Fixed-Point-Free Representations of Alternating Groups of Twin-Prime Degree: A Partial Answer to a Question of M. Lee

M. Lee asked at the Oberwolfach workshop “Computational Group Theory” (June 2025) whether, for a prime \(p\) such that \(p-2\) is also prime, the alternating group \(A_p\) has a faithful irreducible representation over a field of some characteristic \(r>0\) in which all elements of orders \(p\) and \(p-2\) act without nonzero fixed vectors, and remarked that this is easily seen to fail for \(r>p\). We give a partial answer: yes for \(p=5\), no for \(p=7,13,19\) (\(13\) and \(19\) conditional on published decomposition matrices); open for \(p\ge31\). For \(p=5\) such a representation exists exactly in characteristic \(2\) (\(A_5\cong\mathrm{SL}_2(4)\) on \(\mathbb{F}_4^2\)). For \(p=7\) the answer is no in every characteristic, without tables, with explicit certificates in \(\mathbb{F}_2A_7\) and \(\mathbb{F}_3A_7\). For \(p=13\) with \(r=2,3,5\) and \(p=19\) with \(r=2,3,5,7\) the answer rests on the decomposition matrices of symmetric groups published by the Modular Atlas project, generated from the character table library of GAP; the other characteristics are excluded without tables. For every such \(p\) the question reduces to \(S_p\) and to primes \(r<p/2\): \(r=p\) and \(r=p-2\) are impossible, \(r>p\) is Lee's remark (its short proof also covers characteristic \(0\)), and \(p/2<r<p-2\) is excluded by a standard Brauer-tree argument, which for a \(p\)-cycle alone Zalesski attributes to Thompson. For odd \(r\) the question reduces further to the simple modules of \(S_{p-2}\) with a fixed-point-free \((p-2)\)-cycle, a case of an open problem of Zalesski. No printed answer to the question was found in the sources searched; a search that finds nothing is not a proof of novelty. This is an unrefereed note.