Dual involutions in finite Coxeter groups

Abstract

There is a well-known classification of conjugacy classes of involutions in finite Coxeter groups, in terms of subsets of nodes of their Coxeter graphs. In many cases, the product of an involution with the longest element $w_o$ is again an involution. We identify the conjugacy class of this product involution in terms of said classification.

Communicated by Mandi Schaeffer Fry

KEYWORDS:

• Conjugacy class
• Dynkin diagram
• Euclidean reflection group
• folding
• Weyl group

2020 Mathematics Subject Classification:

• 20F55

Acknowledgments

The author has previously discussed some of these ideas on mathoverflow. The results presented are motivated by previous results on the real K-theory of certain homogeneous spaces [1], which they make more explicit.

Notes

1 We consider root systems to be essential and reduced by definition, but not necessarily crystallographic.