Ideals and quotients of diagonally quasi-symmetric functions

TL;DR

This paper constructs a Hilbert basis for the quotient of the algebra of diagonally quasi-symmetric functions in infinitely many variables, providing new insights into its structure and specific Hilbert matrix entries.

Contribution

It introduces a Hilbert basis for the quotient algebra of diagonally quasi-symmetric functions with infinitely many variables, advancing understanding of its algebraic structure.

Findings

Constructed a Hilbert basis for the quotient algebra with infinitely many variables.

Applied the basis construction to finitely many variables case.

Computed the second column of the Hilbert matrix for the finite variable case.

Abstract

In 2004, J-C. Aval, F. Bergeron and N. Bergeron studied the algebra of diagonally quasi-symmetric functions $\operatorname{\mathsf{DQSym}}$ in the ring $\mathbb{Q}[\mathbf{x},\mathbf{y}]$ with two sets of variables. They made conjectures on the structure of the quotient $\mathbb{Q}[\mathbf{x},\mathbf{y}]/\langle\operatorname{\mathsf{DQSym}}^+\rangle$, which is a quasi-symmetric analogue of the diagonal harmonic polynomials. In this paper, we construct a Hilbert basis for this quotient when there are infinitely many variables i.e. $\mathbf{x}=x_1,x_2,\dots$ and $\mathbf{y}=y_1,y_2,\dots$. Then we apply this construction to the case where there are finitely many variables, and compute the second column of its Hilbert matrix.