Note on the Calculation of Groebner-Shirshov Bases for Affine Weyl Groups

In this work we will consider the calculation of Groebner-Shirshov bases of Coxeter groups. This will be the main focus of the work. In \cite{Bokut-Shiao}, Bokut & Shiao gave the Groebner-Shirshov bases of positive definite classical Coxeter groups $A_l, B_l, D_l$ by using the techniques of Elimination of Leading Word. We will give a counter example to a hypothesis which is introduced by Bokut & Shiao in \cite{Bokut-Shiao} and we will calculate the Groebner-Shirshov bases of the positive degenerate infinite affine Weyl group $\widetilde{A}_n $ which is isomorphic to semi-direct product group $\Sigma_n \ltimes {\mathbb Z}^{n-1}$, and further we classify all the reduced elements of the group by using the Composition Diamond Lemma.