The q-deformation of symmetric functions and the symmetric group

The q-deformation of symmetric functions is introduced leading to q-analogues of many well known relationships in the theory of symmetric functions. q-deformed scalar products are developed and used to define q-dependent symmetric functions. The symmetric functions commonly associated with the names Hall-Littlewood, Schur and Jack are all special cases of the q-deformation of Macdonald's (1988) new symmetric functions Plambda (s,t). A q-analogue of the spin and ordinary characters of Sn is given and illustrated by the explicit calculation of examples of q-deformed characters. The methods used are closely parallel to those of quantum groups.