Certain subgroups of Weyl groups are split

TL;DR

This paper proves that for certain subgroups of finite Weyl groups, all complex representations can be realized over the rational numbers, extending to Sylow 2-subgroups, highlighting their rational representability.

Contribution

It establishes that specific centralizers and Sylow 2-subgroups within Weyl groups have all complex representations realizable over the rationals, a new insight into their structure.

Findings

Complex representations of certain centralizers are rationally realizable.

Sylow 2-subgroups of these centralizers also have rational representations.

Supports broader understanding of representation theory in Weyl groups.

Abstract

Let C be the centralizer in a finite Weyl group of an elementary abelian 2-subgroup. We show that every complex representation of C can be realized over the field of rational numbers. The same holds for a Sylow 2-subgroup of C.