Last update: 11 September 2013
In this section we introduce a formalism for which allows us to work in the universe of identities for Weyl group symmetric functions.
We return to the setup of section 2. Let be the set of dominant integral weights, the Weyl characters and the monomial symmetric functions (2.7). Let be a subset of
Let be a vector space of dimension and let be linearly independent functions on Let and, for each define by where denotes taking the coefficient of in Let be the set of such that is not identically Define and define a pairing between and by defining
Suppose that is finite and let and be bases of and indexed by the elements of Then the following conditions are equivalent.
The proof is exactly as in [Mac1979] Chapt. I (4.6) p.34.
|1)||"Jacobi-Trudi" identity. For each|
|2)||Raising operator identity. Allow the raising operators (2.4) to act upon the by defining far each sequence Then for all|
|4)||For all and all|
|5)||If is finite and then and are bases of and respectively and with respect to the pairing between and defined in (5.1)|
1) follows exactly as in (4.6).
2) follows from 1) in the same fashion as in the proof of Theorem (2.15).
By definition By (2.10), giving 3).
4) follows from 3) and the definition of the monomial symmetric functions (using the fact (2.3) that the Weyl group orbits of elements in are all distinct).
It follows from 1) that is a finite linear combination of Thus the span Since the form a basis of That now follows from (5.2), completing the proof of 5).
1. Let and be commuting variables. Let denote the set of partitions of length be a partition and let and be the Schur functions associated to in the and the variables respectively. Consider the identity Then, for each define Then where for each integer We may define a pairing between symmetric functions in the variables and symmetric functions in the variables by defining for all Then for all where denotes the monomial symmetric function. For each where
2. Keeping the same notation as in example 1, let be the set of partitions such that and such that the conjugate partition Consider the identity Then, for each define Then where for each integer Define a pairing between the symmetric functions of degree in the variables and the symmetric functions of degree in the variables by defining for all Then we have that for all and that
3. Keeping the same notation as in example set and let be the set of partitions such that and such that i.e., Consider the identity where denotes the power symmetric function corresponding to the partition and is the irreducible character of the symmetric group Then for define Then is the character of determined by inducing the trivial character of the Young subgroup to Define a pairing between the symmetric functions in the variables and characters of the symmetric group by defining for all Then we have that for all and that for all
4. Let and be commuting variables let and let be the hyperoctahedral group. For each define Then consider the identity where denotes the Schur function in the variables. For each define Define a bilinear pairing between functions in the symmetric under the hyperoctahedral group and classical symmetric functions in the variables by for all Then we have that for all and that where is defined by
5. In a similar fashion the identity where the set is as in example 2, gives the formula where is defined by
6. Let and be two sets of variables and let and be defined as in example 5. For a partition we shall write if is of length and the conjugate partition is of length Given a partition and define the opposite partition by The Morris identity gives the formula where is defined by and
Remarks. Notice that the only case in the above examples where the "Jacobi-Trudi" formula is an alternating sum over a Weyl group other than the symmetric group is in example 6. The "Jacobi-Trudi" identity in example 6 can be obtained easily from the "Jacobi-Trudi" identity of example 5 and is thus given in [KTe1987]. The idea of deriving this identity from the Morris identity is, I believe, new.
The "Jacobi-Trudi" formulas given in examples 4 and 5 are analoguous to the classical ones given in examples 1 and 2. There are known analogues of the formulas in the above examples for the Weyl characters corresponding to all types A,B,C,D. The appropriate bicharacter identities and the corresponding "Jacobi-Trudi" formulas for these analogues as well as the above examples can be found with proofs in the forthcoming paper [RamSTLD]. Many of these results have been in the literature for a long time.
This is an excerpt of a paper entitled Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras, written by Arun Ram.