Fusion of Baxter’s Elliptic R-Matrix and the Vertex-Face Correspondence

The matrix elements of the 2 × 2 fusion of Baxter’s elliptic R-matrix, R^(2,2)(u), are given explicitly. Then a gauge equivalence between R^(2,2)(u) and Fateev’s R-matrix for the 21-vertex model is shown. This part is based on an unpublished note by Jimbo. We then derive the crossing symmetry formula for R^(2,2)(u). 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. Communicated by Vincent Rivasseau To the memory of Daniel Arnaudon Submitted: January 27, 2006; Accepted: April 30, 2006