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Europ . J . Combinatorics (1996) 17 , 647 – 655

A Combinatorial Approach to the Double Cosets of the Symmetric Group with respect to Young Subgroups

A NDREW R . J ONES

Combinatorial methods are employed to study the double cosets of the symmetric group S n with respect to Young subgroups H and K . The current paper develops a correspondence between these double cosets and certain lists of integers . This approach leads naturally to an

algorithm for computing the number of ( H , K )-double cosets of S n . ÷ 1996 Academic Press Limited

1 . I NTRODUCTION

The double cosets of the symmetric group S n by Young subgroups play an important role in the representation theory of the symmetric group . It was shown in [3] that there is a bijective correspondence between these double cosets and certain n ϫ n matrices over the natural numbers , with further results appearing in [4] . The theory is re-developed in this paper by adopting a dif ferent approach to derive similar results concerning these double cosets (see also [2 , 7]) . In Section 2 , the notation is introduced and a brief exposition of material on double cosets is given . The main results in this paper are then developed in Section 3 . An

algorithm is presented for determining the number of ( H , K )-double cosets of S n , which takes advantage of certain recurrence formulae . Finally , in Section 4 , these ideas

are illustrated by considering an interesting example : the ( R ⌳ r , C ⌳ r )-double cosets of S n are examined , where R ⌳ r and C ⌳ r are the row and column subgroups corresponding to a certain array of numbers ⌳ r (a Young tableau) . In this case , it is also proved that there is a unique element of minimum length in each double coset . The study of these double cosets was motivated by the problem of determining a simplified form for the projective analogue of the Young symmetrizers for the symmetric group recently introduced by M . L . Nazarov [5] .

2 . D OUBLE C OSETS FOR S n OVER Y OUNG S UBGROUPS

ϩ Let S n ( n ෈ ޚ ) be the symmetric group of permutations on the symbols ͕ 1 , 2 , . . . , n ͖ , taking the composition of elements from right to left as our convention . It is convenient

to consider permutations in list notation , i . e . to represent ␴ ෈ S n by the list of integers [ ␴ (1) , ␴ (2) , . . . , ␴ ( n )] . For any ␴ ෈ S n , define length( ␴ ) ϭ 4 ͕ inversions in the list [ ␴ (1) , ␴ (2) , . . . , ␴ ( n )] ͖ . This agrees with the conventional definition for length as the number of basic transpositions in a reduced word .

In this paper , we consider a special class of subgroups of S n called Young subgroups .

D EFINITION 2 . 1 . Let ␣ ϭ ( ␣ 1 , ␣ 2 , . . . , ␣ l ) ٛ n (a partition of n ) and let A ϭ ͕ A 1 , A 2 , . . . , A l ͖ be a partition of the set ͕ 1 , 2 , . . . , n ͖ such that ͉ A i ͉ ϭ ␣ i for each i ϭ 1 , 2 , . . . , l .

647 0195-6698 / 96 / 070647 ϩ 09 $18 . 00 / 0 ÷ 1996 Academic Press Limited 648 A . R . Jones

Now let S A i ( i ϭ 1 , 2 , . . . , l ) denote the subgroup of S n consisting of the ␣ i ! A i . Then the گ ͖ permutations which fix every symbol in the complement ͕ 1 , 2 , . . . , n internal direct product

S A ϭ S A 1 ϫ S A 2 ϫ и и и ϫ S A l is called the Young subgroup corresponding to the partition A . Note that all Young subgroups corresponding to a specific partition ␣ are conjugate in S n .

Let H and K be Young subgroups of the symmetric group S n (to remain fixed for the remainder of this section) . Then a combinatorial approach may be adopted to investigate the ( H , K )-double cosets of S n . Each Young subgroup may be written as a direct product of symmetric subgroups

H ϭ S A 1 ϫ S A 2 ϫ и и и ϫ S A l , K ϭ S B 1 ϫ S B 2 ϫ и и и ϫ S B k for certain partitions A ϭ ( A 1 , A 2 , . . . , A l ) and B ϭ ( B 1 , B 2 , . . . , B k ) of the set ͕ 1 , 2 , . . . , n ͖ . Let ␣ and ␤ be the partitions of the integer n corresponding to A and B , respectively .

