Putting the squeeze on the Noether gap. -- The case of the alternating groups \(A_n\)
Let \(\rho : G \hookrightarrow \text{GL}(n, \mathbb F)\) be a faithful representation of the finite group \(G\) over the field \(\mathbb F\) of characteristic \(p\). In 1916, E. Noether showed that if \(p=0\), then the ring of invariants \(\mathbb F[V]^G\) is generated as an algebra by the invariant polynomials of degree at most \(|G|\). This upper bound on the degrees of the generators in a minimal generating set is known as Noether's bound. For \(p>0\) the representation \(\rho\) is called modular if \(p\) divides \(|G|\) and non-modular otherwise. It is known that Noether's bound holds when \(p> |G|\) and when \(\rho\) is a non-modular representation of a solvable group. The bound may fail in the modular case, but for non-modular representations with \(p<|G|\) it is not known whether or not Noether's bound holds. This collection of primes is referred to as the Noether gap in the title. The paper under review establishes (i) if Noether's bound fails in the non-modular case, then there is a finite non-abelian simple group for which it fails, and (ii) Noether's bound holds for the alternating group on \(n\) elements in the non-modular case.
scientific article; zbMATH DE number 1384378