  We identify the algorithm for constructing steady states of the $n$-species
totally asymmetric simple exclusion process (TASEP) on $L$ site periodic chain
by Ferrari and Martin with a composition of combinatorial $R$ for the quantum
affine algebra $U_q(\widehat{sl}_L)$ in crystal base theory. Based on this
connection and the factorized form of the $R$ matrix derived recently from the
tetrahedron equation, we establish a new matrix product formula for the steady
state of the TASEP which is expressed in terms of corner transfer matrices of
the $q$-oscillator valued five-vertex model at $q=0$.
