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

Adem Kilicman, Erol Yilmaz, Cenap Ozel

Publication date: 24 February 2010

Abstract: 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

w i d e t i l d e A_{n}

which is isomorphic to semi-direct product group

S i g m a_{n} l t i m e s {m a t h b b Z}^{n - 1}

, and further we classify all the reduced elements of the group by using the Composition Diamond Lemma.

Mathematics Subject Classification ID

Reflection and Coxeter groups (group-theoretic aspects) (20F55) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) (13P10) Loop groups and related constructions, group-theoretic treatment (22E67) Reflection groups, reflection geometries (51F15)

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