Prym–Tyurin varieties using self-products of groups

Given Prym–Tyurin varieties of exponent q with respect to a finite group G, a subgroup H and a set of rational irreducible representations of G satisfying some additional properties, we construct a Prym–Tyurin variety of exponent [G:H]q in a natural way. We study an example of this result, starting from the dihedral group Dp for any odd prime p. This generalizes the construction of [H. Lange, S. Recillas, A.M. Rojas, A family of Prym–Tyurin varieties of exponent 3, J. Algebra 289 (2005) 594–613] for p=3. Finally, we compute the isogeny decomposition of the Jacobian of the curve underlying the above mentioned example.