GROBNER-SHIRSHOV BASES OF SOME WEYL GROUPS

In this paper, we obtain Crobner-Shirshov (non-commutative) bases for the n-extended affine Weyl group (W) over tilde of type A(1), elliptic Weyl groups of types A(1)((1,1)) A(1)((1,1))* and the 2-extended affine Weyl group of type A(2)((1,1)) with a generator system of a 2-toroidal sense. It gives a new algorithm for getting normal forms of elements of these groups and hence a new algorithm for solving the word problem in these groups.