Some results concerning the ( H , K )-double cosets of S n are given in [3] . It is shown that the double coset H ␴ K is characterized by the array of integers

z i j ϭ ͉ A i ʝ ␴ ( B j ) ͉ , 1 р i р l , 1 р j р k .

If i Ͼ l or j Ͼ k , then set z i j ϭ 0 . Denote by M ␣ , ␤ the set of n ϫ n matrices over ގ ϭ ͕ 0 , 1 , 2 , . . . ͖ with row sums ( ␣ 1 , ␣ 2 , . . . , ␣ l , 0 , . . . , 0) and column sums ( ␤ 1 , ␤ 2 , . . . , ␤ k , 0 , . . . , 0) .

S n / K ) 5 M ␣ , ␤ defined by گ T HEOREM 2 . 2 . There is a bijecti␷ e correspondence f : ( H

f ( H ␴ K ) ϭ ( z i j ) n ϫ n .

C OROLLARY 2 . 3 . The number of ( H , K )- double cosets of S n is equal to ͉ M ␣ , ␤ ͉ .

t The implication Z ෈ M ␣ , ␤ ï Z ෈ M ␤ , ␣ leads to the following corollary .

C OROLLARY 2 . 4 . Suppose that H , H Ј are Young subgroups corresponding to the partition ␣ ٛ n and K , K Ј are Young subgroups for ␤ ٛ n . Then

. ͉ S n / H Ј گ S n / K ͉ ϭ ͉ K Ј گ H ͉

More results concerning the set of matrices M ␣ , ␤ are given in [4] . These include the bijective correspondence that exists between the matrices ( z i j ) ෈ M ␣ , ␤ and pairs of standard tableaux (a standard bitableau ) , together with the insertion / cancellation algorithms of D . E . Knuth (see also [6]) .

3 . A N A LTERNATIVE A PPROACH TO THE ( H , K )-D OUBLE C OSETS

Fix Young subgroups H , K of the symmetric group S n and let A , B be the corresponding partitions of the set ͕ 1 , 2 , . . . , n ͖ as defined in Section 2 . Some further definitions are required before presenting our main results .

D EFINITION 3 . 1 . Given a permutation ␴ ϭ [ ␴ (1) , ␴ (2) , . . . , ␴ ( n )] in list form , let A ,B ⌫ ␴ ϭ ⌫ ␴ denote the list of positive integers obtained from ␴ by replacing each entry Double cosets of the symmetric group 649

x in the list by the class (subset) A i in the partition A which contains the symbol x . For convenience , the class A i ෈ A may be labelled by the integer i .

The partition B ϭ ( B 1 , B 2 , . . . , B k ) of the set ͕ 1 , 2 , . . . , n ͖ gives rise , in a natural way , to a corresponding decomposition of the list ⌫ ␴ into k components . Namely , the i th component ⌫ i consists of the entries in the list ⌫ ␴ which occupy the positions given by the subset B i ෈ B .

1 2 l D EFINITION 3 . 2 . For each i ϭ 1 , 2 , . . . , k , define the sequence ␥ i ϭ ( ␥ i , ␥ i , . . . , ␥ i ) j where ␥ i is the number of symbols j which occur in the component ⌫ i .

D EFINITION 3 . 3 . Let ⌫ *␴ denote the list of integers obtained from ⌫ ␴ by permuting the symbols in each component of ⌫ ␴ into increasing order .

It turns out that the ( H , K )-double coset containing ␴ may be labelled by the list ⌫ *␴ , which implies that the lists ⌫ * may be used to enumerate the ( H , K )-double cosets of

S n . Let ⌬ A , B denote the set of all lists of the form ⌫ * . Then , as we have just shown , there is a mapping f A ,B : S n 5 ⌬ A , B given by f A ,B ( ␴ ) ϭ ⌫ *␴ . The next result shows that this function is constant on any ( H , K )-double coset .

