The Structure of Some Permutation Modules for the Symmetric Group of Infinite Degree

193, 122]143 Ž1997. JA966963

JOURNAL OF ALGEBRA ARTICLE NO.

The Structure of Some Permutation Modules for the Symmetric Group of Infinite Degree Darren G. D. Gray School of Mathematics, Uni¨ ersity of East Anglia, Norwich NR4 7TJ, England Communicated by G. D. James Received June 20, 1996

Suppose that V is an infinite set and k is a natural number. Let w V x k denote the set of all k-subsets of V and let F be a field. In this paper we study the FSymŽ V .-submodule structure of the permutation module F w V x k . Using the representation theory of finite symmetric groups, we show that every submodule of F w V x k can be written as an intersection of kernels of certain FSymŽ V .-homomorphisms F w V x k ª F w V x l for 0 F l - k, and give a simple algorithm to determine the complete submodule structure of F w V x k . Q 1997 Academic Press

1. INTRODUCTION AND NOTATION 1.1. Introduction In this paper we shall investigate the submodule structure of certain permutation modules for the symmetric group SymŽ V ., where V is an infinite set. If k is a natural number, then we can form F w V x k , the vector space over a field F with basis elements the subsets of size k from V. This vector space has a natural SymŽ V .-action, giving F w V x k the structure of an FSymŽ V .-module. When the field F has characteristic zero, the submodule structure of F w V x k is explicitly known Žsee w1x.. The main result of this paper is an algorithm which enables us to effectively compute the submodule structure of F w V x k when F is of prime characteristic p. The algorithm presented here basically consists of checking whether or not p divides certain binomial coefficients. However, many of the results presented in this paper are independent of the characteristic of the field, so we can compute the submodule structure of F w V x k for any field F. As is well known, F w V x k is FSymŽ V .-isomorphic to M l, a module defined using a particular partition l of V. For the case when V is a finite 122 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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set, a great deal is known about M l and its submodules for arbitrary partitions l, and we refer the reader to the works of James Žsee, for example, w4]6x.. We will make use of these finite case results to prove analogous results when V is infinite. In Section 1.2 we introduce our notation and definitions, in particular, we introduce our concept of an infinite partition. Most of the definitions here have been adapted from their finite counterparts, and the formal definitions of these can be found in w6x. The most important definition is that of the Specht module S l, a certain submodule of M l. Section 2 contains some general results about M l and S l when l is an infinite partition. These are analogues or consequences of results for finite V. Section 3 introduces the module F w V x k and the connection with partitions. Most of the results needed to compute the submodule structure of F w V x k are found in this section. We show here that the Specht module for F w V x k is irreducible whatever the characteristic of the field, and we also show that the composition factors of F w V x k are precisely the Specht modules of F w V x l for l s 0, 1, . . . , k, each appearing with multiplicity one. We describe each submodule of F w V x k as an intersection of kernels of certain FSymŽ V .-homomorphisms. We look at some special cases in Section 4, and then proceed in Section 5 to give a description of an algorithm which computes the submodule structure of F w V x k when the characteristic of the field is a prime p. This paper does not seem to overlap with the existing body of work studying representations of the finitary infinite symmetric group SŽ`., that is, the group of all finite permutations of a countably infinite set. Extensive work has been undertaken to classify irreducible representations of SŽ`. over the complex numbers. This work was initiated by Thoma w9x who described the finite characters of SŽ`.. Since then, other classes of representations have been discovered; see, for example, the works of Lieberman w7x, Ol’shanskii w8x, Vershik and Kerov w10]12x, and Hirai w3x. This paper takes a different approach, starting with a particular permutation representation Ždefined over an arbitrary field. and finding all the irreducible representations associated with this permutation representation. 1.2. Definitions and Notation We begin by introducing our notion of a partition. DEFINITION 1.1. Let L be any set and r a natural number. Then we say that l s Ž l1 , l2 , . . . , l r . is a partition of L, written l & L, if there exist

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DARREN G. D. GRAY

pairwise disjoint subsets A1 , A 2 , . . . , A r of L such that: 1. l i s < A i < Ž i s 1, 2, . . . , r .; 2. l i is finite for i s 2, . . . , r; and r 3. D is1 A i s L. If further, l1 G l 2 G ??? G l r ) 0, then we say that the partition is proper, otherwise it is improper. It will be assumed throughout that all partitions are proper unless stated otherwise. We say that the partition l s Ž l1 , l2 , . . . , l r . has r parts. If < L < s n - ` then we say that l is a partition of n, written l & n, and we have that Ý ris1 l i s n. Without loss of generality, if L is infinite, we will take L to be N, the natural numbers, and if L is finite of size n, we will take L to be the set  1, 2, . . . , n4 . We shall occasionally refer to a partition of n as a finite partition and a partition of N as an infinite partition. We can represent partitions diagrammatically. For example, if we consider the partition l s Ž4, 2, 2, 1. of 9, then the diagram of l is ) w lx s ) ) )

) ) )

)

)

We can also represent infinite partitions in this way. For example, the diagram of the partition l s Ž`y8, 5, 2, 1. is ) w lx s ) ) )

) ) )

) )

) )

) )

)

)

???

Let l be any partition. Then by replacing each entry of the diagram of l by an element of L, allowing no repeats and using all the elements of L, we obtain a l-tableau. EXAMPLE 1.2. The following is a Ž` y 4, 3, 1.-tableau: 4 7 10

8 1

5 2

3

6

9

???

We have that SymŽ L . acts transitively on the set of l-tableaux in the natural way; if t is a l-tableau, p g SymŽ L ., and a the ijth node of t, then ap is the ijth node of tp .

THE SYMMETRIC GROUP OF INFINITE DEGREE

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For a l-tableau t, we define its row stabilizer, R t , to be that subgroup of SymŽ L . which fixes the rows of t setwise, and similarly its column stabilizer, Ct , to be that subgroup of SymŽ L . which fixes the columns of t setwise. Note that, since l2 is finite, Ct is a finite subgroup of SymŽ L .. We can define an equivalence relation on the set of l-tableaux by t 1 ; t 2 if and only if t 1p s t 2 for some p g R t 1, and we define a tabloid  t 4 to be the equivalence class of t with respect to this equivalence relation. So a tabloid can be considered as a tableau with unordered row entries. SymŽ L . acts transitively on the l-tabloids by  t 4p s  tp 4 . If we now let F be an arbitrary field, and let M l be the vector space over F whose basis elements are the various l-tabloids, then the action of SymŽ L . on the l-tabloids turns M l into an FSymŽ L .-module. Associated with each l-tableau t we have a polytabloid, e t , defined by et s  t 4

Ý Ž sgn p . p ,

pgC t

where sgn p s

½

q1 y1

if p is an even permutation, if p is an odd permutation.

