The constant term algebra of type $A$: the Structure

TL;DR

This paper introduces the constant term algebra of type A, a new noncommutative algebra with a basis related to forests, arising from residue computations in Ehrhart series of the Birkhoff polytope.

Contribution

It provides the first structural characterization of this algebra, establishing an explicit basis and connecting it to combinatorial objects like forests.

Findings

Defined the constant term algebra of type A.

Established an explicit basis in terms of forests.

Linked the algebra to Ehrhart series and residue computations.

Abstract

In this paper, we discover a new noncommutative algebra. We refer this algebra as the constant term algebra of type $A$, which is generated by certain constant term operators. We characterize a structural result of this algebra by establishing an explicit basis in terms of certain forests. This algebra arises when we apply the method of the iterated Laurent series to investigate Beck and Pixton's residue computation for the Ehrhart series of the Birkhoff polytope. This algebra seems to be the first structural result in the area of the constant term world since the discovery of the Dyson constant term identity in 1962.