L EMMA 3 . 4 . For any ␴ ෈ S n , then f A , B is constant on the double coset H ␴ K .

P ROOF . Take ␳ ෈ H ␴ K . Then ␳ ϭ h ␴ k for some h ෈ H , k ෈ K . The action of the permutation h ෈ H on the left of ␴ has the ef fect of permuting amongst themselves the symbols in each class A i . In particular , the corresponding list ⌫ is invariant under this action , i . e . ⌫ h ␴ ϭ ⌫ ␴ . On the other hand , the action of the permutation k ෈ K on the right of h ␴ permutes the positions in the list representing h ␴ . This carries over to the permutation by k of the positions in the list ⌫ h ␴ . Therefore ⌫ h ␴ k and ⌫ ␴ dif fer by a rearrangement of the symbols within each of their components . Then , permuting the symbols in each component of ⌫ ␳ and ⌫ ␴ into increasing order gives

⌫ *␳ ϭ ⌫ *␴ . ᮀ

This last result implies that , with a slight abuse of notation , f A ,B may be defined on . S n / K 5 ⌬ A , B , where f A , B ( H ␴ K ) ϭ ⌫ *␴ گ the ( H , K )-double cosets of S n , i . e . f A ,B : H This mapping f A ,B is bijective , which gives the following .

P ROPOSITION 3 . 5 . There is a one - to - one correspondence between the ( H , K )- double cosets of S n and the associated lists ⌫ * .

. ͉ S n / K ͉ ϭ ͉ ⌬ A ,B گ C OROLLARY 3 . 6 . ͉ H

Any list ⌫ * ෈ ⌬ A ,B uniquely determines sequences ␥ 1 , ␥ 2 , . . . , ␥ k by Definition 3 . 2 . These sequences constitute the columns of an l ϫ k matrix the row sums of which give the partition ␣ ϭ ( ␣ 1 , ␣ 2 , . . . , ␣ l ) and the column sums of which give the partition ␤ ϭ ( ␤ 1 , ␤ 2 , . . . , ␤ k ) . The matrix may be embedded into the set M ␣ , ␤ in the natural way ; namely , define Z ϭ ( z i j ) n ϫ n ෈ M ␣ , ␤ , where ϭ ␥ i , 1 р i р l , 1 р j р k , z ϭ i j j i j ͭ 0 , otherwise . 650 A . R . Jones

This demonstrates the bijective correspondence between the ( H , K )-double cosets of S n and the matrices Z ෈ M ␣ , ␤ which was introduced earlier . It also illustrates the link between this approach and the earlier method given in [3] .

3 . 1 . The Number of ( H , K )- Double Cosets of S n .

We have shown that the ( H , K )-double cosets of S n are in a one-to-one correspon- dence with the sequences ⌫ * ෈ ⌬ A , B . More precisely , ⌫ * is a list containing ␣ 1 copies of the symbol 1 , ␣ 2 copies of 2 , . . . , ␣ l copies of the symbol l ; which has been decomposed by the partition B such that the symbols in each component of ⌫ * are in a (weak) increasing order . Equivalently , such a list ⌫ * may be considered as an array of shape ␤ with the rows in the array corresponding to the components in ⌫ * . In this case , the symbols (weakly) increase along each row of the array . An array of shape ␤ constructed in this way from a list ⌫ * ෈ ⌬ A ,B will be called a permissible array , and denoted by ⌳ .

D EFINITION 3 . 7 . An array of symbols is said to have content ␮ ϭ ( ␮ 1 , ␮ 2 , . . . , ␮ j ) if the array consists of ␮ 1 copies of the symbol 1 , ␮ 2 copies of 2 , . . . , ␮ j copies of the symbol j .

Hence the permissible arrays ⌳ are precisely those arrays with shape ␤ and content ␣ such that the symbols increase (weakly) along each row . Thus the ( H , K )-double cosets of S n are in one-to-one correspondence with these permissible arrays ⌳ , and the problem of determining the number of these double cosets reduces to that of enumerating all the permissible arrays .

