Low-Dimensional Representations of the Special Linear Group in Cross-Characteristics
Robert M. Guralnick and Pham Huu Tiep
The low-dimensional projective irreducible representations in cross characteristics of the projective special linear group $L_{n}(q)$ are investigated. If $n \geq 3$ and $(n,q) \neq (3,2)$, $(3,4)$, $(4,2)$, $(4,3)$, all such representations of the first degree (which is $(q^{n}-q)/(q-1) - \kappa_n$ with $\kappa_n = 0$ or $1$) and the second degree (which is $(q^{n}-1)/(q-1)$) come from Weil representations. We show that the gap between the second and the third degree is roughly $q^{2n-4}$.