A statistic on the roots of a finite reflection group and a correspondence between the height function and Bruhat order

TL;DR

This paper explores the roots of finite reflection groups, establishing a new representation-based statistic and suggesting potential applications to Costas Arrays, thereby linking algebraic structures with combinatorial designs.

Contribution

It introduces a novel induced representation of the root action and defines a new root statistic related to the length function, connecting algebraic and combinatorial concepts.

Findings

Representation of roots as induced from parabolic subgroups

Definition of a new root statistic based on the length function

Potential application to Costas Arrays

Abstract

The action of a finite reflection group (type A) on its set of roots is understood as a permutation representation or group action. We show that this representation is an induced representation from a certain kind of parabolic subgroup. Furthermore, we use this representation to define a statistic (derived from the length function) on the set of roots. A possible application to Costas Arrays is hinted at in a proposition.