Let ␭ ϭ ( ␭ 1 , ␭ 2 , . . . , ␭ i ) ٛ m and let ␮ ϭ ( ␮ 1 , ␮ 2 , . . . , ␮ j ) be a sequence of natural numbers the sum of which is equal to m . Then define p ( ␭ , ␮ , r ) to be the number of arrays with shape ␭ and content ␮ such that the symbols (weakly) increase along each row , under the additional restriction that the last element in row i is less than or equal to r . Then , using this notation , we have 4 (permissible arrays ⌳ ) ϭ p ( ␤ , ␣ , l ) . The number of permissible arrays ⌳ may be computed by the following algorithm , which recursively reduces an array by one entry at a time until a trivial situation is achieved .

A LGORITHM 3 . 8 . Let ␭ ϭ ( ␭ 1 , ␭ 2 , . . . , ␭ i ) be a partition of m р n and let ␮ ϭ ( ␮ 1 , ␮ 2 , . . . , ␮ j ) be a sequence of natural numbers , the sum of which is equal to m . The algorithm uses the following recurrence formulae , which correspond to the two separate cases ␭ i ϭ 1 and ␭ i ϶ 1 .

( 1 ) ␭ i ϭ 1 : p ( ␭ , ␮ , r ) ϭ ͸ p ( ␭៮ , ␮ ៮ , l ) , (1) s р r ( ␮ s Ͼ 0) where ␭៮ ϭ ( ␭ 1 , ␭ 2 , . . . , ␭ i Ϫ 1 ) is the partition obtained by deleting the entry ␭ i from ␭ ␮ ៮ ϭ ( ␮ 1 , ␮ 2 , . . . , ␮ s Ϫ 1 , ␮ s Ϫ 1 , ␮ s ϩ 1 , . . . , ␮ j ) and l is the length of the partition ␣ .

( 2 ) ␭ i Ͼ 1 : p ( ␭ , ␮ , r ) ϭ ͸ p ( ␭ ៮ , ␮ ៮ , s ) , (2) s р r ( ␮ s Ͼ 0) where ␭៮ ϭ ( ␭ 1 , ␭ 2 , . . . , ␭ i Ϫ 1) and ␮ ៮ is as before . Double cosets of the symmetric group 651

These recurrence formulae represent the deletion of the node in the last position of row i . Furthermore , we have the following results for the numbers p ( ␭ , ␮ , r ) in certain cases : (a) if ␮ ϭ (0 , . . . , 0 , m , 0 , . . . , 0) (i . e . the array contains m copies of one symbol s only) then , clearly , 1 , if s р r , p ( ␭ , ␮ , r ) ϭ ͭ 0 , if s Ͼ r ;

(b) if ␮ 1 ϭ ␮ 2 ϭ и и и ϭ ␮ r Ϫ 1 ϭ ␮ r ϭ 0 , then p ( ␭ , ␮ , r ) ϭ 0 ; (c) if ␭ ϭ ( m ) , then p ( ␭ , ␮ , l ) ϭ 1 . By using the formulae (1) and (2) together with the preceding results , we can evaluate the number of permissible arrays ⌳ with shape ␤ and content ␣ . In practice , this corresponds to the successive deletion of nodes from an array to reach the above standard cases for which the number of such arrays is known .

E XAMPLE . Consider the Young subgroups in S 5 :

H ϭ S ͕ 1 , 2 , 3 ͖ ϫ S ͕ 4 ͖ ϫ S ͕ 5 ͖ , K ϭ S ͕ 1 , 2 ͖ ϫ S ͕ 3 , 4 ͖ ϫ S ͕ 5 ͖ . Here ␣ ϭ (3 , 1 , 1) , ␤ ϭ (2 , 2 , 1) . We compute the number of arrays with shape (2 , 2 , 1) and content (3 , 1 , 1) with the symbols increasing along each row :

