Multiparameter Hall-Littlewood vertex operators

Abstract

We introduce a multiparameter generalization of Hall-Littlewood functions through the action of vertex operators. Properties of these operators, including commutation relations expressed through normal ordered products of quantum fields, are studied. We recover many important properties of the Schur functions, Schur Q-functions, and Hall-Littlewood functions at roots of unity, and we obtain analogues of some of these properties in the multiparameter case. We introduce a deformation of the KP hierarchy and prove that its linear part is integrable. In the process we show that the linear part of the deformed KP hierarchy coincides with the linear part of the usual KP hierarchy up to substitution of variables. When the vertex operators correspond to the Hall-Littlewood functions at a root of unity, we obtain a factorization of the vertex operators and compute commutation relations for the factors. The factorization suggests a possible generalization of the BKP hierarchy.