideal containing the set  I q J ¬ I g K1 , J g K2 4 . Meanwhile, the natural product on I Fq defines a subobject G1 ? G 2 : I Fq, given G1 , G 2 : I Fq, thus defining a product on L Ž I Fq .. The isomorphism of Theorem 1.2 preserves these products on the lattices. Ž2. Since Ž I F .j , I F , twisting by the Frobenius induces an order s q q automorphism of the lattice L Ž I Fq .. Under the isomorphism of Theorem 1.2, this corresponds to the evident automorphism on L Ž I Ž p, s .. induces by cyclically permuting the factors of I Ž p, s . s ŽN s, F.. The indecomposable summand version of Theorem 1.2 is easily stated. There is a decomposition qy1

I Ž p, s . s

@ I Ž p, s . d ds0

into indecomposable posets, where I Ž p, s . 0 s  04 , and, for 1 F d F q y 1, I Ž p, s .d s  I g I Ž p, s . y  04 ¬ dŽ I . ' d mod q y 14 . 2

F is join-irreducible means that F s G q H, with G, H : F, only if F s G or F s H.

216 THEOREM 1.4.

NICHOLAS J. KUHN

For 0 F d F q y 1, there is an isomorphism of lattices L Ž I Ž p, s . d . , L Ž I FqŽ d . . .

Figure 1 shows the lower portion of the infinite poset I Ž2, 2.1 , i.e., the poset that describes the subobject structure of I F4Ž1., the injective envelope of S 1 in F Ž4.. In general, I Ž p, s .d would have a diagram that would look roughly like a s-dimensional cone. An analysis of the posets I Ž p, s .d reveals that for a fixed I g I Ž p, s .d , all but a finite number of J g I Ž p, s .d satisfy J ) I. We thus conclude COROLLARY 1.5. E¨ ery proper subobject of I FqŽ d . is finite. Thus I Fq is an Artinian object in F Ž q .. Various remarks about the results above are in order here. Firstly, parts of the proposition have long been known, as well as the q-restricted highest weight ‘‘name’’ for F I Žsee, e.g., wK:II, Theorem 5.23 and Example 7.6x.. However, we give new and very noncomputational proofs. Secondly, if q s p Ži.e., s s 1., we learn that I FqŽ d . is an infinite uniserial object for all d ) 0. This was already proved by us in wK:II, Sect. 7x. Indeed, our method of proving the proposition follows the strategy uses there. Thirdly, given the proposition, the theorems follow immediately from general lattice theory.

FIG. 1. The poset I Ž2, 2.1 in degrees less than 10.

INVARIANT SUBSPACES

217

Lastly, the corollary gives some slim evidence for the Artinian conjecture of L. Schwartz: for all k, I Fmqk is Artinian.3 ŽSee wK:II, Sect. 3x for a discussion of this conjecture and its implications.. We describe the organization of the rest of the paper. Section 2 is devoted to the general lattice theory we need. In Section 3 we prove various properties about our poset I Ž p, s .. Both the proposition and two theorems are proved in Section 4. In Section 5, we recover the theorems of Doty wDx, Kovacs ´ wKox, and Krop wKr1x, among other related results Žthough we note that when we can conclude something about SL n or GLn lattices, we are generally depending on an elegant theorem of Krop in wKr2x..

2. LATTICE THEORY In this section, we sketch the lattice theory need to identify when the lattice of subobjects of a locally finite F g F Ž q . is distributive, and to then describe the structure of such a distributive lattice. We work in the setting of locally finite AB5 categories: abelian categories with exact direct limits wPox and locally finite objects. Thus, in this section, we let A denote such a category. We begin with some lattice theoretic definitions wGx. DEFINITIONS 2.1. Let L be a lattice. Ž1. L is modular if A k Ž B n C . s Ž A k B . n C whenever C F A, for all A, B, C g L . Ž2. L is distributi¨ e if A k Ž B n C . s Ž A k B . n Ž A k C ., for all A, B, C g L . Ž3. L is complete if one can form joins in L indexed by arbitrary sets. Ž4. In a complete lattice L , A g L is compact if whenever A F Ei g I Bi , there exists a finite subset J : I such that A F Ei g J Bi . Ž5. A complete lattice L is compactly generated if each A g L is the join of the set  B ¬ B F A and is compact4 . If F is an object in A, we let L Ž F . denote its lattice of subobjects. Our hypotheses on A imply that L Ž F . is compactly generated and modular. If k ) 1, it is unreasonable to calculate the complete lattice of subobjects of I Fmk , and it is q no longer true that every proper subobject of an indecomposable summand of I Fmk is finite. q However Powell has a preprint wPx verifying the conjecture when k s q s 2. 3

