Finite SAGBI bases for polynomial invariants of conjugates of alternating groups

Abstract:

It is well-known, that the ring $\mathbb {C}[X_1,\dotsc ,X_n]^{A_n}$ 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 $n \ge 3$. This note proves the existence of a nonsingular matrix $\delta _n \in GL(n,\mathbb {C})$ such that the ring of polynomial invariants $\mathbb {C}[X_1,\dotsc ,X_n]^{A_n^{\delta _n}}$, where $A_n^{\delta _n}$ denotes the conjugate of $A_n$ with respect to $\delta _n$, has a finite SAGBI basis for any $n \geq 3$.