## A quantization of non-commutative symmetric functions

**Mike Zabrocki**

(York University)

The non-commutative and quasi symmetric functions are dual Hopf

algebras that share many of the same properties and are strongly

related to the space of symmetric functions. The quasi-symmetric

functions are attributed to Gessel in 1983 and the non-commutative

symmetric functions have their origins in the early 90's, and hence

are fairly new by comparison to the space of symmetric functions.

Much of the theory that is well understood for the

symmetric functions has yet to be generalized to this pair of

algebras and some interesting questions arise in developing analogs

of constructions that are well known for the symmetric functions.

Much of the recent research in symmetric functions has been on the

properties of two remarkable bases and their generalizations, the

Hall-Littlewood and Macdonald symmetric functions. These bases depend

on a parameter q and by specializing the parameter to various

values they interpolate many of the well known bases of the symmetric

functions. One reason they are of interest is that algebraic identities

involving these functions often encode several well known identities

in the space of symmetric functions at the same time. We define a possible

analog to the Macdonald and Hall-Littlewood bases in the non-commutative

symmetric functions that arises by abstracting a formula for the Hall-Littlewood

functions to the level of Hopf algebras and then demonstrate some of the

surprising properties held by these functions.

This is joint work with Nantel Bergeron.