Geodesic
On the behavior of elements of prime order from a~Zinger cycle in representations of a~special linear group
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 3, pp. 179-186

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Let $G=SL_n(q)$, where $n\geq2$ and $q$ is a power of a prime $p$. A Zinger cycle of the group $G$ is its cyclic subgroup of order $(q^n-1)/(q-1)$. Here absolutely irreducible $G$-modules over a field of the defining characteristic $p$ where an element of a fixed prime order $m$ from a Zinger cycle of $G$ acts freely are classified 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) $n=2$.
Keywords: special linear group, Zinger cycle, absolutely irreducible module, free action of an element. 
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     title = {On the behavior of elements of prime order from {a~Zinger} cycle in representations of a~special linear group},
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A. S. Kondrat'ev; A. A. Osinovskaya; I. D. Suprunenko. On the behavior of elements of prime order from a~Zinger cycle in representations of a~special linear group. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 3, pp. 179-186. http://geodesic.mathdoc.fr/item/TIMM_2013_19_3_a17/