Schur functions and affine Lie algebras (Q1276980)

It is known that irreducible highest weight representations of \(a_{\infty}\), an infinite rank affine Lie algebra, can be realized on the space \(Sym\) of symmetric functions in countably many variables. A canonical basis of weight vectors of \(Sym\) is given by Schur's S-functions. Analogously, the infinite rank affine Lie algebra \(b_{\infty}\), an infinite-dimensional analogue of the Lie algebra \(so_{2n+1}\), has representations contained again in \(Sym\) and a canonical basis of weight vectors given by Schur's P-functions. The authors obtain an explicit intertwining operator which allows them to compute relations between S-functions and P-functions. They also show how to realize the representations of the affine Lie algebra \(A_1^{(1)}\) both in the homogeneous and in the principal picture on \(Sym\) and how to use this situation to get new and different relations between S-functions and P-functions.