The FSymŽ L .-submodule of M l spanned by the various l-polytabloids is of great importance, and is called the Specht module for the partition l, and is denoted by S l. The following useful result follows easily from the transitivity of SymŽ L . on the tabloids. PROPOSITION 1.3. polytabloid.

S l is a cyclic FSymŽ L .-module, generated by any one

Now there is a natural bilinear form on M l defined by ²  t1 4 ,  t 2 4 : s

½

1

if  t 1 4 s  t 2 4 ,

0

if  t 1 4 /  t 2 4

for all l-tabloids  t 1 4 ,  t 2 4 , and, of course, extended linearly to the whole of M l. This bilinear form is clearly symmetric, SymŽ L .-invariant, and nonsingular. We let H denote orthogonality with respect to this form. If V is an FSymŽ L .-submodule of M l, then we write V F M l, and say that V is a submodule of M l if the context is clear. If V is a proper FSymŽ L .-submodule of M l, i.e., V / M l, then we write V - M l. DEFINITION 1.4. If ¨ g M l, then ¨ is a unique linear combination of tabloids; we say that the tabloid  t 4 is in¨ ol¨ ed in ¨ if the coefficient of  t 4

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DARREN G. D. GRAY

in ¨ is nonzero. We define the support of ¨ , SuppŽ ¨ ., to be the set of tabloids involved in ¨ , and the weight of ¨ , denoted weight Ž ¨ ., to be the cardinality of SuppŽ ¨ .. Note that the weight of any element of M l is finite.

2. GENERAL RESULTS Throughout this section let V be any infinite set Žand again, for simplification purposes, we can, without loss of generality, identify V with the set of natural numbers.. Let l s Ž l1 , l2 , . . . , l r . be a partition of V. We will work over an arbitrary field F. Undoubtedly, the most important result is the following theorem which involves both the bilinear form defined earlier and the Specht module: THEOREM 2.1 ŽThe Submodule Theorem.. then either U G S l or U F Ž S l . H .

If U is a submodule of M l,

This powerful result is due to James, who proved it for finite partitions Žsee 4.8 in w6x.. However, the proof can easily be adapted to deal with infinite partitions. We need the following terminology to allow us to restrict to finite partitions and use the results already available. DEFINITION 2.2. Let  t 4 g M l. Then for i s 1, 2, . . . , r, let Ri Ž t 4. be the ith row of  t 4 . Now let n g N be such that n ) 2 l2 q l 3 q ??? ql r . Then we define M lw n x to be the following subspace of M l:

M lw n x [  t 4 g M l :

¦

r

D Ri Ž  t 4 . :  1, . . . , n4 is2

;

.

F

Let Gw n x be the setwise stabilizer of  1, 2, . . . , n4 in SymŽ V ., so in fact Gw n x s SymŽ 1, 2, . . . , n4. = SymŽ V _  1, 2, . . . , n4.. Let l9 be the partition given by l9 s Ž n y w l2 q ??? ql r x, l2 , . . . , l r .. Then Gw n x acts on M l9: any g g Gw n x can be written as g s hk, where h g SymŽ n. and k g SymŽ V _  1, 2, . . . , n4.. So if x g M l9, then the action of Gw n x on x is given by xg s xh. Thus Gw n x acts on M l9 in the same way that SymŽ n. does, and clearly the FGw n x-submodules of M l9 and the FSymŽ n.-submodules of M l9 coincide. Of course, this generalises to arbitrary submodules of M l: if U F M l, then U w n x is that FGw n x-submodule of U defined by U w n x [ U l M lw n x. When U is the Specht module, S l, of M l, then S lw n x is isomorphic to S l9, where l9 is the finite partition above. It is clear that the FGw n x-submodule

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127

lattice of M lw n x and the FSymŽ n.-submodule lattice of M l9 are the same. Thus we will frequently make no distinction between M lw n x and M l9 Žor between submodules of M lw n x and M l9 .. We will characterise the Specht module S l of M l as the intersection of kernels of certain FSymŽ V .-homomorphisms. The following definition is based on concepts introduced in w4x. DEFINITION 2.3. Let l s Ž l1 , l2 , . . . , l r . be a partition of V. Let i be some nonnegative integer satisfying 0 F i - r, let d g N be such that 0 F d F l iq1 , and let the partition m s Ž m 1 , m 2 , . . . , m r . of V be given by: m j s l j for j / i, i q 1; m iq1 s l iq1 y d ; and m i s l i q d . Note that m may be improper. Let  t 4 be any l-tabloid. Then define cˆi, d : M l ª M m by

cˆi , d

 t94 agrees with  t 4 on all ¡the sum of m-tabloidsŽ t94, where . except the ith and i q 1 th rows, the ith row of  t94 :  t 4 ¬~ of the ith row of  t 4 together with d elements ¢consisting Ž from the i q 1.th row of  t 4 .

It is clear that cˆi, d is an FSymŽ V .-module homomorphism. Thus cˆi, d can be viewed as a map which moves d elements up from the Ž i q 1.th row of a tabloid on which it is acting to the ith row. For n g N, we denote the restriction of cˆi, d to M lw n x by cˆi,nd . So from Theorem 9.3 in w4x we have S lw n x s

ry1 l iq1

F F ker cˆin, d . is1 d s1

l

We generalise this to S : THEOREM 2.4. Let V be any infinite set and let l s Ž l1 , l 2 , . . . , l r . be a partition of V. Then Sl s

ry1 l iq1

F F ker cˆi , d . is1 d s1

Proof. Let x g F i F d ker cˆi, d . Then, for large enough n, we have that x g F i F d ker cˆi,nd . But F i F d ker cˆi,nd s S lw n x, and since S lw n x is naturally contained in S l, we have that x g S l. Therefore F i F d ker cˆi, d F S l. Now let t be any l-tableau, and let e t be the corresponding polytabloid in S l. Then for large enough n, we have that e t g S lw n x. Thus, by the ry1 iq 1 ker c ˆi,nd . This completes finite version of this theorem, e t g F is1 F dls1 the proof.

