Last update: 3 March 2013
A finite dimensional is calibrated if it has a basis of simultaneous eigenvectors for the In other words, is calibrated if it has a basis such that for all in the basis and all
Weights. Let be the abelian group generated by the elements and let
The torus can be identified with by identifying the element with the homomorphism given by The symmetric group acts on by permuting coordinates and this action induces an action of on given by
with notation as in (1.2).
Weight spaces. Let be a finite dimensional For each the space of and the generalized space are the subspaces
respectively. From the definitions, and is calibrated if and only if for all If then In general but we do have
This is a decomposition of into Jordan blocks for the action of The set of weights of is the set
An element of is called a weight vector of weight
The operators. The maps defined below are local operators on in the sense that they act on each generalized weight space of seperately. The operators is only defined on the generalized weight spaces such that
Proposition 3.2. Let be such that and let be a finite dimensional Define
(a) Note that is not a well defined element of or since it is a power series and not a Laurent polynomial. Because of this we will be careful to view only as an operator on Let us describe this operator more precisely.
The element acts on by times a unipotent transformation. As an operator on is invertible since it has determinant where Since this determinant is nonzero is a well defined operator on Thus the definition of makes sense.
The operator identities and if now follow easily from the definition of the and the identities in (1.4). These identities imply that maps into
All of the operator identities in part (b) are proved by straightforward calculations of the same flavour as the calculation of 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 For a more pleasant (but less elementary) proof of this identity see Proposition 2.7 in [Ram1998].
Since both and are well defined. Let then
Let Let be a reduced word for and define
Since the satisfy the braid relations the operator is independent of the choice of the reduced word for The operator is a well defined operator on if is such that for all pairs such that One may use the relations in (1.5) to rewrite in the form
where are rational functions in the variables (The functions are analogues of the Harish-Chandra see [Mac1971, 4.1] and [Opd1995, Theorem 5.3].) If is such that for all pairs such that then the expression
is a well defined element of the Iwahori-Hecke algebra If with then
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 case). The proof given below is a of a proof for the case given by S. Fomin [Fom1997].
Proposition 3.6. Let be the longest element of
Let and fix Then
Let Then there is a such that and So
Right multiplying by and using the relation (1.3) gives
The element is a multiple of the minimal central idempotent in corresponding to the representation given by for all Up to multiplication by constants, it is the unique element in such that for all The lemma follows by noting that the coefficients of in and are both 1.
The action of the on weight vectors will be particularly important to the proofs of the results in later sections. Let us record the following facts.
Let be an and let be a weight vector in of weight
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.