Multispecies TASEP and the tetrahedron equation

We introduce a family of layer to layer transfer matrices in a three-dimensional (3D) lattice model which can be viewed as partition functions of the $q$-oscillator valued six-vertex model on $m\times n$ square lattice. By invoking the tetrahedron equation we establish their commutativity and bilinear relations mixing various boundary conditions. At $q=0$ and $m=n$, they ultimately yield a new proof of the steady state formula for the $n$-species totally asymmetric simple exclusion process (TASEP) obtained recently by the authors, revealing the 3D integrability in the matrix product construction.