Constructing modular separating invariants (Q2267456)

Consider a finite dimensional representation \(V\) of a group \(G\) over a field \(F\). The induced action on the dual \(V^*\) extends to an action by algebra automorphisms on the symmetric algebra \(F[V]:=S(V^*)\). The elements of \(F[V]\) represent polynomial functions on \(V\). An invariant polynomial \(f\in F[V]^G\) is constant on \(G\)-orbits. For \(G\) finite, the invariant polynomials can be used to separate orbits: if \(u,v\in V\) with \(Gu\not=Gv\), then there exists \(f\in F[V]^G\) with \(f(v)\not=f(u)\). A subset \(A\subseteq F[V]^G\) is called a \textit{separating} set if for every \(u,v\in V\) with \(f(u)\not=f(v)\) for some \(f\in F[V]^G\), there exists \(h\in A\) with \(h(u)\not=h(v)\). If the characteristic of \(F\) is a prime \(p\) and \(p\) divides the order of \(G\) then \(V\) is a \textit{modular} representation of \(G\). It can be quite difficult to compute a generating set for the ring of invariants of a modular representation. For non-modular representations of finite groups, \(F[V]^G\) is generated by polynomials of degree less than of equal to \(|G|\), the so-called Noether bound. This bound does not hold for modular representations, see for example [\textit{D. R. Richman}, Adv. Math. 124, No. 1, 25--48 (1996; Zbl 0879.13004)]. However, even in the modular case there is a separating subalgebra generated in degrees less than or equal to \(|G|\), see \S 3.9.4 of [\textit{H. Derksen, G. Kemper}, Computational invariant theory. Encyclopaedia of Mathematical Sciences. Invariant Theory and Algebraic Transformation Groups. 130(1). Berlin: Springer. (2002; Zbl 1011.13003)]. In the paper under review the author constructs explicit separating sets for all modular representations of the cyclic group of order \(p\). The separating invariants are all either the orbit-product of a variable or the orbit-sum of a monomial, and all have degree \(p\) or degree \(1\).