Gröbner-Shirshov bases for the Coxeter groups of the types $G_2,F_4,E_6,E_7$ and $E_8$

Abstract

The author is mainly interest in the Gröbner-Shirshov bases of finite Coxeter groups. It is known that the finite Coxeter groups are classified in terms of Coxeter-Dynkin diagrams. Under the fixed order, it is worth mention that the presentation of group determines the Gröbner-Shirshov bases of group. In this paper, the author rearranges the generators, marks them on Coxeter-Dynkin diagrams, and gets a simple presentation of the Gröbner-Shirshov bases for Coxeter groups of types $G_2,F_4,E_6$ and $E_7$. This article also gives the Gröbner-Shirshov basis of Coxeter group of type $E_8$.

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