Restriction of the irreducible representations ofto the symmetric group

SnGLn.

Kyo Nishiyama
Division of Mathematics, Faculty of Integrated Human Studies,
Kyoto University, Kyoto 606-8501, JAPANe-mail:1

kyo@math.h.kyoto-u.ac.jp

ProblemLet  = ( ; ; n) be a dominant integral weight of GLn and consider the nite dimensional irreducible representation  with the highest weight . The permutation matrices
in GLn form a nite subgroup which is isomorphic to the symmetric group Sn. We
identify Sn with this subgroup. Then our problem can be stated as follows.
1 Problem 1.1 Describe the decomposition of jSn when restricted to the subgroup Sn.

We will reduce this problem to the decomposition of plethysms in principle.
2

Main result

Let us review GLn GLm -duality on the symmetric algebra S (C n C m ). It is well-known
that S (C n C m ) is multiplicity free as a representation of GLn GLm and decomposes
as follows (see, e.g. [Howe]):S (C nXC m) ' minfm;ng(n)(m) ;

length( )where  = ( ;  ; ) is a weight (or a partition) and n is the irreducible representationof GLn with highest weight . Put V = C m and assume m n. Then we can reformulate
the left hand side of the above formula:n S (V )S (C n V )
!
1( )

2 



'==n

1
M

k=0
M

Sk (V )

 =(1 ; ;n )2Zn

0

1S1 (V )Sn (V );where Sk (V ) denotes a homogeneous component of degree k in S (V ). For  =( ; ; n) (Z )n, put
1 

20S = S

1  Sn Sjj (  =  + + n):j jThen we haveS1 (V )j j V S :Sn (V ) =





1(2.1)

By the classical Schur duality (cf. [Weyl]), we know the decompositionj j V 'X

jj=j j;length()n

(m)



as a GLm Sjj-module. Therefore the above formula (2.1) becomes


01S

X

@jj=j j;length()n(m)We summarize asm S (V )X''X= Ajj=j j;length()nX( )S :(m)( )S(m) 2Zn0 jj=j j;length()n01X
X
(m) @( )S A :length()n 2Zn0;j j=jj

XHere, ( )S becomes an Sn-module, whose module structure is induced by the original action of GLn. So we obtain
(n)

X

Sn''

 2Zn0;j j=jj( )SXX`jj;length()n 2 Sn  !

( )S :Let  be a representation space on
which GLn acts via n and () its weight space ofXweight . We put (Sn ) = ( ). Then, clearly  (Sn ) is invariant under( )VV

 2Sn VVVSn and we get the following lemma.
Lemma 2.1 As a representation ofSSn, there is an isomorphism: ( n  ) 'VX
 2Sn 2()S :Take a partition j. Consider the normalizer N = NSjj (S ) of S in Sjj ;` jN = s Sjj sSs = S :f2

1jThen there exists a partition () = = ( ;1 N =S S = S'g; k ) of length()  n such that

1 S k:Since N normalizes S, it acts on ( )S . Moreover, by de nition, S acts on ( )Strivially. So we get a representation of S N=S on ( )S .
'Proposition 2.2 With the notations above, we haveX( )S2Sn 
where'IndSSn Sn j j ( )S 1;

1 means the trivial representation of Sn j j.

Proof.

Now we summarize the above results intoTheorem 2.3

M

(n)
where`Sn'`jjIndSSn Sn j j ( )S 1;

length() is determined by  via
NSjj (S )=S ' S :Remark.

Put k =  =  . By the Frobenius reciprocity, we havejjj( )Sj'=HomS (C ;  ) HomSk (IndSSk C ;  );)'dim()S = [ : IndSSk C ]:This number is called the Kostka number (cf. [Macdonald]). By Lemma 2.1, it is equal
to the weight multiplicity of  in the representation (n ; ) of GLn .
( )

3V

3

Relation to the plethysm

It seems diÆcult to determine the action of S on ( )S . The reason why it is diÆcultis as follows. Put k =  (i.e.,  k). Note that the normalizer of S = S1Snin Sk is a wreath productS n S Sk ; S = S 1
Smfor some n. The irreducible representations of a wreath product are well-studied andcompletely classi ed (see [JK, Theorem 4.3.34]).
In our case, for  S^, let ( : 1) be an irreducible representation of S n S which
is obtained by the successive application of the natural projection of S n S S andthe representation  of S . Then we get
HomS (; ()S ) = HomS (( : 1)S ; ()S )= HomS nS (( : 1); )
HomSk (IndSSk nS ( : 1); ):jj` `2!

'If we know ( )S completely as an S -module, then we know HomS (; ()S ), hence
the multiplicity of  in IndSSk nS ( : 1) can be determined. However, the inducedrepresentation
IndSSk nS ( : 1)
is a special case of plethysms and its decomposition is not completely known yet (cf. [JK,5.4]).
Using the above relation, we can rephrase our theorem in terms of the decompositionof plethysm:
M( )S[ : IndSSk nS ( : 1)] :x'Theorem 3.1

(n)M MSn'`jj 2S^

2S^

[ : IndSSk nS ( : 1)] IndSSn Sn j j  1:Take  = (kn) = (k; ; k). In this case, we have n = (det n )k . So, weknow n S = (sgn)k . Moreover, by Lemma 2.1, we get
Example 3.2( )( )( )n( )S =For  = , the normalizer of S = Sk
we haveif  = ,0 otherwise.
Sk in Snk is isomorphic to Sn n (Sk )n. SoC

[(kn ) ; IndSSnk( : 1)] =n n(Sk )n4

1 if  = (sgn)k ,0 if  = (sgn)k .
6 = n and  = (1;    ; 1) = (1n ). Then () = (n) and V () '  . This

Example 3.3 j j

is a well-known result (cf. [Kostant]).

 = kn and  = (k;    ; k) = (kn ). Then () = (n) and

Example 3.4 j j

V()Mh'bn 2Si : IndSSnkn n(Sk )n ( : 1) This is a result of [AMT].
References[AMT] S. Ariki, J. Matsuzawa and I. Terada, Representations of Weyl groups on zeroweight spaces. In \Algebraic and Topological Theories, to the memory of Dr. Takehiko Miyata", Kinokuniya, 1985, pp. 546 { 568.
[Gutkin] E. Gutkin, Representations of the Weyl group in the space of vectors of zeroweight. UspehiMat. Nauka, 28(1973), 237 { 238 (in Russian).
[Kostant] B. Kostant, On Macdonald's -function formula, the Laplacian and generalized
exponents. Adv. in Math., 20(????), 257 { 285.
[King] R. C. King, Branching rules for GL(N ) m and the evaluation of inner
plethysms, J. Math. Phys., 15(1974).
[Howe] R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions
and beyond. In \The Schur lectures (1992)", Israel Mathematical Conference
Proceedings, 8(1995), Bar-Ilan Univ., pp. 1 { 182.
[JK] G. James and A. Kerber, The representation theory of the symmetric group. Encyclopedia of Math. and App. 16, Addison-Wesley, 1981.
[Macdonald] I. Macdonald, Symmetric functions and Hall polynomials. Clarendon Press,
1979.
[Weyl] H. Weyl, The Classical Groups. Princeton University Press, Princeton, N.J., 1946.
First Draft: 1996/10/2
Ver 1.0 : 1996/10/17, Ver 1.1 : 1996/11/5, Ver. 1.2 [00/11/28 10:11]
This version is compiled on November 29, 2000.

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