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We have already noted that the map cˆi, d : M l ª M m can be viewed as a map which moves d elements up from the Ž i q 1.th row of a tabloid on which it is acting to the ith row. We now define a map from M m to M l which moves elements down a row. DEFINITION 2.5. Let l s Ž l1 , l2 , . . . , l r . be a partition of V. Let i be such that 1 - i - r and let d g N satisfy 0 F d F l iq1. Let the partition m s Ž m 1 , m 2 , . . . , m r . of V be given by: m j s l j for j / i, i q 1; m iq1 s l iq1 y d ; and m i s l i q d Žnote that m may be improper.. Then define f i, d : M m ª M l by

fi , d

with  t 4 on all ¡the sum of l-tabloidsŽ  t94., where  t94 agrees Ž . except the ith and i q 1 th rows, the i q 1 th row of  t94 :  t 4 ¬~ consisting of the Ž i q 1.th row of  t 4 together with d ¢elements from the ith row of  t4

for any m-tabloid  t 4 . Yet again it is clear that f i, d is an FSymŽ V .-module homomorphism. Thus f i, d can be viewed as a map which moves d elements down from the ith row of a tabloid on which it is acting to the Ž i q 1.th row. Remark. Note that f i, d is undefined for i s 1, because there are infinitely many ways of choosing d elements from the first row of any m-tabloid. However, if as usual we denote the restriction of f i, d to M mw n x by f i,n d where we insist that im f i,n d F M lw n x, then f i,n d is in fact defined for i s 1 also. The next result is again due to James, and deals with the characterisation of the module orthogonal to the Specht module in the finite case Žsee Corollary 3 in w5x.. THEOREM 2.6. Let l s Ž l1 , l2 , . . . , l r . be a partition of V, let n g N, and consider the f-maps as defined abo¨ e. Then

Ž S lw n x .

Hn

ry1 l iq1

s

Ý Ý

is1 d s1

im f in, d ,

where H n denotes orthogonality in M lw n x. We would like a similar characterisation for Ž S l . H , and indeed we have the following: THEOREM 2.7. Let l s Ž l1 , l2 , . . . , l r . be a partition of V with r ) 2 and consider the f-maps as defined abo¨ e. Then l H

ŽS . s

ry1 l iq1

Ý Ý

is2 d s1

im f i , d .

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THE SYMMETRIC GROUP OF INFINITE DEGREE

Proof. Note firstly that S l s D nG n 0 S lw n x, for some n 0 large enough Žfor example, n 0 G 2 l2 q l2 q ??? ql r .. Then it is clear that Ž S l . H s F nG n 0Ž S lw n x. H . Now Ž S lw n x. H s Ž S lw n x. H n [ M lw V _  1, 2, . . . , n4x and so H

Ž S l. s

F Ž S lw nx .

Hn

[ M l V _  1, 2, . . . , n4

nGn 0

s

F

nGn 0

ry1 l iq1

žÝ Ý

is1 d s1

im f in, d [ M l V _  1, 2, . . . , n4

/

.

Now, to simplify notation, let l iq1

Ýn Ž 1. [

Ý

ds1

im f 1,n d ,

ry1 l iq1

Ýn Ž 2. [

Ý Ý

is2 d s1

im f in, d ,

ry1 l iq1

Ý Ž 2. [

Ý Ý

is2 d s1

im f i , d .

So what we want to prove is Ž S l . H s ÝŽ2.. Let x g ÝŽ2.. Then, for large enough n we have x g Ý n Ž2.. But now nŽ . Ý 2 F Ž S lw n x. H n , and so x g Ž S lw n x. H n for all sufficiently large n. Therefore, since Ž S l . H s F n G n 0 wŽ S lw n x. H n [ M lw V _  1, 2, . . . , n4xx, we have that x g Ž S l . H i.e. Ž S l . H G ÝŽ2.. Now let x g Ž S l . H . Then x g Ý n Ž1. q Ý n Ž2. for some sufficiently large n. We want to show that x g Ý n Ž2., because then x g ÝŽ2. since Ý n Ž2. is contained in ÝŽ2.. So assume, for a contradiction, that x g ŽÝ n Ž1. q Ý n Ž2.. _ Ý n Ž2. and x is of minimal weight. Let xy be the sum of all those tabloids involved in x whose entries in the bottom r y 1 rows are elements of  1, 2, . . . , l2 q ??? ql r 4 . We take the coefficient of such a tabloid in xy to be the same as its coefficient in x. Without loss of generality, we can assume that xy/ 0 Žbecause otherwise we can take a suitable translate of x ., and so weight Ž x y xy. - weight Ž x .. Let AŽ x . [ D u4g S u p pŽ x . j Ž u4., where j maps a tabloid to the subset of V consisting of the entries in the bottom r y 1 rows of the tabloid. Then AŽ x . is a finite subset of V, so we choose l2 elements a1 , . . . , al 2 of V _ AŽ x .. Now let s be any l-tableau whose entries in the bottom r y 1 rows are elements of  1, 2, . . . , l2 q ??? ql r 4 and the first l2 entries in the top row are a1 , . . . , al 2 . Then the polytabloid e s g S l and so ² x, e s : s 0 Žsince x g Ž S l . H. . Now let f s s  s 4 Ý  Ž sgn s . s : s g C s and s fixes a1 , . . . , al 2 4 .

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DARREN G. D. GRAY

Now if s g C s is such that s moves at least one of a1 , . . . , al 2 , then the tabloid  s4s will have at least one element of  a1 , . . . , al 2 4 in its bottom r y 1 rows. Since a1 , . . . , al 2 were chosen to be in V _ AŽ x ., we have that  s4s is not involved in x, and so does not contribute to ² x, e s :. Thus ² x, f s : s 0, and moreover ² xy, f s : s 0 Žbecause ² x y xy, f s : s 0.. $ Now let m s Ž l2 , . . . , l r ., and then for any l-tabloid  t 1 4 g M l, let  t 14 be the m-tabloid formed by deleting the top row of  t 1 4 . Extend this linearly to any element of M l: if ¨ g M l, ¨ s Ý d t t 4 then ¨ˆ s Ý d tˆt 4 . Then fˆs g S Ž l 2 , . . . , l r ., and so, since ˆ s can vary over all m-tableaux with $ entries from  1, 2, . . . , l2 q ??? ql r 4 , we have that xy g Ž S m . H m , where H m denotes orthogonality in M m. So xyg Ý n Ž2. Žby Theorem 2.6.. Thus x y xyg ŽÝ n Ž1. q Ý n Ž2.. _ Ý n Ž2. and weight Ž x y xy. - weight Ž x . which contradicts the minimality of weight Ž x .. Thus x g ÝŽ2., which completes the proof. The case when l has two rows will be dealt with in a more straightforward way in the next section. 3. THE MODULES F w V x k 3.1. k-Sets and b-Maps We shall now turn our attention to a particular class of partitions. Again we will take V to represent any infinite set Žand again, without loss of generality, we will treat V as if it were the set of natural numbers.. We let l be the partition of V with two parts, one of which is finite of size k, that is, l s Ž`yk, k ., and let F be any field. We denote by w V x k the set of all k-sets of V, that is, the set of all subsets of V of cardinality k. Then SymŽ V . acts in the natural way on w V x k :

 a1 , a2 , . . . , ak 4 g s  a1 g , a2 g , . . . , ak g 4

; g g Sym Ž V . .

