# Character correspondences induced by magic representations - Mathematics > Representation Theory

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Abstract: Let G be a finite group, K a normal subgroup of G and H a subgroup such thatG = HK, and set L = H \cap K. Suppose \theta \in Irr K and \phi \in Irr L, and\phi\ occurs in \theta L with multiplicity n > 0. A projective representationof degree n on H-L is defined in this situation; if this representation isordinary, it yields a bijection between IrrG | \theta and IrrH | \phi. Thebehavior of fields of values and Schur indices under this bijection isdescribed. A modular version of the main result is proved. We show that thetheory applies if n and the order of H-L are coprime. Finally, assume that P <=G is a p-group with P \cap K = 1 and PK normal in G, that H = N GP, and that\theta\ and \phi\ belong to blocks of p-defect zero which are Brauercorrespondents with respect to the group P. Then every block of F pG orQ pG lying over \theta\ is Morita-equivalent to its Brauer correspondent withrespect to P. This strengthens a result of Turull Above the Glaubermancorrespondence, Advances in Math. 217 2008, 2170-2205.