On the behavior of elements of prime order from Singer cycles in representations of special linear groups. (Q471002)

Let \(G=SL_n(q)\), where \(n\geq 2\) and \(q\) is a power of a prime \(p\). A Singer cycle of the group \(G\) is a cyclic subgroup of order \((q^n-1)/(q-1)\). In this paper, the authors study the problem of classifying 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) \(n=2\).