This action turns F w V x k , the vector space over F with basis consisting of the elements of w V x k , into an FSymŽ V .-module. Now consider M Ž`yk, k . defined over F. It is easy to see that any tabloid in M Ž`yk, k . is uniquely determined by its bottom row, and so there is an obvious FSymŽ V .-isomorphism between M Ž`yk, k . and F w V x k , given by mapping a tabloid  t 4 to a k-set consisting of the k distinct elements of V in the bottom row of  t 4 . Thus, from now on, we shall not distinguish between k-sets from w V x k and tabloids of M Ž`yk, k .. When explicitly writing tabloids, we shall frequently omit the top row. We shall denote the Specht module of F w V x k by S k Žso S k ( S Ž`yk, k . ..

THE SYMMETRIC GROUP OF INFINITE DEGREE

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In the previous section, we gave a description of the module orthogonal to the Specht module for infinite partitions with more than two parts, and we now investigate what happens for infinite partitions with precisely two parts. The following result not only tells us about Ž S k . H , but also gives us information about the reducibility of S k . LEMMA 3.1. Ž S k . H s  04 . Proof. Assume, for a contradiction, that Ž S k . H /  04 . Let x g Ž S k . H , x / 0. So the inner product of x with any element of S k is zero. We will construct an element of S k which has nonzero inner product with x, giving the required contradiction. Since x / 0, there is a tabloid in SuppŽ x ., say i1 ??? i k Žrecall that a tabloid is identified by its bottom row.. We can assume without loss of generality that the coefficient of this tabloid in x is 1. Now define SŽ x . [ D t4g S u p pŽ x . j Ž t 4. where j Ža1 ??? a k. s  a1 , . . . , a k 4 Ži.e. j can be thought of as the isomorphism between M Ž`yk, k . and F w V x k .. Now x is a finite linear combination of tabloids, thus SŽ x . is a finite subset of the infinite set V. So we can choose j1 , . . . , jk g V _ SŽ x .. Now let t be the tableau whose bottom k entries are i1 , . . . , i k , and the first k entries in the top row are j1 , . . . , jk , i.e. ts

j1 i1

??? ???

jk ik

))) ???

Then let y s e t , so y g S k . By construction, the only tabloid the supports of x and y have in common is i1 ??? i k, and this appears in both x and y with coefficient 1, therefore ² x, y : s 1, which is a contradiction. Combining this with the submodule theorem, we have: THEOREM 3.2. module, S k .

Any nonzero submodule of F w V x k contains the Specht

As an obvious corollary we have: COROLLARY 3.3. The Specht module S k of F w V x k is irreducible. Again, there exist the natural FSymŽ V .-homomorphisms between the F w V x k , as described earlier for M l, but we adopt a change of notation: DEFINITION 3.4. If 0 F j - k, there is a natural FSymŽ V .-homomorphism b k, j : F w V x k ª F w V x j given by b k, j Ž w . s Ý w9: w9 g w w x j 4 for w g w V x k and extended linearly. Note that this homomorphism coincides with the homomorphism cˆ1, kyj when considering the partition Ž` y k, k ..

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DARREN G. D. GRAY

COROLLARY 3.5. Let 0 F l - k. Then F w V x krker b k, l has a unique minimal submodule isomorphic to S l. Proof. We have that F w V x krker b k, l ( im b k, l , and im b k, l F F w V x l. Now b k, l is nonzero and so since S l is the unique minimal submodule of F w V x l , then im b k, l has a unique minimal submodule S l. Using this new notation, we have immediately from Theorem 2.4: THEOREM 3.6 ŽThe Intersection Theorem for k-Sets.. For 0 F i - k, let b k, i be the b-maps as defined abo¨ e. Then we can characterize the Specht module of F w V x k as ky1

Sk s

F ker bk , i . is0

Removing kernels one at a time from this intersection produces a chain of submodules of F w V x k . It turns out that this is precisely the correct thing to do to find all submodules of F w V x k . 3.2. Composition Series for F w V x k Firstly a general fact about finite composition series Žsee, for example, 13.7 in w2x.: THEOREM 3.7 ŽJordan]Holder ¨ .. Let R be any ring. If an R-module M possesses a finite composition series, then any two composition series of M are equi¨ alent Ž where equi¨ alent means that any two composition series ha¨ e the same number of factors, and the factors can be paired off in such a way that the corresponding factors are R-isomorphic.. We shall require that for partitions Ž` y k, k . and Ž` y l, l . of V, where k / l, the corresponding Specht modules S k and S l are not FSymŽ V .-isomorphic. From 11.3 in w6x we have: LEMMA 3.8. Let n be any finite natural number. If l s Ž n y k, k ., m s Ž n y l, l . are partitions of n Ž so 2 k - n, 2 l - n. then S l ( S m « l s m. We can use this to deduce the same result for infinite partitions with two parts. Before we state and prove this result, we need some notation. Recall that w V x k denotes the set of all k-sets of V. If n g N, denote the set of all k-sets of  1, 2, . . . , n4 by w n x k . We note that F w n x k ( M Ž nyk, k .. Let S k w n x denote the Specht module of F w n x k , so S k w n x s S k l F w n x k , where S k is of course the Specht module of F w V x k . More generally, if U F F w V x k , let U w n x be that FSymŽ n.-submodule of F w n x k given by U w n x s U l F w n x k .

