q-Analogs of symmetric function operators

For any homomorphism V on the space of symmetric functions, we introduce an operation which creates a q-analog of V. By giving several examples we demonstrate that this quantization occurs naturally within the theory of symmetric functions. In particular, we show that the Hall-Littlewood symmetric functions are formed by taking this q-analog of the Schur symmetric functions and the Macdonald symmetric functions appear by taking the q-analog of the Hall-Littlewood symmetric functions in the parameter t. This relation is then used to derive recurrences on the Macdonald q,t-Kostka coefficients.Comment: 17 pages - minor revisions to appear in Discrete Mathematics issue for LaCIM'200

q and q,t-Analogs of Non-commutative Symmetric Functions

We introduce two families of non-commutative symmetric functions that have analogous properties to the Hall-Littlewood and Macdonald symmetric functions.Comment: Different from analogues in math.CO/0106191 - v2: 26 pages - added a definition in terms of triangularity/scalar product relations - to be submitted FPSAC'0

Hall-Littlewood vertex operators and generalized Kostka polynomials

A family of vertex operators that generalizes those given by Jing for the Hall-Littlewood symmetric functions is presented. These operators produce symmetric functions related to the Poincare polynomials referred to as generalized Kostka polynomials in the same way that Jing's operator produces symmetric functions related to Kostka-Foulkes polynomials. These operators are then used to derive commutation relations and new relations involving the generalized Kostka coefficients. Such relations may be interpreted as identities in the (GL(n) x C^*)-equivariant K-theory of the nullcone.Comment: 17 page