Degree bounds for separating invariants

TL;DR

This paper investigates degree bounds for separating invariants in algebraic group representations, showing finite bounds exist for finite groups but not for infinite ones, and provides calculations for specific finite groups.

Contribution

It establishes new bounds for degrees of separating invariants for finite groups and demonstrates the non-existence of such bounds for infinite groups.

Findings

Finite groups have bounded degrees for separating invariants.

No degree bound exists for infinite algebraic groups.

Derived inequalities relating bounds for subgroups and quotient groups.

Abstract

If V is a representation of a linear algebraic group G, a set S of G-invariant regular functions on V is called separating if the following holds: If two elements v,v' from V can be separated by an invariant function, then there is an f from S such that f(v) is different from f(v'). It is known that there always exist finite separating sets. Moreover, if the group G is finite, then the invariant functions of degree <= |G| form a separating set. We show that for a non-finite linear algebraic group G such an upper bound for the degrees of a separating set does not exist. If G is finite, we define b(G) to be the minimal number d such that for every G-module V there is a separating set of degree less or equal to d. We show that for a subgroup H of G we have b(H) <= b(G) <= [G:H] b(H)$, and that b(G) <= b(G/H) b(H)$ in case H is normal. Moreover, we calculate b(G) for some specific finite groups.