218

NICHOLAS J. KUHN

THEOREM 2.2. L Ž F . is distributi¨ e if and only if F does not contain any subquotient of the form S [ S, with S a simple object in A. If L is a complete lattice, let I Ž L . s  A g L ¬ A is compact and join irreducible4 , a subposet of L . THEOREM 2.3. If a compactly generated modular lattice L is distributi¨ e, then the assignment that sends an order ideal K : I Ž L . to EA g K A defines an isomorphism of lattices L Ž I Ž L .. , L . If F is an object in A, let I Ž F . s  G : F ¬ G is finite and GrRadŽ G . is simple4 , a poset under inclusion. G : F being finite in A corresponds to G being compact in L Ž F ., and GrRadŽ G . being simple in A corresponds to G being join irreducible in L Ž F .. Thus the previous two theorems combine to give COROLLARY 2.4. If F g A does not contain a subquotient of the form S [ S, with S simple, then there is a lattice isomorphism L Ž I Ž F .. , L Ž F .. To easily identify I Ž F . in this case, we note PROPOSITION 2.5. If L Ž F . is distributi¨ e, then I Ž F . can be characterized as the unique subposet I of the finite subobjects of F such that Ž1. ÝG g I G s F, Ž2. GrRadŽ G . is simple, for all G g I , and Ž3. for all G g I , RadŽ G . s Ý H, summing o¨ er H g I such that H is a proper subobject of G. Versions of both Theorems 2.2 and 2.3 are well known. For example, under the additional hypothesis that L is finite, Theorem 2.3 is called the fundamental theorem for finite distributive lattices in wS, p. 106x. Thus we just sketch their proofs below. Proof of Theorem 2.2. A lattice L is known to be distributive if and only if it has no sublattice with diagram: G G1 G 2

G3

Ž 2.1.

H Žsee wG, p.70x.. p If there exists F ª Q = S [ S, with p epic and S simple, then L Ž F . contains a subdiagram as above, with G s py1 Ž S [ S ., G1 s py1 Ž S [ 0., G 2 s py1 ŽdiagŽ S .., G 3 s py1 Ž0 [ S ., and H s KerŽp .. Thus L Ž F . is not distributive.

219

INVARIANT SUBSPACES

The converse is clear if F is semisimple. But we can reduce to that case: if L Ž F . contains sublattice Ž2.1., then L Ž FrH . contains SocŽ GrH . SocŽ G1rH .

SocŽ G 2rH . SocŽ G 3rH .

Ž 2.2.

0 as a sublattice, where we have used that F is locally finite to be sure that all these socles are nonzero. By the semisimple case, there exists S [ S : FrH, and we are done. Proof of Theorem 2.3. It is convenient to abbreviate ‘‘compact’’ as ‘‘c,’’ and ‘‘join irreducible’’ as ‘‘j.i.’’ Recall that in a lattice L , C F D means that C s C n D, and that C is j.i. means that C s A k B implies that C s A or C s B. It is easy to then verify that in a distributive lattice, C is j.i. and C F A k B implies that C F A or C F B. Furthermore, if C is c.j.i. and K is any subset of L , then C F EB g K B implies that C F B for some B g K. Now suppose that L is a compactly generated distributive lattice. Define Q: L ª L Ž I Ž L . .

C: L Ž I Ž L . . ª L

and

by QŽ A. s  c.j.i. C ¬ C F A4 , and C Ž K . s EB g K B. The theorem will follow once we check that Q is a map of lattices, C Ž QŽ A.. s A for all A g L , and QŽ C Ž K .. s K for all order ideals K : I Ž L .. It is obvious that QŽ A n B . s QŽ A. l QŽ B .. Since L is distributive, we have Q Ž A k B . s  c.j.i C ¬ C F A k B 4 s  c.j.i C ¬ C F A or C F B 4 s Q Ž A. j Q Ž B . . Thus Q is a lattice map. To show that C Ž QŽ A.. s A, we observe that, since L is compactly generated, As

E

c. BFA

Bs

E

c. BFA

ž

E

j.i. CFB

C s

/

E

c.j.i. CFA

C s C Ž Q Ž A. . .