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Let Gw n x s SymŽ 1, 2, . . . , n4. = SymŽ V _  1, 2, . . . , n4., so Gw n x is the setwise stabilizer of  1, 2, . . . , n4 in SymŽ V .. Note that U w n x, defined above, is also an FGw n x-submodule of F w V x k . PROPOSITION 3.9. Let Mi be an FSymŽ V .-submodule of F w V x k . Then for n g N, Mi has a largest finite-dimensional Gw n x-submodule, namely Mi w n x. Proof. Suppose, for a contradiction, that U is a finite-dimensional Gw n x-submodule of Mi with Mi w n x - U. So there exists a tabloid t 4 involved in an element of U _ Mi w n x with one of the bottom row entries of t 4 greater than n. Without loss of generality, to simplify notation, we shall assume t 4 s 1 2 ??? k y 1 n q 1. Now since U is Gw n x-invariant, t 4a g U for all a g Gw n x. Now any element of Gw n x has the form gh, where g g SymŽ 1, 2, . . . , n4. and h g SymŽ V _  1, 2, . . . , n4.. The permutation g will permute the first k y 1 entries in the bottom row of t 4 , and h takes the final entry, which is n q 1, to another element of V _  1, 2, . . . , n4 Žproviding h is nontrivial.. Thus, if h is nontrivial t 4 gh gives another distinct tabloid of U, and since SymŽ V _  1, 2, . . . , n4. is infinite, the set t 4a : a g Gw n x4 is a set of infinitely many distinct tabloids of U. These tabloids must be involved in the basis elements of U. But now any basis element of U has finite support and so there must be infinitely many basis elements of U. This contradicts U being finite-dimensional. PROPOSITION 3.10. Let l s Ž` y k, k . and m s Ž` y l, l . be two distinct partitions of V, i.e. k / l. Then the Specht modules S k and S l of M Ž`yk, k . and M Ž`yl, l . respecti¨ ely are not FSymŽ V .-isomorphic. Proof. Let n ) max 2 k, 2 l 4 and consider Gw n x acting on S l. Then S l has a maximal finite-dimensional Gw n x-submodule S lw n x Žby Proposition 3.9.. Similarly, S m has a maximal finite-dimensional Gw n x-submodule S mw n x. Now if S lw n x and S mw n x are FGw n x-isomorphic, then S Ž nyk, k . and S Ž nyl, l . are FSymŽ n.-isomorphic Žwhere, of course, SymŽ n. denotes SymŽ 1, 2, . . . , n4... So now if S l ( S m Žas FSymŽ V .-modules., then S lw n x and S mw n x are FGw n x-isomorphic, and so S Ž nyk, k . and S Ž nyl, l . are FSymŽ n.-isomorphic, and therefore, by Lemma 3.8, Ž n y k, k . s Ž n y l, l .. That is, k s l and so l s m. From now on we shall relax the notation and drop ‘‘FSymŽ V .-’’ where the context is clear i.e. FSymŽ V .-isomorphic becomes isomorphic, etc. We now show that F w V x k does indeed have a composition series; moreover it is shown that each composition factor is isomorphic to a Specht module, and so by the above result we can distinguish between nonisomorphic factors.

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PROPOSITION 3.11. F w V x k has a finite composition series in which each factor is isomorphic to a Specht module S l , for some l F k, with each S l appearing at least once, and S k appearing exactly once. Proof. Use induction on k. Firstly note that S 0 s F w V x 0 ( F, the trivial module, which completes the proof of the base step of the induction. Now for each l - k, assume F w V x l has a finite composition series in which each factor is isomorphic to a Specht module S m , for some m F l, with each S m appearing at least once, and S l appearing exactly once Žso by the Jordan]Holder theorem, e¨ ery composition series of F w V x l has this ¨ property.. Then by Theorem 3.6, F w V x krS k is isomorphic to a submodule of [0 F l - k F w V x l Žconsider the map which takes x g F w V x k tol Ž bk, 0 Ž x ., . . . , b k, ky1Ž x .... By the induction hypothesis, [0 F l - k F w V x has a finite composition series with each factor isomorphic to S l for some l - k, so by Theorem 3.7, every composition series of [0 F l - k F w V x l has this property. Therefore any submodule of [0 F l - k F w V x l also has this property. As S k is irreducible, it follows that F w V x k has a finite composition series where S k appears once, and the other composition factors are isomorphic to S l , for l - k. Each S l appears at least once by Corollary 3.5. DEFINITION 3.12. Now let l be a partition of any set L. Then a series of submodules of M l is called a Specht series if each factor is isomorphic to a Specht module. The only result we shall require about Specht series is the following, again due to James Žsee 17.17 in w6x.. It concerns the existence of a particular Specht series: LEMMA 3.13. When m F n y m, M Ž nym, m. has a Specht series with factors S Ž n., S Ž ny1, 1., S Ž ny2, 2., . . . , S Ž nym, m. reading from the top. In the next result, we make use of the notation introduced before Proposition 3.10, which enables us to restrict our infinite-dimensional modules to finite-dimensional ones. LEMMA 3.14. Let 0 s M0 - M1 - ??? - Mry1 - Mr s F w V x k be a series of submodules of F w V x k such that MirMiy1 ( S l i for i s 1, . . . , r. Then if n is sufficiently large, F w n x k has a Specht series 0 s M0 w n x - M1 w n x - ??? - Mry1w n x - Mr w n x s F w n x k with Mi w n xrMiy1 w n x ( S l i w n x for i s 1, . . . , r. Proof. Firstly, for i s 1, 2, . . . , r, let g i : MirMiy1 ª S l i denote the isomorphisms in the hypothesis. Then for i s 1, 2, . . . , r let ¨ i g Mi _ Miy1 be such that g i Ž ¨ i q Miy1 . is a polytabloid, and choose n large enough so

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that for all i s 1, 2, . . . , r we have ¨ i g F w n x k . Then ¨ i g Mi w n x _ Miy1 w n x Žfor i s 1, 2, . . . , r ., so we have the following chain of submodules of F w nx k : 0 s M0 w n x - M1 w n x - ??? - Mry1 w n x - Mr w n x s F w n x . k

Ž 1.

Now for i s 1, 2, . . . , r define g iU : Mi w n xrMiy1 w n x ª S l i by g iU Ž x i q Miy1w n x. s g i Ž x i q Miy1 . for all x i g Mi w n x. Clearly g iU is well defined Žbecause g i is.. CLAIM. Ži. g iU is an FGw n x-homomorphism; Žii. g iU is injecti¨ e; Žiii. im g iU s S l i w n x. Proof of Claim. Ži. For all x i g Mi w n x and for all g g Gw n x we have

g iU Ž x i q Miy1 w n x . g s g i Ž x i q Miy1 . g s g i Ž Ž x i q Miy1 . g . s g i Ž x i g q Miy1 .

Ž since Miy1 F F w V x k .

s g iU Ž x i g q Miy1 w n x .

