2016-12-24T18:07:49

Consider the ring of multiplicative invariants, k[A] G , of the group algebra k[A] of a faithful G-lattice A over base field k. Among other things, we will determine the cardinality of the set of all initial algebras, in (k[A] G ), of the ring of multiplicative invariants over all possible admissible orders on A.

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Let [Formula: see text] act multiplicatively on the Laurent polynomial algebra k[x± 1] in n indeterminates x={x1, …,xn}. Consider the initial algebra of the ring of invariants [Formula: see text] with respect to some monomial order. We set a sufficient condition on [Formula: see text] such that each initial algebra is represented by some weight vectors in ℝn. We also show that the condition is necessary in the case where the rank n = 2.