Separating invariants for modular \(p\)-groups and groups acting diagonally
Let \(k\) be a field and \(G\) be a finite group. Suppose that \(G\) acts on \(V\), a finite-dimensional vector space over \(k\). It is well-known that the quotient space \(V/G\) (i.e. the set of the \(G\)-orbit of \(V\)) is an affine variety with coordinate ring \(k[V]^G\), the ring of invariants of \(k[V]\) under the induced action of \(G\). The drawback is that it is not easy to find \(k[V]^G\) explicitly and effectively, in particular, when \(\text{char } k=p > 0\) and the group \(G\) is a \(p\)-group. The notion of separating invariants is invented to remedy this inadequacy (with the price of sacrificing some geometric messages, of course). A finitely generated graded subalgebra \(A\) of \(k[V]^G\) is called a separating algebra if, for any distinct points \([v_1], [v_2] \in V/G\), there is an element \(f \in A\) separating them, i.e. \(f(v_1) \not= f(v_2)\). The goals are to show that (i) separating algebras are still useful in detecting \(V/G\) and \(k[V]^G\), (ii) it is not difficult to find separating algebras effectively in most conceivable cases. The paper under review consists of two part. The first part assumes that \(\text{char }k=p > 0\) and \(G\) is a \(p\)-group. The second part considers the case when \(G\) is a finite abelian group, \(\mathrm{gcd } \text{char } k, |G|\}=1\), and the field \(k\) is algebraically closed. The main result of the first part (Theorem 3) is to show that the norms and the transfers are sufficient to form a separating algebra. The main result of the second part (Proposition 5) provides a construction of a separating algebra consisting of monomials.
scientific article; zbMATH DE number 5707702