220

NICHOLAS J. KUHN

Finally, since L is distributive, we have Q Ž C Ž K . . s c.j.i. C ¬ C F

½

EB

Bg K

5

s  c.j.i. C ¬ C F B for some B g K 4 s K, and the theorem is proved. Proof of Proposition 2.5. It is clear that I Ž F . satisfies Ž1., Ž2., and Ž3.. Conversely, suppose that a poset I satisfies these three properties. Clearly I : I Ž F .; we need to check the reverse inclusion. Given any G g I Ž F ., we need to show G g I . Property Ž1. implies that there exist G1 , . . . , Gr g I such that G : G1 q ??? qGr . Since GrRadŽ G . is simple and L Ž F . is distributive, we conclude that G : Gi for some i. If G s Gi , we are done. Otherwise, G : RadŽ Gi ., by property Ž2. and general properties of the radical. By property Ž3., there exist GX1 , . . . , GXt g I , with each GXj a proper subobject of Gi , and G : GX1 q ??? qGXt . As before, G : GXj for some j. Since Gi is finite, it contains no infinite descending chain. Thus, continuing in this way, we eventually learn that G g I .

3. THE STRUCTURE OF I Ž p, s . In this section we study some purely combinatorial aspects of the partially ordered set I Ž p, s .. It is convenient to add to the definitions and notation of Sect. 1. For I s Ž i 0 , . . . , i sy1 ., let < I < s i 0 q ??? qi sy1. A given nonnegative integer d sy 1 Ž . r can be written uniquely in the form d s Ý rs0 i r d p , with 0 F i r Ž d . F p y 1 for 0 F r F s y 2. Given d G 0, let I Ž d ., J Ž d . g I Ž p, s . and I Ž p, s, d . ; I Ž p, s . be defined by I Ž d . s Ž i 0 Ž d ., . . . , i sy1Ž d .., J Ž d . s Ž d, 0, . . . , 0., and I Ž p, s, d . s  I ¬ dŽ I . s d4 . Immediately from the definitions, we have the next lemma. LEMMA 3.1. Ž p s y 1..

If I ) J, then < I < ) < J <, dŽ I . G dŽ J ., and dŽ I . ' dŽ J . mod

PROPOSITION 3.2. Ž1. For all d G 0, I Ž p, s, d . s  I ¬ I Ž d . F I F J Ž d .4 . Ž2. For all p s y 1 G d G 0, I Ž p, s .d s  I ¬ I Ž d . F I 4 . Proof. To prove Ž1., first note that, using the lemma, it is clear that if dŽ I . F I F J Ž d ., then dŽ I . s d, i.e., I g I Ž p, s, d .. Conversely, suppose that dŽ I . s d.

INVARIANT SUBSPACES

221

By induction on < I <, we show that I Ž d . F I. If I Ž d . s I, we are done. If not, i r G p for at least one r with 0 F r F s y 2. Let I X s I y R r . Then I Ž d . F I X by induction, and I X - I by definition, and we can conclude that I Ž d . - I. Similarly, by downward induction on < I <, we show that I F J Ž d .. If I s J Ž d ., we are done. If not, i r G 1 for at least one r with 1 F r F s y 1. Let I X s I q R r . Then I X F J Ž d . by induction, and I - I X by definition, so that I - J Ž d .. The proof of Ž2. is similar. COROLLARY 3.3. The decomposition p sy1

I Ž p, s . s

@ I Ž p, s . d ds0

is a decomposition of I Ž p, s . into indecomposable Ž i.e., connected. posets. COROLLARY 3.4. Ž1. dŽ I . s dŽ J . if and only if I y J is a Z-linear combination of R1 , . . . , R sy1. Ž2. dŽ I . ' dŽ J . mod Ž p s y 1. if and only if I y J is a Z-linear combination of R 0 , . . . , R sy1. PROPOSITION 3.5. Let I, J g I Ž p, s . y  04 . Then I G J if and only if I y J is an N-linear combination of R 0 , . . . , R sy1. Proof. The ‘‘only if’’ implication is clear. We prove the converse by induction on a0 q ??? qa sy1 , where we suppose that I y J s a0 R 0 q ??? qa sy1 R sy1 ,

Ž 3.1.

with I s Ž i 0 , . . . , i sy1 . and J s Ž j0 , . . . , j sy1 . both elements of I Ž p, s . y  04 , a r G 0 for all r, and at least one a r is positive. We need to show that then J - I. Choose r so that a r G 1, and let I X s I y R r . If I X g I Ž p, s ., we are done: J - I because J F I X by induction, and I X - I by definition. So we can assume that I X f I Ž p, s ., i.e., that i ry1 F p y 1 Žwhere we write subscripts modulo s .. Then Ž3.1. implies that

Ž p y 1 . y jry1 G i ry1 y jry1 s par y ary1 ,

Ž 3.2.

and we conclude that a ry1 G Ž a r y 1 . p q jry1 q 1 G 1.

