On Monomial Orders And Ring Of Multiplicative Invariants

MONOMIAL ORDERS AND RING OF MULTIPLICATIVE INVARIANTSrnMULUGETA HABTE MrnAddis Ababa University, 2018rnFor a finite group G in GL(n,Z) _=rnAut(Zn), action of G on Zn (via automorphism) canrnbe uniquely extended to group algebra of Zn i.e. to Laurent polynomial ring K[X_1]rnover some base field K. M. Lorenz in Lorenz (2001) showed that the invariant algebrarnK[X_1]G = R of this multiplicative action has a form of affine semigroup algebraK[M],rnprovided the group acting is a reflection group. Further he conjured that, if R = K[M]rnan affine semigroup algebra, then must G act as a reflection group? Lorenz (2005). Few partialrnanswer was given, using different restriction and approach. In this dissertation wernshowed that the above conjuncture holds, provided G is taken from a class that satisfiesrncertain linearization conditions. We used monomial ordering of Zn intensivelyrnin relation to the initial algebra of invariant rings. Furthermore M. Tesemma and H.rnWang in Tesemma and Wang (2011), described the initial algebra of invariant rings forrnarbitrary lattice (monomial) ordering can be represented using an archimedean order,rngiving same initial algebra hence SAGBI, provided the invariant algebra is induced byrnaction of reflection group. Further confirmed that for the usual lex ordering all the nonrnreflection groups (4 groups upto conjugacy) in GL(2,Z) do not admit such representationrnof initial algebra. We further show that, such representation (via archimedeanrnorder) of initial algebra for multiplicative invariants of any monomial order is possiblernif and only if the invariant algebra are induced from action of reflection group.

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