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Low-dimensional representations of special linear groups in cross characteristics. (English) Zbl 0974.20014

The low-dimensional projective irreducible representations in cross characteristics of the projective special linear group \(\text{PSL}_n(q)\) 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 \((q^n-q)/(q-1)-\kappa\) with \(\kappa=0\) 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 \(q^{2n-4}\).

MSC:

20C33 Representations of finite groups of Lie type
20G05 Representation theory for linear algebraic groups
20C20 Modular representations and characters
20G40 Linear algebraic groups over finite fields
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