Elements with only negative cycles in Weyl groups of type $B$ and $D$

TL;DR

This paper provides a precise formula and an asymptotic estimate for the proportion of elements with exclusively negative or positive cycles in Weyl groups of types B, C, and D, enhancing understanding of their structure.

Contribution

It introduces a new exact formula and asymptotic analysis for elements with only negative or positive cycles in Weyl groups of types B, C, and D.

Findings

Derived a precise formula for the proportion of such elements.

Established a simple asymptotic function for large group sizes.

Applied results to understand cycle structure in Weyl groups.

Abstract

We give a precise formula and a simple asymptotic function for the proportion of elements with only negative (or only positive) cycles in Weyl groups of type $B,C$ and $D$.