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

JOURNAL OF ALGEBRA 191, 212᎐227Ž. 1997 ARTICLE NO. JA966922

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

Nicholas J. KuhnU

Department of Mathematics, Uniërsity of Virginia, Charlottesïlle, Virginia 22903 View metadata, citation and similar papers at core.ac.uk brought to you by CORE Communicated by Wilberd än der Kallen provided by Elsevier - Publisher Connector Received August 20, 1996

s If Fq is the finite field of characteristic p and order q s p , let FŽ.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 FŽ.q is the injective I defined by Fq

U IVŽ.FV SU Ž.ŽV xq x .. Fq sq s r y

We determine the lattice of subobjects of I . It is the distributive lattice associ- Fq ated to a certain combinatorially defined poset IŽ.p, s whose q connected compo- nents are all infiniteŽ.Ž. with one trivial exception . An analysis of I p, s reveals that every proper subobject of an indecomposable summand of I is finite. Thus I is FFqq Artinian. Filtering I and IŽ.p, s in various ways yields various finite posets, and we Fq recover the main results of papers by Doty, Kovacs,´ and Krop on the structure of UUq SVŽ.Žrx .over Fqp, and SVŽ.over F . ᮊ 1997 Academic Press

1. INTRODUCTION

s Ž. If Fq is the finite field of characteristic p and order q s p , let F q be the category with objects the functors

F: finite dimensional Fqq-vector spaces ª F -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 GVŽ.: FVŽ.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 ᮊ 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 FVŽ. becomes an FqqwGLŽ. V x-module for all F -vector spaces V. The tight rela- tionship between FŽ.q and the categories of FqnqwGL Ž.F x-modules, for all n, makes the study of FŽ.q of great representation theoretic interest.Ž For an overview, see our series of paperswx K:I, K:II, K:III .. U Let I FŽ.q be defined by IVŽ. FV. Thus IVŽ.is the ring of FFqqg sqFq Fq-valued functions on the dual space of V. This is a fundamental object in FqŽ.:I is injective, and, indeed, the collection ÄIkmk< 04 is a set of FFq qG injective cogenerators for FŽ.q wxK:I, Sect. 3 . Inwx K:I , it is noted that there is a decomposition into indecomposable summands I I Ž.0 I Ž.1 иии Iq Ž1. . FFqq, [ F q[ [ Fqy Furthermore, there is a revealing alternate description of I : Fq IVŽ.SU Ž.ŽV xq x .. Fq , r y Here SVUŽ.is the polynomial algebra on V, with dth homogeneous component SVdŽ. ŽVmd.Ž, and x qx.Ždenotes the nonhomoge- s ⌺ d y q neous. ideal generated by elements of the form x y x, x g V. Then IŽ.0 S0 is the constant functor, and, for 1 d q 1, IdŽ.is the Fq s F F y Fq image in I of ϱ S dqrŽqy1.. Since the functors S d are known to have Fq [rs0 only a finite number of simple composition factorsŽ seewx K:I or Section 5. , one concludes that each IdŽ.with 1 d q 1 is a locally finite Fq F F y object1 with an infinite number of composition factors. The main result of this paper is a complete determination of the lattice LŽ.I of subobjects of I and, individually, the lattices L ŽŽ..Id of FFqq Fq subobjects of each of the IdŽ.. Fq Before defining our lattice of subobjects, we comment briefly on other interpretations of our work, and its relation to previous results. Let P FŽ.q be defined by PV Ž . FV. Thus PVŽ.is the FFqqqg sqwx F F-vector space with the set V as basis. Then P is the projective in FŽ.q q Fq dual to I under the duality D: FŽ.q op FŽ.q defined by ŽDF .Ž. V Fq ª s FVŽ U .U. Determining L Ž.I is equivalent to determining the lattice of Fq quotient objects of P . Fq With V Ž.F n, IVŽ.is a module for the multiplicative semigroup nqs Fqn MŽ.F. As will be explained in Section 5, determining L Ž.I allows us to nq Fq immediately determine the lattice of M Ž.F -submodules of IVŽ.for all nq Fq n n.ŽŽ We confess to not completely determining the GLnqF .-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 by Fq polynomial degree and taking the associated graded object, we recover the d main results ofwx Ko : the determination of the submodules of SVŽ.n, Ž. Ž. Ž. UŽ. viewed as either an MnqF , GL nqF ,orSL nqF -module, where SVs U q SVŽ.Žrx.. Letting q ‘‘go to infinity,’’ we recover the main results ofwx D dŽ. andwx Kr1 : the determination of the submodules of SVnn, where V s n Ž.Fpn, viewed as either an M Ž.Fp, GLn Ž.Fp,orSLn Ž.Fp-module. As in all these previous papersŽ and alsowx K1. , 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,’’ i j i j we mean the maps S m S ª S q , and by ‘‘pth powers,’’ we mean the i pi inclusions S␰ ¨ S , where G␰ denotes the functor G twisted by the FrobeniusŽ as inwx K:II. . N s Ž. Let be the additive monoid of s-tuples I s i0 ,...,isy1 of nonnega- Ž.NsŽ. Ýsy1 r tive integers. Given I s i0 ,...,isy1 g , let dI s rs0 ipr , and de- IIId fine S , F , S˜g FŽ.q as follows. First defining F to be the degree d U U p component of FVŽ.sSVŽ.Žrx., we let

