# Macdonald trees and determinants of representations for finite Coxeter groups

## TL;DR

This paper explores the structure of odd-dimensional irreducible representations of symmetric and hyperoctahedral groups, introducing the Macdonald tree, and investigates the relationship between odd and chiral partitions across Coxeter groups.

## Contribution

It describes the Macdonald tree structure for hyperoctahedral groups and extends previous symmetric group results to all Coxeter groups, linking odd and chiral partitions.

## Key findings

- The Macdonald tree is a binary recursive structure for odd-dimensional representations.
- The number of odd and chiral partitions are closely correlated.
- Extensions of the structure to hyperoctahedral groups and all Coxeter groups are provided.

## Abstract

Every irreducible odd dimensional representation of the $n$'th symmetric or hyperoctahedral group, when restricted to the $(n-1)$'th, has a unique irreducible odd-dimensional constituent. Furthermore, the subgraph induced by odd-dimensional representations in the Bratteli diagram of symmetric and hyperoctahedral groups is a binary tree with a simple recursive description. We survey the description of this tree, known as the Macdonald tree, for symmetric groups, from our earlier work. We describe analogous results for hyperoctahedral groups. A partition $\lambda$ of $n$ is said to be chiral if the corresponding irreducible representation $V_\lambda$ of $S_n$ has non-trivial determinant. We review our previous results on the structure and enumeration of chiral partitions, and subsequent extension to all Coxeter groups by Ghosh and Spallone. Finally we show that the numbers of odd and chiral partitions track each other closely.