Matrix product operator representations for the local conserved quantities of the spin-$1/2$ XYZ chain

We present explicit matrix product operator (MPO) representations for the local conserved quantities of the spin-$1/2$ XYZ chain. Through these MPO representations, we simplify the coefficients appearing in the local conserved quantities originally derived by one of the authors, and reveal their combinatorial meaning: the coefficients prove to be a polynomial generalization of the Catalan numbers, given by weighted sums over monotone lattice paths. We also prove that a single recurrence relation for the coefficients guarantees the conservation law. We further show that these MPO representations can also be recovered from a product of two eight-vertex transfer matrices built from Baxter's R-matrix, providing the first direct connection between the R-matrix formalism and closed-form expressions for all local conserved quantities. We also obtain a simple Lax operator for the XYZ chain that, unlike Baxter's R-matrix, involves no elliptic functions.