i I i0 i1 sy 1 S s S m S␰ m иии m S␰ sy 1 and

i I i0 i1 sy 1 F s F m F␰ m иии m F␰ sy 1 .

I I dŽ I . Now let ⌽ : S ª S be the composite

pth powers 6 sy1 multiply 6 ii01 is101ipi pis1dŽI. SmS␰mиии m SS␰syy1 mSmиии m SSy .

⌽I␲ Then S˜II is defined to be the image of the composite S IdS ŽI.I , ; Fq ª ª Fq where ␲ is the inclusion S dŽ I . SU followed by the projection SU I . ¨ ª Fq Zs Let R0 ,...,Rsy10g be the following vectors: if s s 1, R s p y 1, Ž.Ž.Ž and, if s ) 1, R01sy1, 0, . . . , 0, p , R s p, y1, 0, . . . , 0 , R 2s 0, p, .Ž.Ž. y1, 0, . . . , 0 , . . . , and R sy1 s 0,...,0, p,y1 .Let I p, s be the poset s ŽN , F., where ‘‘F ’’ is the partial ordering generated by the inequalities J - I,if IsJqRr , for some r. I PROPOSITION 1.1.Ž. 1 I Ý S˜ . Fq s I I II Ž.2 S˜ rRad ŽS˜ ., F , and is simple. Ž. ŽI . J 3 Rad S˜ s ÝJ - I S˜ . IJ Ž.4F,F if and only if I s J. INVARIANT SUBSPACES 215

To state our main theorem, we need a standard construction from lattice theoryŽ as in, e.g.,wxwx G, p. 72 or S, p. 100. . 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 ÄK : I ¬ K is an order ideal4 . This becomes a distributive lattice using union and intersection for the join and meet lattice operations.

I THEOREM 1.2. The assignment I ¬ S˜ induces an isomorphism of lattices

L Ž.I Ž.p, s L I . , Ž.Fq

Thus the finite join-irreducible subobjects2 of I are precisely the S˜I, Fq IIŽ.p,s, and every subobject G I has a representation g : Fq

