Shuffle algebras, lattice paths and the commuting scheme

TL;DR

This paper establishes an isomorphism between a shuffle algebra and the center of the Hecke algebra, using lattice path models, and applies this to compute the Hilbert series of the commuting scheme with combinatorial methods.

Contribution

It introduces a novel realization of the shuffle algebra as lattice path partition functions and links it to the center of the Hecke algebra, enabling new combinatorial computations.

Findings

Ring isomorphism between shuffle algebra and Hecke algebra center

Representation of algebra elements as lattice path partition functions

Explicit combinatorial formula for the Hilbert series of the commuting scheme

Abstract

The commutative trigonometric shuffle algebra ${\mathrm A}$ is a space of symmetric rational functions satisfying certain wheel conditions. We describe a ring isomorphism between ${\mathrm A}$ and the center of the Hecke algebra using a realization of the elements of ${\mathrm A}$ as partition functions of coloured lattice paths associated to the $R$-matrix of $\mathcal U_{t^{1/2}}(\widehat{gl}_{\infty})$. As an application, we compute under certain conditions the Hilbert series of the commuting scheme and identify it with a particular element of the shuffle algebra ${\mathrm A}$, thus providing a combinatorial formula for it as a "domain wall" type partition function of coloured lattice paths.