The Saxl conjecture and the tensor square of unipotent characters of GL n (q)

We know from 9 that if for some triple of partitions (λ,μ,ν) of n the Kronecker coefficient 〈χ λ ⊗χ μ ,χ ν 〉 is non-zero then the corresponding multiplicity 〈𝒰 λ ⊗𝒰 μ ,𝒰 ν 〉 for the unipotent characters of GL n (𝔽 q ) is also non-zero. A conjecture of Saxl says that if μ is a staircase partition, then all irreducible characters of S |μ| appear non-trivially in the tensor square χ μ ⊗χ μ . Therefore the Saxl conjecture implies its analogue for unipotent characters, i.e. all unipotent characters of GL |μ| (𝔽 q ) appear non-trivially in the tensor square 𝒰 μ ⊗𝒰 μ when μ is a staircase partition. In this paper we prove the analogue of the Saxl conjecture for unipotent characters. In a second part we describe conjecturally the set of all partitions μ for which the tensor square 𝒰 μ ⊗𝒰 μ contains non-trivially all the unipotent characters of GL |μ| (𝔽 q ).