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 >= 31. For p = 5 such a representation exists exactly in characteristic 2 (A_5 ≅ SL_2(4) on F_4^2). For p = 7 the answer is no in every characteristic, without tables, with explicit certificates in F_2A_7 and 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 Sp−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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-14299292-011 (M. Lee, Problem 1 of the problem session, Oberwolfach Report 27/2025, Computational Group Theory, p. 1442).