I G s Ý S˜ IgK for a unique order ideal K : IŽ.p, s . Remark 1.3. Our lattice isomorphism carries two little bits of extra structure. Ž.1 I Žp,s . has a monoid structure compatible with the partial ordering, and this induces a product on L ŽŽJp,s ..:if K12and K are two order ideals of IŽ.p, s , then K12ؒ K is defined to be the smallest order Ä4 ideal containing the set I q J ¬ I g K12, J g K . Meanwhile, the natural product on I defines a subobject G ؒ G I , given G , G I , thus Fqq12: F 12: Fq defining a product on L Ž.I . The isomorphism of Theorem 1.2 preserves Fq these products on the lattices. Ž.2 Since ŽI . I , twisting by the Frobenius induces an order s Fqq␰ , F automorphism of the lattice L Ž.I . Under the isomorphism of Theorem Fq 1.2, this corresponds to the evident automorphism on L ŽŽI p, s ..induces s by cyclically permuting the factors of IŽ.Žp, s s N , F.. The indecomposable summand version of Theorem 1.2 is easily stated. There is a decomposition

qy1 I Ž.p, s I Ž.p, s I s @I d ds0

Ž.Ä4 into indecomposable posets, where I p, s 0 s 0 , and, for 1 F d F q y 1, Ž.Ä Ž.Ä4 Ž. 4 Ip,sdsIgIp,sy0¬dI 'dmod q y 1.

2 Fis join-irreducible means that F s G q H, with G, H : F, only if F s G or F s H. 216 NICHOLAS J. KUHN

THEOREM 1.4. For 0 F d F q y 1, there is an isomorphism of lattices

L I Ž.p, s L Id Ž.. Ž.d, Ž.Fq

Figure 1 shows the lower portion of the infinite poset IŽ.2, 21 , i.e., the poset that describes the subobject structure of I Ž.1 , the injective envelope F4 1 of S in FŽ.4 . In general, IŽ.p, s d would have a diagram that would look roughly like a s-dimensional cone. Ž. Ž. An analysis of the posets I p, s ddreveals that for a fixed I g I p, s , Ž. all but a finite number of J g I p, s d satisfy J ) I. We thus conclude COROLLARY 1.5. E ery proper subobject of IŽ. d is finite. Thus I is an ¨ FFqq Artinian object in FŽ.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.,w K:II, Theorem 5.23 and Example 7.6x. . However, we give new and very noncomputational proofs. Secondly, if q p Ž.i.e., s 1 , we learn that Id Ž.is an infinite s s Fq uniserial object for all d ) 0. This was already proved by us inwx K:II, Sect. 7 . 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 IŽ.2, 21 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 mk is Artinian.3 Ž See K:II, Sect. 3 for a Fq wx 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 IŽ.p, s . Both the proposition and two theorems are proved in Section 4. In Section 5, we recover the theorems of Dotywx D , Kovacs´ w Ko x , and Kropw Kr1 x , among other related results Žthough we note that when we can conclude something about SLnnor GL lattices, we are generally depending on an elegant theorem of Krop in wxKr2. .

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 FŽ.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 wxPo and locally finite objects. Thus, in this section, we let A denote such a category. We begin with some lattice theoretic definitionswx G .

DEFINITIONS 2.1. Let L be a lattice.

Ž.1 Lis modular if A k ŽB n C .s ŽA k B .n C whenever C F A, for all A, B, C g L. Ž.2 Lis distributi¨e if A k ŽB n C .s ŽA k B .n ŽA k C ., for all A, B, C g L. Ž.3 Lis complete if one can form joins in L indexed by arbitrary sets. Ž.4 In a complete lattice L , A g L is compact if whenever A F Eig IiB , there exists a finite subset J : I such that A F EigJiB . Ž.5 A complete lattice L is compactly generated if each A g L is the join of the set Ä4B ¬ B F A and is compact . 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.

