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Finite SAGBI bases for polynomial invariants of conjugates of alternating groups

It is well-known, that the ring C[X1,..., Xn]An of polynomial invariants of the alternating group An has no finite SAGBI basis with respect to the lexicographical order for any number of variables n ≥ 3. This note proves the existence of a nonsingular matrix δn ∈ GL(n, C) such that the ring of polynomial invariants C[X1,..., Xn]Anδn, where Anδn denotes the conjugate of An with respect to δn, has a finite SAGBI basis for any n ≥ 3.