Integer Representations of Classical Weyl Groups

TL;DR

This paper introduces a novel mixed-base number system for classical Weyl groups, specifically type D, establishing a bijection between positive integers and group elements, thereby completing integer representations for all classical Weyl groups.

Contribution

It develops a new mixed-base number system and a bijective correspondence for Weyl groups of type D, extending integer representations to all classical Weyl groups.

Findings

Established a one-to-one correspondence between positive integers and type D Weyl group elements.

Constructed a subexceedant function linking integers to Weyl group elements.

Completed the integer representation framework for all classical Weyl groups.

Abstract

In this paper, we define a mixed-base number system over a Weyl group of type $D$, the group even-signed permutations. We introduce one-to-one correspondence between positive integers and elements of Weyl groups of type $D$ after constructed the subexceedant function associating to the group. Thus, the integer representations of all classical Weyl groups are now completed.