Weights and weight spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 3 March 2013

Weights and weight spaces

A finite dimensional Hn-module is calibrated if it has a basis of simultaneous eigenvectors for the xi,1in. In other words, M is calibrated if it has a basis {vt} such that for all vt in the basis and all 1in,

xivt=tivt, for someti*.

Weights. Let X be the abelian group generated by the elements x1,,xnHn and let

T={group homomorphismsX*} .

The torus T can be identified with (*)n by identifying the element t=(t1,,tn)(*)n with the homomorphism given by t(xi)=ti, 1in. The symmetric group Sn acts on n by permuting coordinates and this action induces an action of Sn on T given by

(wt)(xγ) =xw-1γ, forwSn,γ n,

with notation as in (1.2).

Weight spaces. Let M be a finite dimensional Hn-module. For each t=(t1,,tn)T the t-weight space of M and the generalized t-weight space are the subspaces

Mt = { mM xim=tim for all1in } and Mtgen = { mMfor each 1in, (xi-ti)k m=0for somek >0 } ,

respectively. From the definitions, MtMtgen and M is calibrated if and only if Mtgen=Mt for all tT. If Mtgen0 then Mt0. In general MtTMt, but we do have


This is a decomposition of M into Jordan blocks for the action of [X]= [ x1±1 ,, xn±1 ] . The set of weights of M is the set

supp(M)= {tTMtgen0} . (3.1)

An element of Mt is called a weight vector of weight t.

The τ operators. The maps τi:MtgenMsitgen defined below are local operators on M in the sense that they act on each generalized weight space Mtgen of M seperately. The operators τi is only defined on the generalized weight spaces Mtgen such that titi+1.

Proposition 3.2. Let t=(t1,,tn)T be such that titi+1 and let M be a finite dimensional Hn-module. Define

τi Mtgen Msitgen m ( Ti- (q-q-1)xi+1 xi+1-xi ) m.
  1. The map τi:MtgenMsitgen is well defined.
  2. As operators on Mtgen xiτi = τixi+1, if ji,i+1, xi+1τi = τixi, if ji,i+1, xjτi = τixj, if ji,i+1, τiτi = (qxi+1-q-1xi) (qxi-q-1xi+1) (xi+1-xi) (xi-xi+1) , if 1in-1, τiτj = τjτi, if i-j>1, τiτi+1τi = τi+1τi τi+1, 1in-1, whenever both sides are well defined.


(a) Note that (q-q-1)xi+1/(xi+1-xi) is not a well defined element of Hn or [x1±1,,xn±1] since it is a power series and not a Laurent polynomial. Because of this we will be careful to view (q-q-1)xi+1/(xi+1-xi) only as an operator on Mtgen. Let us describe this operator more precisely.

The element xixi+1-1 acts on Mtgen by titi+1-1 times a unipotent transformation. As an operator on Mtgen, (1-xixi+1-1) =xi+1/ (xi+1-xi) is invertible since it has determinant (1-titi+1-1)d where d=dim(Mtgen). Since this determinant is nonzero (q-q-1)xi+1/ (xi+1-xi)= (q-q-1) (1-xixi+1-1)-1 is a well defined operator on Mtgen. Thus the definition of τi makes sense.

The operator identities xiτi=τi xi+1xi+1 τi=τixi, and xjτj=τixj, if ii,i+1 now follow easily from the definition of the τi and the identities in (1.4). These identities imply that τi maps Mtgen into Msitgen.

All of the operator identities in part (b) are proved by straightforward calculations of the same flavour as the calculation of τiτi given below. We shall not give the details for the other cases. The only one which is really tedious is the calculation for the proof of τi+1τi τi+1=τi τi+1τi. For a more pleasant (but less elementary) proof of this identity see Proposition 2.7 in [Ram1998].