3 If k ) 1, it is unreasonable to calculate the complete lattice of subobjects of Imk, and it is Fq no longer true that every proper subobject of an indecomposable summand of Imk is finite. Fq However Powell has a preprintwx P verifying the conjecture when k s q s 2. 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 IŽ.L s ÄA 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, Ž. then the assignment that sends an order ideal K : I L to EAg K A defines an isomorphism of lattices L ŽŽI L .., L. If F is an object in A, let IŽ.F s ÄG : 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 GrRad Ž.G being simple in A corresponds to Gbeing 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 ŽŽ..I F , L Ž.F . To easily identify IŽ.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 Ž. Ý 1 GgIGsF, Ž.2 GrRad ŽG . is simple, for all G g I, and Ž.3 for all G g I, Rad ŽG .s ÝH, 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 inwx S, p. 106 . 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

GG13G2 Ž.2.1

H Žseewx G, p.70. . ␲ If there exists F ª Q = S [ S, with ␲ epic and S simple, then L Ž.F y1 Ž. y1Ž. contains a subdiagram as above, with G s ␲ S [ S , G1 s ␲ S [ 0 , y1 ŽŽ.. y1Ž. Ž. Ž. G23s␲diag S , G s ␲ 0 [ S , and H s Ker ␲ . Thus L F is not distributive. INVARIANT SUBSPACES 219

The converse is clear if F is semisimple. But we can reduce to that case: if L Ž.F contains sublattice Ž 2.1 . , then L ŽFrH .contains

SocŽ.GrH

Ž.Ž.Ž. Soc G13rH Soc G2rH Soc G rH 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

⌰: L ª L Ž.I Ž.L and ⌿: L Ž.I Ž.L ª L

⌰Ž. Ä4⌿Ž. by A s c.j.i. C ¬ C F A , and K s EB g K B. The theorem will follow once we check that ⌰ is a map of lattices, ⌿⌰ŽŽ..A sAfor all AgL, and ⌰⌿ŽŽ..K sKfor all order ideals K : I ŽL .. It is obvious that ⌰Ž.Ž.Ž.A n B s ⌰ A l ⌰ B . Since L is distributive, we have

⌰Ž.AkBsÄ4c.j.i C ¬ C F A k B sÄ4c.j.i C ¬ C F A or C F B s ⌰Ž.A j ⌰ Ž.B .

Thus ⌰ is a lattice map. To show that ⌿⌰ŽŽ..A sA, we observe that, since L is compactly generated,

A s EEEEB s ž/C s C s ⌿⌰Ž.Ž.A . c. BFA c. BFA j.i. CFB c.j.i. CFA 220 NICHOLAS J. KUHN

Finally, since L is distributive, we have

⌰⌿Ž.Ž.K s½5c.j.i. C ¬ C F EB BgK sÄ4c.j.i. C ¬ C F B for some B g K s K, and the theorem is proved. Proof of Proposition 2.5. It is clear that IŽF .satisfies Ž.Ž. 1 , 2 , and Ž. 3 . Conversely, suppose that a poset I satisfies these three properties. Clearly I:IŽ.F; we need to check the reverse inclusion. Given any G g IŽ.F , we need to show G g I. Property Ž. 1 implies that Ž. there exist G1,...,Gr gI such that G : G1 q иии qGr . Since GrRad G Ž. is simple and L F is distributive, we conclude that G : Gi for some i. Ž. Ž. If G s Gii, we are done. Otherwise, G : Rad G , by property 2 and Ž. XX general properties of the radical. By property 3 , there exist G1,...,Gt g X X X I, with each Gjia proper subobject of G , and G : G1q иии qGt.As X before, G : Gj 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 IŽ.p, s

In this section we study some purely combinatorial aspects of the partially ordered set IŽ.p, s . It is convenient to add to the definitions and notation of Sect. 1. For Ž.<< иии Isi0 ,...,isy10, let I s i q qi sy1. A given nonnegative integer d Ýsy1Ž.r Ž. can be written uniquely in the form d s rs0idprr, with 0 F idFp y1 for 0 F r F s y 2. Given d G 0, let IdŽ.,Jd Ž.gI Žp,s .and Ž . Ž . Ž. ŽŽ. Ž.. Ž. Ip,s,d;Ip,s be defined by Ids id0 ,...,idsy1 ,Jds Ž.Ž.d,0,...,0, and I p,s,d sÄI¬dIŽ.sd4. Immediately from the definitions, we have the next lemma.

