SAGBI bases for rings of invariant Laurent polynomials

Abstract

Let k be a field, let L n = k[x ±1 1 , x ±1 n ] be the Laurent polynomial ring in n variables and let G be a finite group of k-algebra automorphisms of L n . We give a necessary and sufficient condition for the ring of invariants L G n to have a SAGBI basis. We show that if this condition is satisfied, then L G n has a SAGBI basis relative to any choice of coordinates in L n and any term order.