The higher transvectants are redundant

Abstract

Let A, B denote generic binary forms, and let ur = (A, B) r denote their τ-th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the {ur}. As a consequence, we show that each of the higher transvectants {ur: r ≥ 2} is redundant in the sense that it can be completely recovered from u0 and 1. This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of SL2- representations, and the notion of a 9-j symbol from the quantum theory of angular momentum. We give explicit computational examples for SL3, g2 and G5 to show that this result has possible analogues for other categories of representations.