Rings of polynomial invariants of the alternating group have no finite SAGBI bases with respect to any admissible order

Abstract

It is well known, that the invariant ring C[X 1,X 2,X 3] A 3 of the alternating group A 3 is the “smallest” ring of polynomial invariants of a permutation group with respect to the number of variables and the number of generators, which has no finite SAGBI basis with respect to any admissible order. We show in this note that for any number of variables n≥3 the invariant ring C[X 1,…,X n] A n has no finite SAGBI basis with respect to any admissible order.