RECENT RESULTS AND CONJECTURES CONCERNING QUASI-SYMMETRIC POLYNOMIALS

A very classical result, with broad implications, is that the quotient of
the ring of polynomials in $n$ variables, by the ideal generated by
constant term free symmetric polynomials, is of dimension $n$!.  We will
discuss a similar problem concerning the quotient of ring $QSym$ of
quasi-symmetric polynomials by the ideal (in $QSym$) generated by constant
term free symmetric polynomials, and extensions of all these questions to
a diagonal setup. Together with $C$.  Reutenauer, we have given a whole
set of conjectures boiling down to stating that $QSym$ is a free $Sym$
module.  This has been shown to be true by Garsia and Wallach, but we will
show that this is not the end of the question.  We will then extend this
question and others to the context of diagonally symmetric and diagonally
quasi-symmetric polynomials.