Ž 3.3.

Continuing in this way, under our inductive hypothesis, either J - I or a r G 1 for all r. Noting that R 0 q ??? qR sy1 s Ž p y 1, . . . , p y 1., and recalling that J / 0, this latter case implies that i t G p for some t. But then, letting I X s I y R tq1 , we will have J F I X by induction, and I X - I by definition, so that J - I.

222

NICHOLAS J. KUHN

PROPOSITION 3.6. Gi¨ en I g I Ž p, s .d , all but a finite number of J g I Ž p, s .d satisfy J G I. To prove this proposition, it is convenient to define some notation. Let E0 , . . . , Esy1 be the standard basis of R s. Given I s Ž i 0 , . . . , i sy1 ., let d 0 Ž I ., . . . , d sy1Ž I . be defined by d r Ž I . s i r q pi rq1 q p 2 i rq2 q ??? qp sy1 i ry1

Ž with indices taken modulo s . , and let Dr Ž I . s d r Ž I . Er . We say that a vector B is an Rq-linear combination of A 0 , . . . , A sy1 if B s a0 A 0 q ??? qa sy1 A sy1 , with a r a nonnegative s real number, for all r. Let Rq denote the Rq-linear combinations of E0 , . . . , Esy1. LEMMA 3.7. Ž1. Er is an Rq-linear combination of R 0 , . . . , R sy1 for all r. s Ž2. If I g Rq , Dr Ž I . y I is an Rq-linear combination of R 0 , . . . , R sy1 for all r. Proof. By symmetry, it suffices to prove each statement for a single value of r. Then we have Esy1 s p sy 1r Ž p s y 1 . R 0 q p sy2r Ž p 2 y 1 . R1 q ??? q1r Ž p s y 1 . R sy1 , which proves Ž1., and D 0 Ž I . y I s i sy1 R sy1 q Ž pi sy1 q i sy2 . R sy2 q ??? q Ž p sy2 i sy 1 q ??? qpi 2 q i1 . R1 , which proves Ž2.. Proof of Proposition 3.6. The proposition is obvious when d s 0, so assume d / 0. By Proposition 3.5, we need to show that, given I g I Ž p, s .d , for all but finitely many J g I Ž p, s .d , J y I is an N-linear combination of R 0 , . . . , R sy1. By statement Ž1. of the last lemma, the R r are linearly independent. Thus J y I is an N-linear combination of R 0 , . . . , R sy1 if and only if J y I is both a Z-linear combination and an Rq-linear combination of R 0 , . . . , R sy1. By Corollary 3.4, the former is true for all J g I Ž p, s .d . Thus it suffices to show that for all but a finite number of J g I Ž p, s ., J y I is an Rq-linear combination of R 0 , . . . , R sy1. This follows from the two parts of the lemma, which combine to show s that if J s Ž j0 , . . . , j sy1 . g Rq , and J y I is not an Rq-linear combination of R 0 , . . . , R sy1 , then 0 F jr - d r Ž I ., for all r.

223

INVARIANT SUBSPACES

COROLLARY 3.8.

E¨ ery proper order ideal K ; I Ž p, s .d is finite.

4. PROOF OF THE MAIN RESULTS Recall that, for F g F Ž q ., DF is defined by Ž DF .Ž V . s F Ž V U .U . In wK:IIx we proved that simple functors are self-dual. This has the following corollary. LEMMA 4.1 wK:II, Corollary 7.5x. Let F be a finite functor. If Hom F Ž q.Ž F, DF . , Fq , generated by a : F ª DF, then KerŽ a . s RadŽ F ., and FrRadŽ F . , ImŽ a . is simple. Proposition 1.1 will be proved by applying this lemma to the case when F s S I or S˜I. In preparation for this, let T i g F Ž q . be defined by T i Ž V . s V m i, and then, if I s Ž i 0 , . . . , i sy1 ., define T I, SI , and S˜I by i

sy 1 1, T I s T i 0 m Tji1 m ??? m Tj sy

SI s DS I, and S˜I s DS˜I. Furthermore, let S I be the group S i 0 = ??? = S i sy 1, so that ŽT I . S I s S I and ŽT I . S I s S I . Finally, note that the norm map, the sum of the permutations