LEMMA 3.1. If I ) J, then<< I ) <

By induction on <

COROLLARY 3.3. The decomposition s p y1 IŽ.p, s I Ž.p, s I s @I d ds0 is a decomposition of IŽ.p, s into indecomposableŽ. i.e., connected posets. COROLLARY 3.4.Ž. 1 dI Ž .sd Ž J . if and only if I y JisaZ-linear combination of R1,...,Rsy1. s Ž.2 dI Ž .'dJ Ž .mod Žp y 1. if and only if I y JisaZ-linear combination of R0,...,Rsy1. PROPOSITION 3.5. Let I, J g IŽ.p, s y Ä40 . Then I G J if and only if N I y Jisan -linear combination of R0,...,Rsy1. Proof. The ‘‘only if’’ implication is clear. We prove the converse by иии induction on a0 q qasy1, where we suppose that иии I y J s aR00q qaRsy1sy1,3.1Ž. Ž.Ž. Ž. with I s i0 ,...,isy10and J s j ,..., jsy1both elements of I p, s y Ä4 0,arrG0 for all r, and at least one a is positive. We need to show that then J - I. XXŽ. Choose r so that arrG 1, and let I s I y R .If IgI p,s, we are X X done: J - I because J F I by induction, and I - I by definition. So we X Ž. Ž can assume that I f I p, s , i.e., that iry1 F p y 1 where we write subscripts modulo s.Ž.. Then 3.1 implies that

Ž.p y 1 y jry1G iry1y jry1s parry a y1,3.2Ž. and we conclude that

ary1G Ž.arry 1 p q j y1q 1 G 1.Ž. 3.3 Continuing in this way, under our inductive hypothesis, either J - I or иии Ž. ar G1 for all r. Noting that R0 q qR sy1 s p y 1,..., py1 , and recalling that J / 0, this latter case implies that it G p for some t. But X X X then, letting I s I y Rtq1, we will have J F I by induction, and I - I by definition, so that J - I. 222 NICHOLAS J. KUHN

Ž. PROPOSITION 3.6. Gi¨en I g I p, s d, all but a finite number of J g Ž. Ip,sd satisfy J G I. To prove this proposition, it is convenient to define some notation. Let R s Ž. E0 ,...,Esy10be the standard basis of . Given I s i ,...,isy1, let Ž. Ž. dI0 ,...,dIsy1 be defined by

2 иии sy1 dIrrrŽ.siqpi q1q p irq2q qp iry1 Ž.with indices taken modulo s ,

Ž. Ž. R and let DIrrrsdIE. We say that a vector B is an q-linear combina- иии tion of A0 ,..., Asy100if B s aAq qaAsy1sy1, with ara nonnegative R s R real number, for all r. Let qqdenote the -linear combinations of E0,...,Esy1. EMMA Ž. R L 3.7. 1 Eisanr q-linear combination of R0 ,...,Rsy1 for all r. Ž. R s Ž. R 2 If I g q, DIr yIisan q-linear combination of R0,...,Rsy1 for all r. Proof. By symmetry, it suffices to prove each statement for a single value of r. Then we have

sy1 s sy2 2 иии s Esy101s p rŽ.p y 1 R q p rŽ.p y 1 R q q1rŽ.p y 1 R sy1, which provesŽ. 1 , and

иии DI0Ž.yIsiRsy1 sy1qŽ.pisy1q iRsy2 sy2q sy2 иии qŽ.p isy1211q qpi q iR, which provesŽ. 2 .