{1, 1, 1, 2, 3},

p ([2 , 2 , 1] , [3 , 1 , 1] , 3) ϭ p ([2 , 2] , [2 , 1 , 1] , 3) ϩ p ([2 , 2] , [3 , 0 , 1] , 3) ϩ p ([2 , 2] , [3 , 1] , 3) ϭ p ([2 , 1] , [1 , 1 , 1] , 1) ϩ p ([2 , 1] , [2 , 0 , 1] , 2) ϩ p ([2 , 1] , [2 , 1] , 3) ϩ p ([2 , 1] , [2 , 0 , 1] , 1) ϩ p ([2 , 1] , [3] , 3) ϩ p ([2 , 1] , [2 , 1] , 1) ϩ p ([2 , 1] , [3] , 2) ϭ p ([2] , [0 , 1 , 1] , 3) ϩ p ([2] , [1 , 0 , 1] , 3) ϩ p ([2] , [1 , 1] , 3)

ϩ p ([2] , [2] , 3) ϩ p ([2] , [1 , 0 , 1] , 3) ϩ p ([2] , [2] , 3)

ϩ p ([2] , [1 , 1] , 3) ϩ p ([2] , [2] , 3)

ϭ 3 p ([2] , [2] , 3) ϩ 2 p ([2] , [1 , 1] , 3) ϩ 2 p ([2] , [1 , 0 , 1] , 3)

ϩ p ([2] , [0 , 1 , 1] , 3)

ϭ 3 ϩ 2 ϩ 2 ϩ 1 i . e . p ( ␤ , ␣ , 3) ϭ 8 .

R EMARK . This approach to the double cosets leads to a simple combinatorial formula for calculating their orders . Recall Definition 3 . 2 , which defines the sequences 652 A . R . Jones

␥ 1 , ␥ 2 , . . . , ␥ k for any double coset H ␴ K . Then the order may be computed using the formula ͉ H ␴ K ͉ ϭ ͉ H ͉ ͉ K ͉ r ( H ␴ K ) , where k 1 r ϭ . ( H ␴ K ) ͹ ͩ 1 2 l ͪ i ϭ 1 ␥ i ! ␥ i ! и и и ␥ i !

4 . T HE ( R ⌳ r , C ⌳ r )-D OUBLE C OSETS OF S n In this final section , a special example is developed which illustrates the ideas introduced previously . The combinatorial concepts involved appear in many standard texts [1 , 6] .

Let ␭ ϭ ( ␭ 1 , ␭ 2 , . . . , ␭ l ) ٛ n and denote the conjugate partition by ␭ Ј . Construct the left-justified array of n boxes with ␭ i boxes in the i th row ( i ϭ 1 , 2 , . . . , l ) . This array is called the Young diagram of shape ␭ . The conjugate partition ␭ Ј represents the columns in the Young diagram of shape ␭ . Let ⌳ r denote the ␭ -tableau obtained by inserting the symbols ͕ 1 , 2 , . . . , n ͖ by rows into the Young diagram for ␭ . Then ⌳ r is called the row tableau of shape ␭ . Define r R ⌳ r р S n to be the row subgroup for ⌳ consisting of the permutations which permute r r symbols within each row of ⌳ ; and let C ⌳ r р S n be the column subgroup for ⌳ , i . e . the subgroup of elements which permute the symbols within each column of ⌳ r . The groups R ⌳ r and C ⌳ r are Young subgroups for the partition ␭ and the conjugate partition ␭ Ј , respectively . For the remainder of this section , specialize H ϭ R ⌳ r , K ϭ C ⌳ r . Then the previous result concerning the number of ( H , K )-double cosets becomes

. ( S n / C ⌳ r ͉ ϭ p ( ␭ Ј , ␭ , l ) ϭ P ( ␭ گ R ⌳ r ͉ . The numbers P ( ␭ ) have been calculated for every ␭ ٛ n (2 р n р 7) using Algorithm 3 . 8 These results are presented in Table 1 . In particular , note that P ( ␭ Ј ) ϭ P ( ␭ ) .