Ž on noting that

s g iU Ž Ž x i q Miy1 w n x . g .

Ž Miy1

x i g g Mi w n x .

is an FG w n x -module. ,

which proves that g iU is an FGw n x-homomorphism. Žii. Let x i , yi g Mi w n x be such that g iU Ž x i q Miy1 w n x. s g iU Ž yi q Miy1w n x.. Then g i Ž x i q Miy1 . s g i Ž yi q Miy1 ., so we have that x i q Miy1 s yi q Miy1 since g i is injective. Thus x i y yi g Miy1. But also we have x i y yi g Mi w n x, and so x i y yi g Mi w n x l Miy1 s Miy1w n x. Thus x i q Miy1w n x s yi q Miy1w n x and so g iU is injective. Žiii. Recall that ¨ i was chosen such that g i Ž ¨ i q Miy1 . is a polytabloid. Now ¨ i q Miy1 w n x lies in a finite Gw n x-orbit and so g iU Ž ¨ i q Miy1 w n x. also lies in a finite Gw n x-orbit. Thus we have that g iU Ž ¨ i q Miy1w n x. g F w n x l i l S l i s S l i w n x. So im g iU F S l i w n x. Now g i Ž ¨ i q Miy1 . is a polytabloid, so g iU Ž ¨ i q Miy1w n x. is a polytabloid, and so ²g iU Ž ¨ i q Miy1w n x.:F Gw nx s S l i w n x. Thus im g iU s S l i w n x. This completes the proof of the claim. So we have that g iU is a well-defined bijective FGw n x-homomorphism from Mi w n xrMiy1 w n x to S l i w n x. This completes the proof of the lemma. We are now in a position to prove the main result, which gives us the composition factors of F w V x k .

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THEOREM 3.15. F w V x k has a composition series in which the composition factors are isomorphic to S l , where l s 0, 1, . . . , k, and each S l appears exactly once. Proof. We know that any composition series of F w V x k has each factor isomorphic to S l , for some l F k, with each l appearing at least once. Let 0 - M1 - M2 - ??? - Mry1 - Mr s F w V x k be such a composition series, so MirMiy1 ( S l i for some l i F k Žand each l i appearing at least once.. Then by the previous lemma, for large enough n, the chain of FGw n xsubmodules 0 - M1 w n x - ??? - Mry1w n x - Mr w n x s F w n x k is such that Mi w n xrMiy1 w n x ( S l i w n x for all i. Now each Mi w n x is a finite-dimensional FGw n x-submodule of F w n x k , so dim Ž F w n x

k

. s dim

ž

F w nx

k

Mry1 w n x

/ ž q dim

Mry1 w n x Mry2 w n x

/

q ??? qdim

ž

M1 w n x 0

/

,

and so we have dim Ž F w n x

k

. s dim Ž S l w n x . q dim Ž S l r

ry 1

w n x . q ??? qdim Ž S l1 w n x . . Ž 2 .

But now F w n x k has a Specht series Žfor k - n y k . S 0 w n x, S 1w n x, . . . , S k w n x Žsee Lemma 3.13., and so dimŽ F w n x k . s dimŽ S 0 w n x. q dimŽ S 1 w n x. q ??? qdimŽ S k w n x.. Thus in Ž2., dimŽ S l w n x. appears at most once, for every l F k. But already in Ž2. each S l w n x appears at least once, and so each S l w n x appears exactly once. i.e.  l 0 , l 1 , . . . , l r 4 s  0, 1, . . . , k 4 and so r s k. Thus our original composition series for F w V x k has k factors, namely S 0 , S 1, . . . , S k. Remarks. Ž1. By the Jordan]Holder theorem, e¨ ery composition series ¨ of F w V x k has length k q 1 and the factors of any composition series are Žin no particular order. S 0 , S 1, . . . , S ky1, S k . Ž2. Now we have the existence of a composition series of F w V x k , what Corollary 3.5 tells us is that for l satisfying 0 F l - k, every composition series of F w V x krker b k, l has a bottom composition factor S l. COROLLARY 3.16. Let U - F w V x k and let l be an integer satisfying 0 F l - k. Then S l is a composition factor of F w V x krU if and only if U F ker b k, l . Proof. Assume U F ker b k, l . Now by remark Ž2. above, we know that S l is the unique bottom composition factor of F w V x krker b k, l . Clearly the composition factors of F w V x krker b k, l appear amongst the composition factors of F w V x krU, thus S l is a composition factor of F w V x krU. Conversely, assume that U g ker b k, l . Then ŽU q ker b k, l .rker b k, l is nonzero, so it has a unique bottom composition factor S l by remark Ž2.

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above. But now ŽU q ker b k, l .rker b k, l ( UrŽU l ker b k, l ., and so by Theorem 3.15, S l is not a composition factor of F w V x krU. We can now give a precise description of the submodules of F w V x k : COROLLARY 3.17. Let U - F w V x k . Then U s F ker b k, l : S l is a composition factor of F w V x krU 4 . Proof. Assume the composition factors of F w V x krU are S l 1 , . . . , S l r . Then by the previous corollary, U F ker b k, l i for i s 1, 2, . . . , r and so r r r U F F is1 ker b k, l i . But now if U - F is1 ker b k, l i , then F is1 ker b k, l irU lt has S as a composition factor, with t g  1, 2, . . . , r 4 , which implies that S l t r is not a composition factor of F w V x krF is1 ker b k, l i , which means that S l t k is not a composition factor of F w V x rker b k, l t , which is a contradiction. r Thus U s F is1 ker b k, l i . Remark. What this result tells us is that every submodule of F w V x k is an intersection of some of the kernels ker b k, l where 0 F l - k. COROLLARY 3.18. The maximal proper submodules of F w V x k are kernels of the b-maps. Proof. Let V be a maximal proper submodule of F w V x k , so F w V x kr V ( S l , for some l F k. Thus, by Corollary 3.17, V s ker b k, l . With the aid of the next technical result, we can give a result about the surjectivity of the b-maps. PROPOSITION 3.19. Let F be a field of characteristic p. Let 0 F m - l - k, and consider the maps b k, l : F w V x k ª F w V x l and b l, m : F w V x l ª F w V x m . y m. Then im b k, l F ker b l, m if and only if p di¨ ides Ž kl y m . Proof. Let x g w V x k , and let y s b k, l Ž x .. Then y g im b k, l . Now b l, m maps y to a sum of m-sets. Now the coefficient of a typical m-set z in b l, mŽ y . is equal to the number of ways of choosing an l-set containing z y m. y m. Ž kl y from x, which is Ž kl y then y g ker b l, m so m . Thus if p divides m y m. im b k, l F ker b k, m , and if p does not divide Ž kl y then y f ker b l, m so m im b k, l g ker b l, m .

