Matrix invariants of binary forms
Let \(k\) be a field of characteristic 0, let \(S=k[x_1,x_2]\) be the polynomial ring in two commuting indeterminates \(x_1\), \(x_2\), and let \(S_n\) be the vector space given by the homogeneous part of \(S\) of degree \(n\). The group \(\text{SL}_2\) acts on \(S\) canonically and the \(\text{SL}_2\)-invariant polynomial maps from the space \(S_d\) to the matrix algebra \(\text{End }S_n\) form an algebra \(A_d(n)\) by matrix multiplication. \(A_d(0)\) is the algebra of invariants of the \(d\)-ic form of classical invariant theory. This paper shows that \(A_d(n)\) is a deformation of a factor of the algebra of covariants of the \(d\)-ic form. Consequently, generators for \(A_d(n)\) can be derived from the generators for the algebra of covariants.
scientific article; zbMATH DE number 808301