The partition A of ͕ 1 , 2 , . . . , n ͖ corresponding to the Young subgroup R ⌳ r is

A ϭ ( ͕ 1 , 2 , . . . , ␭ 1 ͖ , ͕ ␭ 1 ϩ 1 , . . . , ␭ 1 ϩ ␭ 2 ͖ , . . . , ͕ n Ϫ ␭ l ϩ 1 , . . . , n ͖ ) .

T ABLE 1

The number of ( R ⌳ r , C ⌳ r )-double cosets of S n

n ␭ P ( ␭ ) ␭ P ( ␭ ) ␭ P ( ␭ )

2 [2] 1 [1 , 1] 1 3 [3] 1 [2 , 1] 2 [1 , 1 , 1] 1 4 [4] 1 [3 , 1] 3 [2 , 2] 3 [2 , 1 , 1] 3 [1 , 1 , 1 , 1] 1 5 [5] 1 [4 , 1] 4 [3 , 2] 5 [3 , 1 , 1] 7 [2 , 2 , 1] 5 [2 , 1 , 1 , 1] 4 [1 , 1 , 1 , 1 , 1] 1 6 [6] 1 [5 , 1] 5 [4 , 2] 8 [4 , 1 , 1] 13 [3 , 3] 7 [3 , 2 , 1] 12 [3 , 1 , 1 , 1] 13 [2 , 2 , 2] 7 [2 , 2 , 1 , 1] 8 [2 , 1 , 1 , 1 , 1] 5 [1 , 1 , 1 , 1 , 1 , 1] 1 7 [7] 1 [6 , 1] 6 [5 , 2] 12 [5 , 1 , 1] 21 [4 , 3] 13 [4 , 2 , 1] 25 [4 , 1 , 1 , 1] 34 [3 , 3 , 1] 19 [3 , 2 , 2] 19 [3 , 2 , 1 , 1] 25 [3 , 1 , 1 , 1 , 1] 21 [2 , 2 , 2 , 1] 13 [2 , 2 , 1 , 1 , 1] 12 [2 , 1 , 1 , 1 , 1 , 1] 6 [1 , 1 , 1 , 1 , 1 , 1 , 1] 1 Double cosets of the symmetric group 653

Let A ( k ) denote the class in A containing the symbol k . Then the partition A satisfies the order - preser␷ ing property

( ء ) . ( i р j é A ( i ) р A ( j The next result follows as a consequence of this property .

P ROPOSITION 4 . 1 . There is a unique element of minimum length in each ( R ⌳ r , C ⌳ r )- double coset of S n .

P ROOF . The proof is constructive . Given any ( R ⌳ r , C ⌳ r )-double coset with represen- tative element ␴ , then a method is presented for determining the double coset element which has the minimum number of inversions when expressed in list form . It shall be proved that the following algorithm constructs this element τ of minimum length . Furthermore , the row and column permutations required to derive the minimum element τ from the permutation ␴ may also be obtained directly from this procedure . ᮀ

A LGORITHM 4 . 2 . ( 1 ) Reduction by the column subgroup C ⌳ r : for each class in the column partition B , permute the corresponding positions in the list ␴ to eliminate e␷ ery in␷ ersion in␷ ol␷ ing the rele␷ ant symbols . Let the permutation gi␷ en by this first step be denoted by ␴ 1 . ( 2 ) Reduction by the row subgroup R ⌳ r : for each class in A , permute these symbols in the list ␴ 1 into increasing order .

