Shuffle algebras, lattice paths and the commuting scheme

Abstract

The commutative trigonometric shuffle algebra A is a space of symmetric rational functions satisfying certain wheel conditions. We describe a ring isomorphism between A and the center of the Hecke algebra using a realization of the elements of A as partition functions of coloured lattice paths associated to the R-matrix of U1/2(ĝlt ∞). 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, thus providing a combinatorial formula for it as a “domain wall” type partition function of coloured lattice paths.