Gröbner-Shirshov bases for non-crystallographic Coxeter groups

Area of Science:

• Algebraic Combinatorics
• Group Theory
• Geometric Group Theory

Background:

• The H4 Coxeter group is a finite non-crystallographic group with significant geometric and algebraic properties.
• Understanding the structure of group algebras is crucial for studying symmetries and combinatorial objects.

Purpose of the Study:

• To construct the Gröbner-Shirshov basis for the group algebra of the H4 Coxeter group.
• To determine the standard monomials associated with this basis.
• To explicitly describe the symmetries of the 120-cell and generate its operation table.

Main Methods:

• Utilizing Gröbner-Shirshov bases theory for constructing algebraic structures.
• Applying computational methods to derive standard monomials.
• Analyzing the geometric interpretation of the algebraic results in relation to the 120-cell.

Main Results:

• A Gröbner-Shirshov basis for the H4 Coxeter group algebra has been successfully constructed.
• The corresponding standard monomials are identified, providing a basis for the algebra.
• Explicit descriptions of symmetries acting on the 120-cell are obtained.
• A natural operation table for the 14400 elements of the H4 group is generated.

Conclusions:

• The Gröbner-Shirshov basis and standard monomials offer a comprehensive algebraic framework for the H4 Coxeter group.
• This work provides new insights into the symmetries of the 120-cell and its underlying group structure.
• The generated operation table facilitates further computational and theoretical studies of this group.