τ We prove that this algorithm determines a unique element in the ( R ⌳ r , C ⌳ r )-double coset , where τ is the element of minimum length . The first step uses a column permutation to remove as many inversions as possible from the list ␴ . Equivalently , by this minimizes the number of reversals in the associated list ⌫ ␴ with , ( ء ) the property respect to the column subgroup , to give the configuration ⌫ *␴ . The list ⌫ *␴ depends only on the ( R ⌳ r , C ⌳ r )-double coset of S n and is invariant under the left action of any row permutation r ෈ R ⌳ r . The second step in the procedure uses a row permutation to τ minimize the number of inversions in the list ␴ 1 . This determines a unique element in the double coset . It remains to prove that this permutation τ is the unique element of minimum length in the ( R ⌳ r , C ⌳ r )-double coset . This follows as a consequence of the order-preserving of the partition A . We establish that any remaining element in the double ( ء ) property coset has length strictly greater than that of the element τ . τ Clearly , any permutation obtained from by the left action of a row element r ෈ R ⌳ r has a greater number of inversions when expressed in list form . Therefore , it is only necessary to consider those permutations which have associated lists ⌫ ϶ ⌫ * and for any fixed list ⌫ , it is suf ficient to consider only the permutation ␳ ⌫ for which the symbols from each row in ⌳ r occur in an increasing order . We prove that length( τ ) Ͻ length( ␳ ⌫ ) , ᭙ ⌫ ϶ ⌫ * . Any list ⌫ may be obtained from ⌫ * by the right action of some column element c ෈ C ⌳ r which permutes the relevant positions in ⌫ * . Such a column permutation may be expressed as a reduced word c ϭ c 1 c 2 и и и c k in the ‘adjacent’ transpositions ( i , j ) , i . e . i Ͻ j are adjacent symbols in the same column of ⌳ r . Applying the transpositions 0 c 1 , c 2 , . . . , c k successively to the list ⌫ * ϭ ⌫ creates a sequence of lists 1 2 k ⌫ , ⌫ , . . . , ⌫ ϭ ⌫ . In fact , for any list ⌫ , there exists a column permutation c ෈ C ⌳ r 654 A . R . Jones

and a reduced decomposition c ϭ c 1 c 2 и и и c k such that for each m ϭ 0 , 1 , . . . , k Ϫ 1 , if m m c m ϩ 1 ϭ ( i , j ) ( i Ͻ j ) , then ⌫ ( i ) Ͻ ⌫ ( j ) . This property of the decomposition will lead to the result

length( ␳ ⌫ m ) Ͻ length( ␳ ⌫ m ϩ 1 ) , m ϭ 0 , 1 , . . . , k Ϫ 1 .

m m m Consider the list ⌫ and suppose that ⌫ ( i ) ϭ a , ⌫ ( j ) ϭ b , where c m ϩ 1 ϭ ( i , j ) . By the above property of the decomposition , we have a Ͻ b . Let p , q denote the number of symbols a , b respectively which occupy positions between i and j in the list ⌫ m ; and let r denote the number of symbols a Ͻ d Ͻ b which occur between the positions i and j :

i j ⌫ m ϭ [ и и и a и и и b и и и ] .

Applying the transposition c m ϩ 1 ϭ ( i , j ) interchanges the symbols a , b to give the list ⌫ m ϩ 1 . m m ϩ 1 Next , consider the corresponding permutations ␳ ⌫ and ␳ ⌫ which are obtained by inserting the symbols from each row in ⌳ r into the relevant positions in the lists ⌫ m , m ϩ 1 m ⌫ m ϩ 1 in increasing order . ␳ ⌫ is obtained from ␳ ⌫ by interchanging the symbols in positions i , j and then reordering the symbols (by some row permutation) into the increasing order . The first step increases the number of inversions in the list by 1 ϩ 2( p ϩ q ϩ r ) while reordering the resulting list decreases the number by p ϩ q . Therefore length( ␳ ⌫ m ϩ 1 ) ϭ length( ␳ ⌫ m ) ϩ 1 ϩ p ϩ q ϩ 2 r . Thus at each stage in the procedure , the length of the resulting permutation increases . In particular , length( ␳ ⌫ * ) Ͻ length( ␳ ⌫ ) ; i . e .

length( τ ) Ͻ length( ␳ ⌫ ) ᭙ ⌫ ϶ ⌫ * . ᮀ

N OTES . (a) A row permutation r ෈ R ⌳ r and column permutation c ෈ C ⌳ r such that τ ϭ r ␴ c may be directly obtained from the algorithm . These elements r , c are not uniquely determined in general .

