On the Linear Combinants of a Binary Pencil

TL;DR

This paper derives explicit formulas to recover higher linear combinants of binary forms from the first two, using classical invariant theory and quantum angular momentum, with potential extensions to other groups.

Contribution

It provides explicit formulas for all higher combinants based on the first two, expanding understanding of binary form pencils and their invariants.

Findings

Explicit formulas for C_{2r-1} in terms of C_1 and C_3

Application of symbolic invariant theory and quantum angular momentum

Extension example for SL_3 group

Abstract

Let A,B denote binary forms of order d, and let C_{2r-1} = (A,B)_{2r-1} be the sequence of their linear combinants for r between 1 and (d+1)/2. It is known that C_1 and C_3 together determine the pencil generated by A and B, and hence indirectly the higher C_{2r-1}. In this paper we exhibit explicit formulae for all r>2, which allow us to recover C_{2r-1} from the knowledge of C_1 and C_3. The calculations make use of the symbolic method of classical invariant theory, as well as the quantum theory of angular momentum. Our theorem pertains to the second exterior power representation of S_d, for the group SL_2. We give an example for the group SL_3 to show that such a result may hold for other categories of representations.