Spin Representations of Finite Coxeter Groups and Generalisations of Saxl's Conjecture

TL;DR

This paper generalizes Saxl's conjecture to finite Coxeter groups, utilizing spin representations to connect Lie theory and symmetric group tensor decompositions, and verifies the conjecture for exceptional and non-crystallographic types.

Contribution

It introduces a uniform spin representation approach to generalize Saxl's conjecture across finite Coxeter groups, including non-crystallographic cases.

Findings

Verified the conjecture for exceptional types

Connected spin representations to tensor product decompositions

Provided an alternative description of the cuspidal family

Abstract

This paper presents a natural generalisation of Saxl conjecture from a Lie-theoretical perspective, which is verified for the exceptional types. For classical types, progress is made using spin representations, revealing connections to certain tensor product decomposition problems in symmetric groups. We provide an alternative uniform description of the cuspidal family (in the sense of Lusztig) through spin representations, offering an equivalent conjecture formulation. Additionally, we generalise Saxl conjecture to finite Coxeter groups and prove it for the non-crystallographic cases.