Shuffle algebras, lattice paths and the commuting scheme

Abstract

The commutative trigonometric shuffle algebra A \mathrm {A} is a space of symmetric rational functions satisfying certain wheel conditions. We describe a ring isomorphism between A \mathrm {A} and the center of the Hecke algebra using a realization of the elements of A \mathrm {A} as partition functions of coloured lattice paths associated to the R R -matrix of U t 1 / 2 ( g l ^ ∞ ) \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 A \mathrm {A} , thus providing a combinatorial formula for it as a “domain wall” type partition function of coloured lattice paths.