Shuffle algebras, lattice paths and 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.