Abstract
Let G = SL n (q), where n ≥ 2 and q is a power of a prime p. A Singer cycle of the group G is its cyclic subgroup of order (q n − 1)/(q − 1). We classify absolutely irreducible G-modules over a field of characteristic p where an element of fixed prime order m from a Singer cycle of G acts freely in the following three cases: (a) the residue of q modulo m generates the multiplicative group of the field of order m (in particular, this holds for m = 3); (b) m = 5; (c) n = 2.
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Original Russian Text © A.S. Kondrat’ev, A.A. Osinovskaya, I.D. Suprunenko, 2013, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2013, Vol. 19, No. 3.
To Aleksandr Alekseevich Makhnev on the occasion of his 60th birthday
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Kondrat’ev, A.S., Osinovskaya, A.A. & Suprunenko, I.D. On the behavior of elements of prime order from Singer cycles in representations of special linear groups. Proc. Steklov Inst. Math. 285 (Suppl 1), 108–115 (2014). https://doi.org/10.1134/S0081543814050113
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DOI: https://doi.org/10.1134/S0081543814050113