A quantization of non-commutative symmetric functions
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.