Ý

s: TIªTI

sgS I NI

factors as T I ª S I ª SI ª T I. Starting from the observation that Hom F Ž q. Ž T 1 , Ž T 1 . j r . ,

½

Fq

if r s 0,

0

otherwise,

the methods of wK:III, Sect. 4.4x formally imply LEMMA 4.2. Ž1. Hom F Ž q.ŽT I, T I . , Fq w S I x as Fq-algebras. Ž2. Hom F Ž q.ŽT I, T J . , 0 if I / J. COROLLARY 4.3. Ž1. Hom F Ž q.Ž S I, SI . , Fq generated by NI . Ž2. Hom F Ž q.Ž S I, S J . , 0 if I / J. Let K I denote the kernel of the projection S I ª F I. Elementary inspection of our definitions Žcompare with wK:II, Example 7.6x. reveals LEMMA 4.4. KerŽ NI . s K I. We need one more observation before proving Proposition 1.1. Given FI

˜ I: S I ª I Fq denote the composite S I ª S dŽ I . ª I Fq, I s Ž i 0 , . . . , i sy1 ., let F I where F is as in the introduction.

224

NICHOLAS J. KUHN

˜ I Ž K I . s Ý J - I S˜J. LEMMA 4.5. F ˜ I. Thus S˜I Ž V . is the span of Proof. By definition, S˜I is the image of F ˜ I Ž x Ž0. m x Ž1.j m ??? m x Ž s y 1.j sy 1 ., with x Ž r . g S i r Ž V ., and elements F IŽ I Ž .. ˜ F K V ; S˜I Ž V . is the span of such elements for which at least one of the x Ž r . is in the ideal of pth powers. Now observe that if x Ž r . g SU Ž V . can be written in the form yz p , then IŽ Ž . ˜ ˜ Iq R r Ž y Ž0. m y Ž1.j m ??? m y Ž s y F x 0 m x Ž1.j m ??? m x Ž s y 1.j sy 1 . s F . . 1 j sy 1 , where, with indices written mod s, y Ž r X . s x Ž r X . if r X / r, r q 1, sy1 ˜Iq R r ˜ I Ž K I . s Ý rs0 y Ž r . s y, and y Ž r q 1. s zx Ž r q 1.. It follows that F S s J ˜ ÝJ- I S . Proof of Proposition 1.1. Statement Ž1. is clear by inspection. Using Lemma 4.1, Corollary 4.3 and Lemma 4.4 imply that S IrRadŽ S I . s F I ˜ I: and is simple, RadŽ S I . s K I, and F I , F J if and only if I s J. Since F I I I I I I ˜ ˜ ˜ ˜ S ª S is onto, we conclude that S rRadŽ S . s F , and RadŽ S . s ˜ I Ž K I .. The last lemma identifies F ˜ I Ž K I . with Ý J - I S˜J. F Proof of Theorem 1.2. By Proposition 1.1, all the composition factors of I Fq are distinct. Thus, by Corollary 2.4, L Ž I Fq . is distributive, and there is a lattice isomorphism L Ž I Ž I Fq .. , L Ž I Fq .. Using Proposition 2.5, Proposition 1.1 precisely shows that the assignment I ¬ S˜I defines an isomorphism of posets I Ž p, s . ª I Ž I Fq .. Proof of Theorem 1.4. This follows from Theorem 1.2 and Corollary 3.3. Proof of Corollary 1.5. This follows from Theorem 1.4 and Corollary 3.8.

5. RELATED RESULTS In this section we filter the results of Section 1 in various interesting ways. In wK:Ix we noted that finite functors are precisely the F g F Ž q . such that the growth function n ¬ dim FqF Ž Vn . is a polynomial function of n, and that these functors are polynomial in the sense of wEMx. We let Fd Ž q . ; F Ž q . be the full subcategory generated by finite functors whose growth functions are polynomial of degree no more than d.