Proof of Proposition 3.6. The proposition is obvious when d s 0, so assume d / 0. Ž. By Proposition 3.5, we need to show that, given I g I p, s d, for all but Ž. finitely many J g I p, s d, J y I is an N-linear combination of Ž. R0 ,...,Rsy1. By statement 1 of the last lemma, the Rr are linearly N independent. Thus J y I is an -linear combination of R0 ,...,Rsy1 if Z R and only if J y I is both a -linear combination and an q-linear combination of R0 ,...,Rsy1. By Corollary 3.4, the former is true for all Ž. JgIp,sd. Thus it suffices to show that for all but a finite number of Ž. R JgIp,s,JyIis an q-linear combination of R0,...,Rsy1. This follows from the two parts of the lemma, which combine to show Ž.Rs R that if J s j0 ,..., jsy1 g qq, and J y I is not an -linear combination Ž. of R0 ,...,Rsy1, then 0 F jrr- dI, for all r. INVARIANT SUBSPACES 223

Ž. COROLLARY 3.8. E¨ery proper order ideal K ; I p, sd is finite.

4. PROOF OF THE MAIN RESULTS

UU Recall that, for F g FŽ.q , DF is defined by ŽDF .Ž. V s FV Ž ..In wxK:II we proved that simple functors are self-dual. This has the following corollary.

LEMMA 4.1wx K:II, Corollary 7.5 . Let F be a finite functor. If Ž. Ž.Ž. Hom FŽq. F, DF , Fq , generated by ␣: F ª DF, then Ker ␣ s Rad F , and FrRadŽ.F , Im Ž.␣ is simple. Proposition 1.1 will be proved by applying this lemma to the case when I Iii FsSor S˜ . In preparation for this, let T g FŽ.q be defined by TVŽ.s miIŽ. ˜ V, and then, if I s i0 ,...,isy1 , define T , SII, and S by i Ii0i1 sy 1 TsTmT␰ mиии m T␰ sy 1 , SDS I, and S˜˜DS I. Furthermore, let ⌺ be the group ⌺ = иии = IIs s I i0 III⌺I ⌺, so that ŽT .⌺ S and ŽT .S . Finally, note that the norm map, i sy 1 I s s I the sum of the permutations I I Ý ␴ : T ª T ␴g⌺I N III I factors as T ª S ª SI ª T . Starting from the observation that

Fq if r s 0, 11r Hom FŽq.Ž.T , Ž.T ␰, ½0otherwise, the methods ofwx K:III, Sect. 4.4 formally imply Ž. Ž II. LEMMA 4.2. 1 Hom FŽq. T , T , FqIwx⌺ as F q-algebras. Ž. Ž IJ. 2 Hom FŽq. T , T , 0 if I / J. Ž. Ž I . COROLLARY 4.3. 1 Hom FŽq. S , SIq, F generated by N I. Ž. Ž I . 2 Hom FŽq. S , SJ , 0 if I / J. I I I Let K denote the kernel of the projection S ª F . Elementary inspection of our definitionsŽ compare withwx K:II, Example 7.6. reveals Ž. I LEMMA 4.4. Ker NI s K . We need one more observation before proving Proposition 1.1. Given ⌽ I IŽ.i,...,i , let ⌽˜II: S I denote the composite S IS dŽI. I , s0 sy1 ª FFqª ª q where ⌽ I is as in the introduction. 224 NICHOLAS J. KUHN

