Finite SAGBI basis for polynomial invariants of conjugates of alternating groups

It is well-known, that the ring C [X-1,...,X-n](An) of polynomial invariants of the alternating group A(n) has no finite SAGBI basis with respect to the lexicographical order for any number of variables n greater than or equal to 3. This note proves the existence of a nonsingular matrix delta(n) is an element of GL(n, C) such that the ring of polynomial invariants C [X-1,...,X-n]A(n)(deltan), where A(n)(deltan) denotes the conjugate of A(n) with respect to delta(n), has a finite SAGBI basis for any n greater than or equal to 3.