THEOREM 3.20. Let F be a field of characteristic p. Let 0 F l - k, and consider the map b k, l : F w V x k ª F w V x l. Then b k, l is onto if and only if p y 1. Ž k y 1l q 1 .. does not di¨ ide Ž kl . Ž kl y 1 ???

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Proof. We know that b k, l is onto if and only if im b k, l s F w V x l. Now since the maximal proper submodules of F w V x l are kernels of b-maps, im b k , l s F w V x m im b k , l g ker b l , m , l

m p does not divide m p does not divide

0 F m - l, kym , lym

m s 0, 1, . . . , l y 1,

ž / ž /ž / ž k l

ky1 kylq1 ??? . ly1 1

/

B

Remarks. Ž1. This product of binomial coefficients can be simplified to a product of fractions: b k, l is onto if and only if p does not divide ly1 ŽŽ Ł is0 k y i .rŽ l y i .. iq1. Ž2. From the above two proofs we see that b k, l is onto if and only if all y m. the binomial coefficients Ž kl y m for m s 0, 1, . . . , l y 1 are nonzero in the field. Thus when the field F has characteristic zero, the map b k, l : F w V x k ª F w V x l is always onto. From this theorem we can obtain a corresponding result involving the finite b-maps: y 1. Ž k y 1l q 1 . then, for COROLLARY 3.21. If p does not di¨ ide Ž kl . Ž kl y 1 ??? n k l large enough n, the map b k, l : F w n x ª F w n x is onto. Con¨ ersely, if b k,n l is onto for sufficiently large enough n, then p does not di¨ ide Ž kl . y 1. Ž kl y Ž k y 1l q 1 .. 1 ??? y 1. Ž k y 1l q 1 .. Then the b-map Proof. Suppose p does not divide Ž kl . Ž kl y 1 ??? k l l b k, l : F w V x ª F w V x is onto. Let w g w V x . Then there exists an x g F w V x k such that b k, l Ž x . s w. Now choose n 0 g N large enough so that SuppŽ x . : w n x k for all n G n 0 . Then, by definition, b k,n l Ž x . s w, so w g w n x l. So b k,n l : F w n x k ª F w n x l is onto for all n G n 0 . y 1. Ž k y 1l q 1 .. Then b k, l is not Conversely, suppose p divides Ž kl . Ž kl y 1 ??? l onto. So there exists w g F w V x such that there is no element x g F w V x k with b k, l Ž x . s w. Let n1 be large enough so that SuppŽ w . : w n x l and SuppŽ x . : w n x k for all n G n1. Then b k,n l : F w n x k ª F w n x l is not onto for all n G n1 .

Remark. It would be interesting to have an explicit bound for n 0 here. We conclude this section with a result regarding the order in which the factors appear in a particular composition series of F w V x k . To prove this,

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we need the following result: PROPOSITION 3.22.

Let K j denote ker b k, j . Suppose that

K iŽ1. l ??? l K iŽ r . s K iŽ1. l ??? K iŽ r . l K n for some iŽ1., . . . , iŽ r ., n g  0, 1, . . . , k y 14 . Then K iŽ j. F K n for some j. Proof. Suppose, for a contradiction, that K iŽ j. g K n for any j. So for all j, we have that S n is not a composition factor of F w V x krK iŽ j. Žby Corollary 3.16.. As F w V x krŽ K iŽ1. l ??? l K iŽ r . . embeds into [ jF r F w V x krK iŽ j. , it follows that S n is not a composition factor of F w V x krŽ K iŽ1. l ??? l K iŽ r . ., that is, S n is not a composition factor of F w V x krŽ K iŽ1. l ??? l K iŽ r . l K n .. But S n is a composition factor of F w V x krK n and K iŽ1. l ??? K iŽ r . l K n F K n , so we have a contradiction. LEMMA 3.23. F w V x k has a composition series with factors in the order Ž from the top. S 0 , S 1 , . . . , S k , and the corresponding composition series is ky1

0-

F ls0

ky2

ker b k , l -

F ker bk , l - ??? - ker bk , 0 l ker bk , 1 ls0

- ker b k , 0 - F w V x . k

Proof. This result follows easily from Corollary 3.17 and Theorem 3.15 if we can show that, for any d satisfying 0 - d F k y 1, we have that the dy 1 d ker b k, lrF ls0 ker b k, l is nonzero. quotient module F ls0 dy 1 d So assume, for a contradiction, that F ls0 ker b k, l s F ls0 ker b k, l . Then by Proposition 3.22, ker b k, j F ker b k, d for some j satisfying 0 F j d. So by Corollary 3.16, S d is a composition factor of F w V x krker b k, j . But F w V x krker b k, j is isomorphic to im b k, j F F w V x j, and j - d, so by Theorem 3.15, F w V x krker b k, j cannot have a composition factor isomorphic to dy1 d S d. This is a contradiction. Thus we have that F ls0 ker b k, lrF ls0 ker b k, l is nonzero. dy 1 d In fact, by Corollary 3.17, we have that F ls0 ker b k, lrF ls0 ker b k, l is d isomorphic to S , and so by Theorem 3.15, we have the required composition series.

4. SPECIAL CASES Before we give the algorithm to compute the submodule structure of F w V x k , we look at some special cases where the submodule structure can

