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

Shuffle algebras, lattice paths and the commuting scheme