### Abstract:

MONOMIAL ORDERS AND RING OF MULTIPLICATIVE INVARIANTS
MULUGETA HABTE M
Addis Ababa University, 2018
For a finite group G in GL(n,Z) _=
Aut(Zn), action of G on Zn (via automorphism) can
be uniquely extended to group algebra of Zn i.e. to Laurent polynomial ring K[X_1]
over some base field K. M. Lorenz in Lorenz (2001) showed that the invariant algebra
K[X_1]G = R of this multiplicative action has a form of affine semigroup algebraK[M],
provided the group acting is a reflection group. Further he conjured that, if R = K[M]
an affine semigroup algebra, then must G act as a reflection group? Lorenz (2005). Few partial
answer was given, using different restriction and approach. In this dissertation we
showed that the above conjuncture holds, provided G is taken from a class that satisfies
certain linearization conditions. We used monomial ordering of Zn intensively
in relation to the initial algebra of invariant rings. Furthermore M. Tesemma and H.
Wang in Tesemma and Wang (2011), described the initial algebra of invariant rings for
arbitrary lattice (monomial) ordering can be represented using an archimedean order,
giving same initial algebra hence SAGBI, provided the invariant algebra is induced by
action of reflection group. Further confirmed that for the usual lex ordering all the non
reflection groups (4 groups upto conjugacy) in GL(2,Z) do not admit such representation
of initial algebra. We further show that, such representation (via archimedean
order) of initial algebra for multiplicative invariants of any monomial order is possible
if and only if the invariant algebra are induced from action of reflection group.