Adjacency for special representations of a Weyl group

TL;DR

This paper explores the structure of special representations of Weyl groups by representing them as vertices in a graph with an involution, revealing relationships based on a-functions differing by one.

Contribution

It introduces a novel graph-theoretic framework for understanding special representations of Weyl groups and their involution properties.

Findings

Vertices form a graph with specific involution properties.

Edges connect vertices with a-functions differing by 1.

Provides new insights into the structure of special representations.

Abstract

The special representations of a Weyl group can be regarded as the vertices of a graph with an involution i such that any edge e has the following property: either e or i(e) joins two vertices whose a-functions differ by 1.