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no. 4
The Saxl conjecture and the tensor square of unipotent characters of $\mathrm{GL}_n(q)$
Letellier, Emmanuel1; Nam, GyeongHyeon2
1Université Paris Cité IMJ-PRG CNRS 45 Rue des Saints-Pères Paris 75006 (France)
2Ajou University Department of Mathematics 206 World cup-ro Suwon 16499 (Republic of Korea)
Algebraic Combinatorics, Volume 8 (2025) no. 4, pp. 1119-1140

We know from [9] that if for some triple of partitions $(\lambda ,\mu ,\nu )$ of $n$ the Kronecker coefficient $\langle \chi ^\lambda \otimes \chi ^\mu ,\chi ^\nu \rangle $ is non-zero then the corresponding multiplicity $\langle {\mathcal{U}}^\lambda \otimes {\mathcal{U}}^\mu ,{\mathcal{U}}^\nu \rangle $ for the unipotent characters of $\mathrm{GL}_n(\mathbb{F}_q)$ is also non-zero. A conjecture of Saxl says that if $\mu $ is a staircase partition, then all irreducible characters of $S_{|\mu |}$ appear non-trivially in the tensor square $\chi ^\mu \otimes \chi ^\mu $. Therefore the Saxl conjecture implies its analogue for unipotent characters, i.e. all unipotent characters of $\mathrm{GL}_{|\mu |}(\mathbb{F}_q)$ appear non-trivially in the tensor square ${\mathcal{U}}^\mu \otimes {\mathcal{U}}^\mu $ when $\mu $ 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 $\mu $ for which the tensor square ${\mathcal{U}}^\mu \otimes {\mathcal{U}}^\mu $ contains non-trivially all the unipotent characters of $\mathrm{GL}_{|\mu |}(\mathbb{F}_q)$.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.434
Classification: 20C33, 20C30, 05E05
Keywords: tensor product, unipotent character, Kronecker coefficient, symmetric function

Letellier, Emmanuel  1 ; Nam, GyeongHyeon  2

1 Université Paris Cité IMJ-PRG CNRS 45 Rue des Saints-Pères Paris 75006 (France)
2 Ajou University Department of Mathematics 206 World cup-ro Suwon 16499 (Republic of Korea)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Letellier, Emmanuel; Nam, GyeongHyeon. The Saxl conjecture and the tensor square of unipotent characters of $\mathrm{GL}_n(q)$. Algebraic Combinatorics, Volume 8 (2025) no. 4, pp. 1119-1140. doi: 10.5802/alco.434

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