Quadratic Involutions on Binary Forms

TL;DR

This paper characterizes a broad class of involutions on binary forms defined via quadratic forms and transvectants, explores their invariant varieties, and introduces a new recoupling formula for transvectants.

Contribution

It provides a complete characterization of involutions on binary forms using compound transvectant formulas and studies their invariant varieties, including a new recoupling formula for transvectants.

Findings

Characterization of involutions using transvectant formulas

Identification of varieties preserved by these involutions

Development of a recoupling formula for transvectants

Abstract

There is a classical geometric construction which uses a binary quadratic form to define an involution on the space of binary d-ics. We give a complete characterization of a general class of such involutions which are definable using compound transvectant formulae. We also study the associated varieties of forms which are preserved by such involutions. Along the way we prove a recoupling formula for transvectants, which is used to deduce a system of equations satisfied by the coefficients in these involutions.