Last update: 3 March 2013
Restriction to Let be the subalgebra of generated by The elements form a basis of Since is not a root of unity the subalgebra of is semisimple. The irreducible representations of are indexed by the partitions and these representations are of the irreducible representations of the symmetric group The following result describes the decomposition of the restriction to of the irreducible
Theorem 6.1. Let be the irreducible representation of the affine Hecke algebra which is defined in Theorem 4.1. Then
where is the classical Littlewood-Richardson coefficient and is the irreducible indexed by the partition
If is a composition of let where (in cycle notation). Let be the trace of the action of the element on the With notations as in Theorem 4.1
and one can copy (without change) the proof of Theorem 2.20 in [HRa1996] and obtain
where the sum is over all sequences such that and the factor is given by
In the formula for a border strip is a skew shape with at most one box in each diagonal, is the set of connected components of is the number of connected components of is the number of rows of and is the number of columns of
Let denote the Schur function (see [Mac1995]) and define
By Proposition 6.11(a) in [HRa1995],
Letting one can inductively apply this formula to obtain
Thus, with notations as in [Mac1995] Chapt. I,
where denotes the irreducible character of evaluated at the element The result follows since, by [Ram1991-2] Theorem 5.1, the characters of are determined by their values on the elements
Classically, the Littlewood-Richardson coefficients describe
Theorem 6.2 gives an exciting new way of interpreting these coefficients. They describe
Induction from “Young subalgebras”. Let and be such that Let
In this way is naturally a subalgebra of
Let be a placed skew shape with boxes and let be a placed skew shape with boxes. Number the boxes of with (as in Section 2, along diagonals from southwest to northeast) and number the boxes of with in order to match the imbeddings of and in Let and be the corresponding representations of and as defined in Theorem 4.1.
Let (resp. be the skew shape obtained by placing and adjacent to each other in such a way that of is immediately above (resp. to the left of) of Let be the content function given by
Theorem 6.2. With notations as above,
in the Grothendieck ring of finite dimensional representations of
Let be the set of minimal length coset representatives of the cosets of in The module has basis
By repeatedly applying the relations (1.4) we obtain
for some constants From this we can see that the action of on is an upper triangular matrix with eigenvalues where runs over the standard tableaux of shapes and It follows that has
distinct weights. Since this number is exactly the dimension of it follows that every generalized weight space of is one dimensional and thus that M is calibrated. By Theorem 4.1, all irreducible calibrated representations are of the form for some placed skew shape and by Lemma 2.2 this placed skew shape is completely determined by any one of the weights of the module Thus, our analysis of the weights of implies that both and are composition factors of The result follows since
A ribbon is a skew shape which has at most one box in each diagonal.
Corollary 6.3. Let be the content function given by for Let and let be the one dimensional module for given by
In the Grothendieck ring of finite dimensional
where the sum is over all connected ribbons with boxes.
Since this result can be obtained by repeatedly applying Theorem 6.2.
Theorem 6.2 and Corollary 6.3 are realizations of the Schur function identities in [Mac1995] I §5 Ex 21 (a),(b). The module is a principal series module for (see [Kat1981]). The identity in Corollary 6.3 describes the composition series of this principal series module. Using the methods of [Ram1998] and [Ram1998-2, (1.2) Ex. 2] one can obtain a generalization of this identity (and thus of [Mac1995] I §5 Ex 21 (b)) which holds for affine Hecke algebras of arbitrary Lie type.
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.