(b) R ⌳ r ʝ C ⌳ r ϭ ͕ e ͖ . (c) Proposition 4 . 1 may be generalised to the ( H , K )-double cosets of S n , where H is In particular , the . ( ء ) any Young subgroup satisfying the order - preser␷ ing property r result applies to the ( R ⌳ ៮ r , C ⌳ ៮ r )-double cosets for the shifted row tableau ⌳ ៮ . These double cosets play an important role in the study of the projective Young symmetrizers [5] .

: (E XAMPLE . ␭ ϭ (4 , 3 , 2) ٛ 9 , ␭ Ј ϭ (3 , 3 , 2 , 1

1 234 r Λ = 567 R ⌳ r ϭ S ͕ 1 , 2 , 3 , 4 ͖ ϫ S ͕ 5 , 6 , 7 ͖ ϫ S ͕ 8 , 9 ͖ ,

89 C ⌳ r ϭ S ͕ 1 , 5 , 8 ͖ ϫ S ͕ 2 , 6 , 9 ͖ ϫ S ͕ 3 , 7 ͖ ϫ S ͕ 4 ͖ .

Suppose that ␴ ϭ (1 6 8 5 2 4 9 3 7) ෈ S 9 (in disjoint cycle notation) . Then in list form we have ␴ ϭ [6 , 4 , 7 , 9 , 2 , 8 , 1 , 5 , 3] . So ⌫ ␴ ϭ [2 , 1 , 2 , 3 , 1 , 3 , 1 , 2 , 1] . Applying the reduction Algorithm 4 . 2 gives τ ␴ 1 ϭ [2 , 3 , 1 , 9 , 5 , 4 , 7 , 6 , 8] ; ⌫ *␴ ϭ [1 , 1 , 1 , 3 , 2 , 1 , 2 , 2 , 3] ; ϭ [1 , 2 , 3 , 8 , 5 , 4 , 6 , 7 , 9] . Double cosets of the symmetric group 655

Converting τ to cycle notation gives τ ϭ (4 8 7 6) . The row and column permutations to perform this reduction are r ϭ (1 3 2)(6 7)(8 9) and c ϭ (1 5 8)(2 9 6)(3 7) . Then τ ϭ r ␴ c is the element of minimum length in the ( R ⌳ r , C ⌳ r )-double coset containing ␴ .

A CKNOWLEDGEMENTS I wish to thank my supervisor , Professor A . O . Morris , for his guidance and support over the past three years . I am also grateful to Dr T . P . McDonough and Dr M . Nazarov for helpful discussions and to the referee for providing useful comments concerning the original draft of this article . Finally , I must thank the EPSRC for their financial support during my period of postgraduate research .

R EFERENCES

1 . G . E . Andrews , The Theory of Partitions , Encyclopedia of Mathematics , Volume 2 , Addison-Wesley , Reading , Massachusetts , 1976 . 2 . G . D . James , The Representation Theory of the Symmetric Groups , Lecture Notes in Mathematics 682 , Springer-Verlag , New York , 1978 . 3 . G . D . James and A . Kerber , The Representation Theory of the Symmetric Group , Encyclopedia of Mathematics , Volume 1 6 , Addison-Wesley , Reading , Massachusetts , 1981 . 4 . A . Kerber , Algebraic Combinatorics ␷ ia Finite Group Actions , Bibliographisches Institut Wissenschafts- verlag , Mannheim , 1991 . 5 . M . L . Nazarov , Young’s symmetrizers for projective representations of the symmetric group , Ad␷ . Math . , to appear . 6 . B . E . Sagan , The Symmetric Group : Representations , Combinatorial Algorithms and Symmetric Functions , Wadsworth & Brooks / Cole , Pacific Grove , California , 1991 . 7 . K . H . Zimmermann , On weight spaces of polynomial representations of the general linear group as linear codes , J . Combin . Theory , Ser . A , 67 (1994) , 1 – 22 . Recei␷ ed 2 3 September 1 9 9 4 and accepted 2 4 July 1 9 9 5

A NDREW R . J ONES Department of Mathematics , Uni␷ ersity of Wales , Aberystwyth , Dyfed SY 2 3 3 BZ , U .K . e - mail : awj ê aber .ac .uk