On alternating invariants and Hilbert ideals

In a paper on F-rationality [J. Algebra 176 (1995) 824–860] Donna Glassbrenner showed that over a field of odd characteristic p the Hilbert ideals of the tautological representations of the symmetric group Σn and alternating group An coincide if . She asked if this was always the situation in the modular case. We answer this in the affirmative using Macaulayʹs theory of irreducible ideals in polynomial algebras: a somewhat forgotten bywater of commutative algebra. As a bonus, the method yields applications back to the original question of F-rationality studied in [J. Algebra 176 (1995) 824–860].