Saxl conjecture and the tensor square of unipotent characters of GL(n,q)
Emmanuel Letellier, GyeongHyeon Nam
TL;DR
The paper develops and proves an analogue of Saxl’s conjecture for unipotent characters of GL$_n(\mathbb{F}_q)$, showing that when the first two partitions form a staircase, the unipotent tensor square $U_{(\xi_d,\xi_d,\tau)}(q)$ is nonzero for every $\tau$. It builds a general framework linking tensor multiplicities to root systems of star-shaped graphs and to generalized Littlewood–Richardson coefficients, enabling a nonvanishing criterion via imaginary roots (and a trace-based LR-style analysis). A central conjecture is proposed: $U_{(\mu,\mu,\tau)}(q)$ is nonzero for all $\tau$ exactly when the first part $\mu_1$ satisfies $\mu_1\le \lceil n/2\rceil$, with a complementary vanishing statement when $\mu_1>\lceil n/2\rceil$; this is verified for $n\le 8$ and illustrated with explicit examples. The results hint at a deeper reduction of unipotent-character tensor problems to Kronecker coefficients, offering a pathway to broader understanding of tensor squares in finite groups of Lie type.
Abstract
We know from Letellier that if for some triple of partitions the corresponding Kronecker coefficient is non-zero then the corresponding multiplicities for unipotent characters of GL(n,q) is also non-zero. A conjecture of Saxl says that the tensor square of an irreducible character of the symmetric group corresponding to a staircase partition contains all the irreducible characters. Therefore Saxl conjecture implies its analogue for unipotent characters. In this paper we prove the analogue of Saxl conjecture for unipotent characters and we describe conjecturally the set of all partitions for which the tensor square of the associated unipotent character contains all the unipotent characters.