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Journal of Lie Theory 17 (2007), No. 4, 731--750
Copyright Heldermann Verlag 2007

Lie Group Invariants of Inhomogeneous Polynomial Vector Spaces

Joshua T. Horwood
Dept. of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, England

We present a method which efficiently generates Lie group invariants in the classical invariant theory of polynomials and its extensions to vector spaces of inhomogeneous polynomials under the actions of the general affine group and pseudo-Euclidean subgroups. Our derivation of the invariants uses the classical Cartan method of moving frames and requires no assumption on the degree of the polynomial or the number of variables. Consequently, we are able to express the invariants in a compact indicial notation. We employ our results to solve the equivalence and canonical forms problems for the vector space of inhomogeneous cubic polynomials in two real variables under the action of the Euclidean group. We show that the space partitions into twelve distinct classes of canonical forms, each admitting a system of invariants which globally separates its associated orbits.

Keywords: Classical invariant theory, Cartan geometry, invariants, inhomogeneous polynomials, cubic polynomials.

MSC: 16W22, 20C33, 53A45

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