INVARIANT SUBSPACES

225

Let pd I Fq be the largest subobject of I Fq in Fd Ž q .. This is an injective in Fd Ž q .. Let IFdŽ p, s . s  I g I Ž p, s . ¬ < I < F d4 . Since F I has a growth function of degree < I <, Theorem 1.2 has the following corollary. COROLLARY 5.1.

L Ž pd I Fq . , L Ž IFdŽ p, s ...

Remark 5.2. Note that pd I Fqrpdy1 I Fq is semisimple. This is not a general phenomenon. Indeed, there can be nontrivial extensions between simple functors of the same degree, with the simplest example perhaps being a nontrivial extension between the two simple functors of degree 4 in F Ž2.. Recall that S d is defined by SU Ž V . s SU Ž V .rŽ x q .. Note that S d s S˜d if d F q y 1, and S d s S˜drS˜dyŽ qy1. if q F d. Recalling that I Ž p, s, d . s  I ¬ dŽ I . s d4 , Theorem 1.2 and Proposition 3.2 imply COROLLARY 5.3. simple head F J Ž d..

L Ž S d . , L Ž I Ž p, s, d ... S d has simple socle F IŽ d. and

This is equivalent to the main result of wKox. To make the translation, we need some notation. Given F g F Ž q ., let LM nŽ F ., LG L nŽ F ., and LS L nŽ F . denote the lattices of subobjects of F Ž Vn ., regarded respectively as an MnŽFq .-module, GLnŽFq .-module, and SL nŽFq .-module. Now we make two observations. The first is that, by general principles Žas discussed in wK:IIx., the map L Ž F . ª LM Ž F . that sends G ; F to n GŽ Vn . ; F Ž Vn . is onto, for any F g F Ž q .. Thus LM nŽ F . will be the quotient lattice of L Ž F . under the equivalence relation generated by saying that H ; G if H ; G ; F and Ž GrH .Ž Vn . s 0. In our case this goes as follows. Given I s Ž i 0 , . . . , i sy1 ., let nŽ I . s min n ¬ n G i rrŽ p y 1. for all r 4 . Then nŽ I . F n if and only if F I Ž Vn . / 0. Let IM nŽ p, s . s  I ¬ nŽ I . F n4 , and let IM nŽ p, s, d . s IM nŽ p, s . l I Ž p, s, d .. Theorem 1.2 implies COROLLARY 5.4. Ž1. LM nŽ I Fq . , L Ž IM nŽ p, s ... Ž2. LM Ž S d . , L Ž IM Ž p, s, d ... n

n

The second observation is that in wKr2, Theorem 1x Krop gave an easy-to-check criterion on a functor F ensuring that LM nŽ F . s LG L nŽ F . s LS L nŽ F . . Roughly put, it says that this is the case if F is the restriction of a polynomial functor defined on Fp-vector spaces and with q restricted weights. This criterion does not hold for I Fq, but does for S d Žas Krop points out.. We conclude that L Ž IM nŽ p, s, d . . , LM nŽ S d . s LG L nŽ S d . s LS L nŽ S d . , which is the main result of wKox.

226

NICHOLAS J. KUHN

Remark 5.5. A simple example showing that LM nŽ I Fq . / LG L nŽ I Fq . occurs when n s q s 2. In positive degrees, SU Ž V2 .rŽ x 2 y x . has two composition factors: F 1 Ž V2 . s V2 and F 2 Ž V2 . s L2 Ž V2 .. As M2 ŽF2 .-modules, there is a nontrivial extension between these. This extension splits when viewed as an extension of GL2 ŽF2 .-modules. In spite of example like this, we can still conclude that LG L nŽ I FqŽ d G 0.. and LS L nŽ I FqŽ d ) 0.. are both distributive, and thus satisfy the structure theorems of Section 2. Finally, we note that, for d F q y 1, S d s S˜d, and that S d is defined on Fp-vector spaces. These observations allow us to determine LF pŽ S d ., the lattice of subobjects of S d, viewed as an object in the category F ŽFp . of functors F : finite dimensional Fp-vector spaces ª Fp-vector spaces. To explain this, we need to introduce yet another category of functors: F Ž q, Fp . will denote the category whose objects are functors F : finite dimensional Fq-vector spaces ª Fp-vector spaces. Given F g F Ž q ., let FFp g F Ž q, Fp . be defined by FFpŽ V . s F Ž V . mFqFp . Given G g F ŽFp ., let ResŽ G . g F Ž q, Fp . be defined by ResŽ G .Ž V . s F Ž V mFqFp .. LEMMA 5.6.