IIŽ. J LEMMA 4.5. ⌽˜ K s ÝJ - I S˜. Proof. By definition, S˜I is the image of ⌽˜ I. Thus SV˜IŽ.is the span of IŽŽ. Ž. Ž . . Ž.i r Ž . elements ⌽˜ x 0 m x 1 ␰ m иии m xsy1␰ sy1 , with xr gSV, and II I ⌽˜ ŽŽ..Ž.KV ;SV˜is the span of such elements for which at least one of the xrŽ.is in the ideal of pth powers. U p Now observe that if xrŽ.gSVŽ.can be written in the form yz , then IŽŽ. Ž. Ž . .IqR r ŽŽ. Ž. Ž ⌽˜ x0mx1␰mиии m xsy1␰␰sy1 s⌽˜ y0my1 mиии m ysy .. ŽX.ŽX. X 1␰ sy1 , where, with indices written mod s, yr sxr if r / r, r q 1, Ž. Ž.Ž. ⌽˜ IIŽ.Ý sy1˜IqRr yr sy, and yrq1szx r q 1 . It follows that K s rs0 S s J ÝJ - I S˜ . Proof of Proposition 1.1. StatementŽ. 1 is clear by inspection. Using I I I Lemma 4.1, Corollary 4.3 and Lemma 4.4 imply that S rRadŽS . s F II IJ I and is simple, RadŽ.S s K , and F , F if and only if I s J. Since ⌽˜: I I I II I SªS˜ is onto, we conclude that S˜ rRadŽ.S˜ s F , and RadŽ.S˜s IIŽ. IIŽ. J ⌽˜˜K. The last lemma identifies ⌽ K with ÝJ - I S˜. Proof of Theorem 1.2. By Proposition 1.1, all the composition factors of Iare distinct. Thus, by Corollary 2.4, L Ž.I is distributive, and there is a FFq q lattice isomorphism L ŽŽI I ..L ŽI .. Using Proposition 2.5, Proposi- FFqq, I tion 1.1 precisely shows that the assignment I ¬ S˜defines an isomorphism of posets IŽ.p, s I Ž.I . ª 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. Inwx K:I we noted that finite functors are precisely the F g FŽ.q such that the growth function

n dim FVŽ. ¬ Fq n is a polynomial function of n, and that these functors are polynomial in Ž. Ž. the sense ofwx EM . We let Fd q ; F q be the full subcategory generated by finite functors whose growth functions are polynomial of degree no more than d. INVARIANT SUBSPACES 225

Let pI be the largest subobject of I in F Ž.q . This is an injective in d FFqqd FŽ.q. Let I Žp, s . ÄI IŽ.p, s ¬ <

COROLLARY 5.1. L Ž.ŽŽ..pI L I p,s . d Fqd, F Remark 5.2. Note that pI pIis semisimple. This is not a d Fqqrdy1 F 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 FŽ.2. dqddUU Recall that S is defined by SVŽ.sSV Ž.Ž.rx. Note that S s S˜if dddŽq1. dFqy1, and S s S˜˜rS yyif q F d. Recalling that IŽ.p, s, d s ÄI ¬dIŽ.sd4, Theorem 1.2 and Proposition 3.2 imply ddIŽd. COROLLARY 5.3. L Ž.S , L ŽŽI p, s, d ... S has simple socle F and simple head F JŽd.. This is equivalent to the main result ofwx Ko . To make the translation, we need some notation. Given F FŽ.q , let L Ž.F , L Ž.F , and L Ž.F g MGLSLnn n denote the lattices of subobjects of FVŽ.n , regarded respectively as an MnqŽ.F-module, GLnqŽ.F -module, and SLnqŽ.F -module. Now we make two observations. The first is that, by general principles Žas discussed in K:II.Ž.Ž. , the map L F L F that sends G F to wx ª M n ; GVŽ.FV Ž.is onto, for any F FŽ.q . Thus L Ž.F will be the quotient nn; g Mn lattice of L Ž.F under the equivalence relation generated by saying that Ž.Ž. H;Gif H ; G ; F and GrHVn s0. Ž.Ž. In our case this goes as follows. Given I s i0,...,isy1 , let nI s Ä Ž. 4 Ž. IŽ. min n ¬ n G irr p y 1 for all r . Then nI Fnif and only if FVn/0. Let I Ž.p, s ÄI ¬ nIŽ. n4, and let I Ž.Ž.Žp, s, d I p, s I p, s, MMns F nns Ml d.. Theorem 1.2 implies

COROLLARY 5.4.Ž. 1 L ŽI .L ŽI Žp, s ... M nqF , M n Ž.2 L ŽSd .L ŽI Žp,s,d ... MMnn, The second observation is that inwx Kr2, Theorem 1 Krop gave an easy-to-check criterion on a functor F ensuring that L Ž.F L Ž.F L Ž.F . MGnnns LSs L 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 , but does for S d Žas Krop Fq points out. . We conclude that LIŽp,s,d .L Ž.SdddL Ž.SL Ž.S, Ž.MMGnnnn,sLSsL which is the main result ofwx Ko . 226 NICHOLAS J. KUHN

