ABSTRACT


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 quasisymmetric 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 quasisymmetric polynomials. Return to Applied Math Colloquium home page 