Suppose that F g F Ž q . and G g F ŽFp . satisfy

Ž1. F is locally finite, Ž2. L Ž F . is distributi¨ e, Ž3. FF , ResŽ G ., and p Ž4. e¨ ery H g I ŽResŽ G .. Ž notation as in Section 2. is of the form H s ResŽ K . for some K ; G. Then L FpŽ G . , L Ž F .. Proof. The first point is that Fq is a splitting field for F Ž q . wK:II, Sect. x 5 , i.e., all simple functors in F Ž q . are absolutely simple. Assuming Ž1. and Ž2., this implies that I Ž F . , I Ž FFp ., and so L Ž F . , L Ž I Ž F .. , L Ž I Ž FFp .. , L Ž FF p .. Now notice that Res induces a monic map L Ž G . ª L ŽResŽ G .. , L Ž F Fp .. Under assumption Ž4., this monic map will also be epic. This lemma applies in the case G s S d, F s S d, and d - q. Then Theorem 1.2 determines L Ž S d ., and thus L FpŽ S d .. The details go as follows. Let I Ž p, `. be the poset ŽN` , F., where N` is the set of sequences Ž i 0 , i1 , . . . . of nonnegative integers that are eventually 0, and F is

INVARIANT SUBSPACES

227

generated by the inequalities I q R r ) I for r G 1. ŽHere, as in Section 1, R r is the vector with p in the r th place, and y1 in the Ž r y 1.st.. Let I Ž p, `, d . s  I g I Ž p, `. ¬ dŽ I . s d4 . Then I Ž p, ` . s

@ I Ž p, `, d . dG0

is a decomposition of the poset I Ž p, `. into indecomposable posets. COROLLARY 5.7.

LF pŽ S d . , L Ž I Ž p, `, d ...

As before, one can immediately read off the lattice of subobjects of S d Ž Vn ., viewed as an MnŽFp .-module, thus recovering the results of wKr1, Sect. 2x. Again, using wKr2x, this agrees with the GLnŽFp . and SL nŽFp . lattices. Thus we recover the main result of wDx. Remark 5.8. Note that I Ž p, s . s I Ž p, `.rŽ;., where ; is the equivs " !# alence relation generated by Ž i 0 , i1 , . . . . ; Ž 0, . . . , 0 , i 0 , i1 , . . . .. REFERENCES wDx

S. R. Doty, The submodule structure for certain Weyl modules for groups of type A n , J. Algebra 95 Ž1985., 373]383. wEMx S. Eilenberg and S. MacLane, On the groups H Žp , n., II. Ann. Math. 60 Ž1954., 49]139. wGx G. Gratzer, ‘‘Lattice Theory: First Concepts and Distributive Lattices,’’ W. H. ¨ Freeman, San Francisco, 1971. wKox L. G. Kovacs, ´ Some representations of special linear groups, Amer. Math. Soc. Proc. Symp. Pure Math. 47 Ž1987., 207]218. wKKx P. Krason and N. J. Kuhn, On embedding polynomial functors in symmetric powers, J. Algebra 163 Ž1994., 281]294. wKr1x L. Krop, On the representations of the full matrix semigroup on homogeneous polynomials, J. Algebra 99 Ž1986., 370]421. wKr2x L. Krop, On comparison of M-, G-, and S-representations, J. Algebra 146 Ž1992., 497]513. wK:Ix N. J. Kuhn, Generic representation theory of the finite general linear groups and the Steenrod algebra: I, Amer. J. Math. 116 Ž1994., 327]360. wK:IIx N. J. Kuhn, Generic representation theory of the finite general linear groups and the Steenrod algebra: II, K-Theory 8 Ž1994., 395]428. wK:IIIx N. J. Kuhn, Generic representation theory of the finite general linear groups and the Steenrod algebra: III, K-Theory 9 Ž1995., 273]303. wK1x N. J. Kuhn, The Morava K-theories of some classifying spaces, Trans. Amer. Math. Soc. 304 Ž1987., 193]205. wPox N. Popescu, ‘‘Abelian Categories with Applications to Rings and Modules,’’ Academic Press, London, 1973. w Px G. M. L. Powell, The Artinian conjecture for I m2 , preprint, 1996. wSx R. P. Stanley, ‘‘Enumerative Combinatorics,’’ Vol. 1, Wadsworth & BrooksrCole, Monterey, CA, 1986.