Remark 5.5. A simple example showing that L Ž.I / L Ž.I oc- M nqF GL nqF UŽ.Ž2 . curs when n s q s 2. In positive degrees, SV2 rxyxhas two com- 1Ž. 2Ž. 2Ž. Ž. position factors: FV22sVand FV 2s⌳V 2.As M 22F -modules, there is a nontrivial extension between these. This extension splits when viewed as an extension of GL22Ž.F -modules. In spite of example like this, we can still conclude that L ŽŽId 0 .. and L ŽŽId)0 .. are both GLnqF G SLnqF distributive, and thus satisfy the structure theorems of Section 2.

d dd Finally, we note that, for d F q y 1, S s S˜ , and that S is defined on F-vector spaces. These observations allow us to determine L Ž.S d , the p Fp d lattice of subobjects of S , viewed as an object in the category FŽ.Fp of functors

F : finite dimensional Fpp-vector spaces ª F -vector spaces. To explain this, we need to introduce yet another category of functors:

FŽ.q,Fp will denote the category whose objects are functors

F : finite dimensional Fqp-vector spaces ª F -vector spaces. Given F FŽ.q , let F F Žq, F .be defined by FVŽ.FV Ž. F. g Fppg p F s mFqp Ž. Ž. Ž . Ž.Ž. Ž Given G g F Fpp, let Res G g F q, F be defined by Res GVsFV F.. mFqp Ž. Ž . LEMMA 5.6. Suppose that F g F q and G g F Fp satisfy Ž.1 F is locally finite, Ž.2 L ŽF . is distributi¨e, Ž.3 F Res ŽG ., and Fp , Ž.4 e¨ery H g IŽRes ŽG ..Ž notation as in Section 2.is of the form HsResŽ.K for some K ; G. Then L Ž.G L Ž.F . Fp ,

Proof. The first point is that Fq is a splitting field for FŽ.q wK:II, Sect. 5x , i.e., all simple functors in FŽ.q are absolutely simple. Assuming Ž. 1 andŽ. 2 , this implies that IŽ.F I ŽF ., and so L Ž.F L ŽŽ..I F , Fp , , LŽŽIF ..L ŽF .. Fpp, F Now notice that Res induces a monic map L Ž.G ª L ŽRes Ž..G , LŽ.F. Under assumptionŽ. 4 , this monic map will also be epic. Fp d d This lemma applies in the case G s S , F s S , and d - q. Then Theorem 1.2 determines L Ž.S d , and thus L Ž.S d . The details go as Fp follows. ϱ ϱ Let IŽ.p, ϱ be the poset ŽN , F., where N is the set of sequences Ž. i01,i, . . . of nonnegative integers that are eventually 0, and F is INVARIANT SUBSPACES 227

Ž generated by the inequalities I q Rr ) I for r G 1. Here, as in Section 1, Ž.. Rr is the vector with p in the rth place, and y1 in the r y 1 st. Let IŽ.p,ϱ,dsÄIgIŽ.Ž.p,ϱ¬dI sd4. Then I Ž.p, ϱ s @I Žp, ϱ, d . dG0 is a decomposition of the poset IŽ.p, ϱ into indecomposable posets. d COROLLARY 5.7. L Ž.S L ŽŽI p, ϱ, d ... Fp , As before, one can immediately read off the lattice of subobjects of d SVŽ.nn, viewed as an M Ž.Fp-module, thus recovering the results ofw Kr1,

Sect. 2xw . Again, using Kr2x , this agrees with the GLnpŽ.F and SLnp Ž.F lattices. Thus we recover the main result ofwx D . Remark 5.8. Note that IŽ.p, s s I Žp, ϱ .Ž.r ; , where ; is the equiv- s !#" alence relation generated by Ž.Ži01, i ,... ; 0,...,0,i01,i ,... . .

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