Frobenius reciprocity

Let V be a finite-dimensional representation of a finite groupMathworldPlanetmath G, and let W be a representation of a subgroupMathworldPlanetmathPlanetmath HG. Then the charactersPlanetmathPlanetmath of V and W satisfy the inner product relationPlanetmathPlanetmath


where Ind and Res denote the induced representationMathworldPlanetmath IndHG and the restriction representation ResHG.

The Frobenius reciprocity theorem is often given in the stronger form which states that Res and Ind are adjoint functorsMathworldPlanetmathPlanetmathPlanetmath between the categoryMathworldPlanetmath of G–modules and the category of H–modules:


or, equivalently

Title Frobenius reciprocity
Canonical name FrobeniusReciprocity
Date of creation 2013-03-22 12:17:51
Last modified on 2013-03-22 12:17:51
Owner djao (24)
Last modified by djao (24)
Numerical id 7
Author djao (24)
Entry type Theorem
Classification msc 20C99