On the invariant theory of finite groups: Orbit polynomials and splitting principles
Let \({\mathbb{F}}_ p\) denote the field of p elements, \(V=\oplus_{n}{\mathbb{F}}_ p\) an n-dimensional \({\mathbb{F}}_ p\) vector space, and \(G<GL(V)\). The action of G on V extends to P(V) the algebra of polynomials on V (the symmetric algebra); and there is the subalgebra \(P(V)^ G\) of invariants. In this paper we address the problem of constructing generators for \(P(V)^ G\) out of the combinatorics of the action of G on the finite set V. Briefly, we associate to an orbit \(B\subset V\) of G a sequence of elements \(c_ i(B)\in P(V)^ G\), \(i=1,...,| B|\), where \(| B|\) denotes the cardinality of B. If p is sufficiently large we show that these classes suffice to generate \(P(V)^ G\) as an algebra. This result indicates the interplay of the orbit structure of V and the invariant theory of G in P(V). We introduce the subgroup of GL(V) preserving the G orbits and study its relation to G and its invariant theory in P(V). We devote particular attention to the groups of order prime to p generated by pseudoreflections. These are the groups with \(P(V)^ G\) a polynomial algebra and of order relatively prime to p. We show that the introduction of one further invariant associated to an orbit, the Euler class, suffices in all but finitely many cases to construct polynomial generators for \(P(V)^ G\).
scientific article; zbMATH DE number 4063384
scientific article