Linearizations of binary and ternary forms
A homogeneous form \(f(x_ 1,...,x_ n)\) of degree \(d\) with coefficients in a field \(F\) has a finite linearization if for some \(m\) there are \(m\times m\) matrices \(\alpha_ 1,...,\alpha_ n\) with entries in \(F\) such that \((\alpha_ 1x_ 1+...+\alpha_ nx_ n)^ d=f(x_ 1,...,x_ n)I\), where \(I\) is the identity matrix. The order \(m\) of \(I\) is called the ``rank'' of the linearization. Finite linearizations of \(f\) corresponding to matrix representations of the generalized Clifford algebra are denoted by \(C_ f\). After proofs of needed propositions, a corollary, lemmas, the author obtains two main theorems: Theorem 1. If \(f\) is a form in two variables and degree \(>3\) such that \(f\) is not divisible by a square then there are irreducible representations of \(C_ f\) of arbitrary high rank. Theorem 2. If \(f\) is a ternary form of degree three such that the projective curve \(f(X_ 1,X_ 2,X_ 3)=0\) is non-singular then \(f\) has a linearization of rank 3.
matrix representations of the generalized Clifford algebra
scientific article; zbMATH DE number 4025558