Separating invariants and finite reflection groups

Separating Invariants and Finite Reflection Groups

Roughly speaking, a separating algebra is a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a more geometric notion of separating algebra. This allows us to prove that when there is a polynomial separating algebra, the group is generated by reflections, and when there is a complete int...

Invariants of Euclidean Reflection Groups

Separating Invariants for Modular P -groups and Groups Acting Diagonally

We study separating algebras for rings of invariants of finite groups. We describe a separating subalgebra for invariants of p-groups in characteristic p using only transfers and norms. Also we give an explicit construction of a separating set for invariants of groups acting diagonally. Let F be an algebraically closed field and let G be a finite group. Consider a faithful representation ρ : G ...

Explicit separating invariants for cyclic P-groups

We consider a finite dimensional indecomposable modular representation of a cyclic p-group and we give a recursive description of an associated separating set: We show that a separating set for a representation can be obtained by adding to a separating set for any subrepresentation some explicitly defined invariant polynomials. Meanwhile, an explicit generating set for the invariant ring is kno...

Finite Unitary Reflection Groups

Introduction. Any finite group of linear transformations on n variables leaves invariant a positive definite Hermitian form, and can therefore be expressed, after a suitable change of variables, as a group of unitary transformations (5, p. 257). Such a group may be thought of as a group of congruent transformations, keeping the origin fixed, in a unitary space Un of n dimensions, in which the p...