FINITE SAGBI BASES FOR POLYNOMIAL INVARIANTS OF CONJUGATES OF ALTERNATING GROUPS

Abstract. 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] Aδn n,whereA δn n denotes the conjugate of An with respect to δn, has a finite SAGBI basis for any n ≥ 3. 1