An Elliptic Determinant Transformation

Determinant evaluations play an important role in mathematics, perhaps most notably in combinatorics, see Krattenthaler’s surveys [K2] and [K3]. Many useful determinant evaluations are rational identities, which rises the question of finding generalizations to the elliptic level. In recent work with Schlosser [RS2], we gave an approach to elliptic determinant evaluations that encompasses most results in the literature, from the classical Frobenius determinant to the Macdonald identities for non-exceptional affine root systems. As an example of an elliptic determinant evaluation, we mention Warnaar’s determinant [W, Corollary 5.4], which we write as