2008/01/10 by Abdelmalek Abdesselam, Abdesselam, Abdelmalek, Jaydeep Chipalkatti +1 · 2 citations
Mathematics · #13A50 #22E70 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT #msc:13A50 #msc:22E70
paper · pdf · doi:10.48550/arxiv.0801.1533
LaTeX, 38 pages
arxiv created 2008/01/10 · openalex publication_date 2008/01/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A, B denote generic binary forms, and let ur = (A,B)r denote their r-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>1, is redundant in the sense that it can be completely recovered from u0 and u1. 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 S5 to show that this result has possible analogues for other categories of representations.