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be explicitly calculated independently of the algorithm. The results in this section make use of the following: PROPOSITION 4.1. Let F be a field of characteristic p Ž p a prime., and consider the maps b k, i : F w V x k ª F w V x i and b k, j : F w V x k ª F w V x j. Then y j. ker b k, i F ker b k, j if and only if p does not di¨ ide Ž ki y j , where 0 F j - i - k. Proof. Consider the composition of maps bi, j b k, i . As in Proposition y j. Žk y j. 3.19, bi, j b k, i s Ž ki y j b k, j . Thus if p does not divide i y j then ker b k, i F ker b k, j . y j. Now assume that p does divide Ž ki y j and suppose, for a contradiction, y j. that ker b k, i F ker b k, j . Now since p divides Ž ki y j we have that im b k, i F j ker bi, j . Now S is a composition factor of F w V x irker bi, j , and so by Theorem 3.15, S j is not a composition factor of im b k, i . Now clearly im b k, i / 0, and im b k, i ( F w V x krker b k, i . Once again, we note that S j is a composition factor of F w V x krker b k, j , and, by assumption, ker b k, i F ker b k, j . So S j is a composition factor of F w V x krker b k, i , that is, S j is a composition factor of im b k, i , which is a contradiction. 4.1. The Case char F s 0 In w1x, it is shown that when char F s 0, the submodule structure of F w V x k is uniserial with composition factors Žfrom the top. S 0 , S 1, . . . , S k , the corresponding composition series being 0 - ker b k, ky1 - ker b k, ky2 - ??? - ker b k, 0 - F w V x k . We now deduce this result from our results so far: First we note that, as in the proof of Proposition 4.1, for j s 0, 1, . . . , k y 2 we have b jq1, j b k, jq1 s Ž k y j . b k, j , and so we have 0 - ker b k, ky1 F ker b k, ky2 F ??? F ker b k, 0 F F w V x k . Now let U F F w V x k , so U s l ker b k, j : j g J 4 for some subset J of  0, 1, . . . , k y 14 . Then, by the nesting of the kernels, we have that U s ker b k, l where l is the minimum element of J. Thus, by Corollary 3.17, the only nontrivial submodules of F w V x k are ker b k, 0 , ker b k, 1 , . . . , ker b k, ky1. But now F w V x k has a composition series 0 - ker b k, ky1 - ker b k, ky2 - ??? - ker b k, 0 - F w V x k Žby Lemma 3.23. and this is clearly unique. 4.2. The Case char F s p, where p ) k y j. If p ) k, then p does not divide Ž ki y for all i and j satisfying j 0 F i - j - k, and so by Proposition 4.1, we have that ker b k, ky1 F ker b k, ky2 F ??? F ker b k, 0 . But now, by Lemma 3.23, F w V x k has a composition series 0 - ker b k, ky1 - ker b k, ky2 - ??? - ker b k, 0 - F w V x k , and since every submodule of F w V x k is an intersection of kernels and all the

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kernels are nested, this is in fact the unique composition series of F w V x k . So when char F s p, with p ) k, F w V x k is uniserial in exactly the same way as F w V x k was for char F s 0. 4.3. The Case k s p, where p s char F y j. Here we let k s p, where p s char F. Now p does not divide Ž ki y j for all i and j satisfying 1 F j - i - k, so from Proposition 4.1 we have ky 1 ker b k, l s ker b k, ky1 F ker b k, ky2 F ??? F ker b k, 1 , and so F ls1 ker b k, ky1. Using Proposition 4.1 with p s k, i s k y 1, and j s 0, we have ker b k, ky1 g ker b k, 0 . Now F w V x krker b k, 0 ( im b k, 0 s S 0 , and S 0 ( F, and so ker b k, 0 has codimension 1 in F w V x k . Therefore the sum of submodules ker b k, 0 q ker b k, ky1 is either equal to ker b k, 0 or F w V x k . But if ker b k, 0 q ker b k, ky1 s ker b k, 0 then ker b k, 0 would have to contain ker b k, ky1 , which is a contradiction. Thus we have ker b k, 0 q ker b k, ky1 s F w V x k . Now we can make use of the results of Section 4.2: by the first isomorphism theorem F w V x krker b k, ky1 ( im b k, ky1 , and by Proposition 3.19, im b k, ky1 s ker b ky1, 0 . When char F s k, F w V x ky 1 is uniserial, as proved in the previous section, and so the composition factors of F w V x krker b k, ky1 are Žfrom the top. S 1, S 2 , . . . , S ky1 , and so the composition factors of ker b ky 1, 0rS k are S 1 , S 2 , . . . , S ky1 Žusing the fact that ker b k, 0 l ker b k, ky1 s S k ..

5. THE ALGORITHM In this section we will present an algorithm to compute the submodule structure of F w V x k , for any nonnegative integer k. Throughout this section, it will be assumed that F is a field of characteristic p, since the submodule structure of F w V x k is explicitly known when char F s 0 Žsee Section 4.. In Section 3.2 we proved that every submodule of F w V x k was an intersection of kernels of the b-maps. In fact, since every kernel is a submodule of F w V x k then every possible intersection of kernels is a submodule. So we know all the possible submodules}just compute all possible subsets of  0, 1, . . . , k y 14 . Then for each such subset I, we have that F i g I ker b k, i is a submodule of F w V x k . Thus we can identify each submodule of F w V x k by a subset of  0, 1, . . . , k y 14 . However, since we have that some kernels are contained in other kernels, some of the submodules are in fact the same. That is, we can have two different subsets of  0, 1, . . . , k y 14 which describe the same submodule. Thus we need a procedure which, given an intersection of kernels X Ži.e., a subset of  0, 1, . . . , k y 14., determines all the kernels which could appear in the

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expression for X. So then, using this procedure, we can assign a unique subset of  0, 1, . . . , k y 14 to each submodule. Once we have all the submodules of F w V x k , each with its unique ‘‘label,’’ it should then be clear how to construct the submodule lattice. Before we present the algorithm, we will describe the ‘‘uniqueness’’ procedure. We are given an intersection of kernels as a subset J of  0, 1, . . . , k y 14 . For each j g J we can calculate all the kernels which ker b k, j is contained in, using Proposition 4.1. Thus we replace ker b k, j by the intersection of all the kernels it is contained in. By Proposition 3.22, doing this will give us the desired description of the original kernel intersection. We now present the ‘‘uniqueness’’ procedure Žwhich we shall call ‘‘procedure unique’’. and the algorithm in a formal language, suitable for adaptation to computer programming languages. Procedure Unique input a subset I of  0, 1, . . . , k y 14 let tempI s I for each i g I do for j s 0 to i y 1 do y j.  4 if p does not divide Ž ki y j then add j to tempI end end output the set tempI Žwhich is also a subset of  0, 1, . . . , k y 14. Algorithm to Compute All the Submodules of F w V x k let P be the power set of  0, 1, . . . , k y 14 let temp P s P for each J in P do apply Procedure unique to J Žand let the output set be tempJ. let temp P s temp P y  J 4 let temp P s temp P j  tempJ4 end When the algorithm terminates, temp P will contain all the proper submodules of F w V x k as subsets of  0, 1, . . . , k y 14 , each one corresponding to an intersection of kernels. ACKNOWLEDGMENTS The material in this paper was written whilst the author was in receipt of an E.P.S.R.C. studentship. The author would also like to thank David Evans for his help and support, particularly in finding the composition series of length k q 1.

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