The matrix elements of the $2\times 2$ fusion of Baxter's elliptic $R$-matrix, $R^{ (2,2)}(u)$, are given explicitly. Based on a note by Jimbo, we give a formula which show that $R^{ (2,2)}(u)$ is gauge equivalent to Fateev's $R$-matrix for the 21-vertex model. Then the crossing symmetry formula for $R^{ (2,2)}(u)$ is derived. We also consider the fusion of the vertex-face correspondence relation and derive a crossing symmetry relation between the fusion of the intertwining vectors and their dual vectors.
Categories: math.QA math-ph math.MP