Since titi+1 both τi:MtgenMsitgen and τi:MsitgenMtgen are well defined. Let mMtgen. then

τiτim = ( Ti- (q-q-1)xi+1 xi+1-xi ) ( Ti- (q-q-1)xi+1 xi+1-xi ) m = ( Ti2-Ti (q-q-1)xi+1 xi+1-xi - (q-q-1)xi+1 xi+1-xi Ti+ (q-q-1)2 xi+12 (xi+1-xi)2 ) m = ( (q-q-1)Ti +1-Ti (q-q-1)xi+1 xi+1-xi -Ti (q-q-1)xi xi-xi+1 -(q-q-1)2 xi+1 xi+1-xi ( xi+1 xi+1-xi - xi xi-xi+1 ) + (q-q-1)2 xi+12 (xi+1-xi)2 ) m = ( (q-q-1)Ti +1-(q-q-1) Ti+(q-q-1)2 xixi+1 (xi+1-xi) (xi-xi+1) ) m = q2xixi+1 -2xixi+1+ q-2xixi+1 -xi2+2xi xi+1- xi+12 (xi+1-xi) (xi-xi+1) m = (qxi+1-q-1xi) (qxi-q-1xi+1) (xi+1-xi) (xi-xi+1) m.

Let wSn. Let w=si1sip be a reduced word for w and define

τw= τi1 τip. (3.3)

Since the τ-operators satisfy the braid relations the operator τw is independent of the choice of the reduced word for w. The operator τ is a well defined operator on Mtgen if t=(t1,,tn) is such that titj for all pairs i<j such that w(i)>w(j). One may use the relations in (1.5) to rewrite τw in the form

τw=uw Twauw (x1,,xn)

where auw(x1,,xn) are rational functions in the variables x1,,xn. (The functions auw(x1,,xn) are analogues of the Harish-Chandra c-function, see [Mac1971, 4.1] and [Opd1995, Theorem 5.3].) If t=(t1,,tn) is such that titj for all pairs i<j such that w(i)>w(j) then the expression

τwt= uwTw auw (t1,,tn) (3.4)

is a well defined element of the Iwahori-Hecke algebra Hn. If w=uv with (w)=(u)+(v) then

τwt= τuvt τvt. (3.5)

The following result will be crucial to the proof of Theorem 5.5. This result is due to D. Barbasch and P. Diaconis [Dia1997] (in the q=1 case). The proof given below is a q-version of a proof for the q=1 case given by S. Fomin [Fom1997].

Proposition 3.6. Let w0 be the longest element of Sn,

w0= ( 12n-1n nn-121 ) .

Let a* and fix t= ( a,aq2,aq4, ,aq2(n-1) ) . Then

τw0t= wSnTw (-q)(w0)-(w) ,

where (w0)=(n2).


Let 1kn. Then there is a vSn such that w0=vsk and (w0)=(v)+1. So

τw0=τvτk =τv ( Tk- (q-q-1)xk+1 xk+1-xk )


τw0t= τvskt ( Tk- (q-q-1)tk+1 tk+1-tk ) =τvskt ( Tk- (q-q-1)q2tk (q2-1)tk ) =τvskt (Tk-q).

Right multiplying by Tk+q-1 and using the relation (1.3) gives

τw0t (Tk+q-1)= τvskt (Tk-q) (Tk+q-1) =0.

The element h=wSnTw(-q)(w0)-(w) is a multiple of the minimal central idempotent in Hn corresponding to the representation ϕ given by ϕ(Tk)=-q-1, for all 1kn. Up to multiplication by constants, it is the unique element in Hn such that h(Tk+q-1)=0 for all 1kn. The lemma follows by noting that the coefficients of Tw0 in h and τw0t are both 1.

The action of the τ-operators on weight vectors will be particularly important to the proofs of the results in later sections. Let us record the following facts.

Let M be an Hn-module and let mt be a weight vector in M of weight t=(t1,,tn).

(3.7a) Iftiti+1 then τimt= ( Ti- (q-q-1)xi+1 xi-xi+1 ) mt= ( Ti- (q-q-1)ti+1 ti-ti+1 ) mt is a weight vector of weightsit. (3.7b) By the second set of identities in Proposition 3.2 (b), τiτimt= (qti+1-q-1ti) (qti-q-1ti+1) (ti+1-ti)-1 (ti-ti+1)-1 mt. Thus Iftiti+1 andti q±2ti+1 thenτimt0.

Notes and References

This is an excerpt of the preprint entitled Skew shape representations are irreducible authored by Arun Ram in 1998.

Research supported in part by National Science Foundation grant DMS-9622985, and a Postdoctoral Fellowship at Mathematical Sciences Research Institute.

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