Affine Weyl grupları için Gröbner-Shirshov tabanları ve indirgenmiş formları

Gröbner-Shirshov basis and reduced forms for the elements in the affine Weyl group of type A_{n} are presented. Starting from the standard defining relations R of A_{n}, a new set of relations R' is founded. Then, by using the algorithm of elimination of leading words with respect to the relations in R', all the forms in the ane Weyl group A_{n} are reduced to an explicit list of forms. Using combinatorial techniques, the number of forms in such a list is found to be equal to the number of elements in A_{n}. Therefore, by the Composition-Diamond Lemma, the relations in R' are Gröbner-Shirshov basis for the affine Weyl group A_{n}.Keywords: A ne Weyl Groups, Composition-Diamond Lemma, Grobner-Shirshov ¨Basis, q-binomials and Basic Partitions.