Mackey’s restriction formula

This fact is related to: linear representation theory
View other facts related to linear representation theory | View terms related to linear representation theory

This fact is related to: Mackey theory
View other facts related to Mackey theory | View terms related to Mackey theory

Statement

Let $G$ be a finite group with subgroups $H$ and $K$. Let $W$ be a linear representation of $H$. Denote $K\backslash G/H:=\{KgH|g\in G\}$ to be the set of double cosets of $G$ with $K$ on the left and $H$ on the right.

Then $\operatorname {Res} _{K}^{G}\operatorname {Ind} _{H}^{G}W\cong \bigoplus _{g\in K\backslash G/H}\operatorname {Ind} _{K\cap gH}^{K}\operatorname {Res} _{^{g}H\cap K}^{^{g}H}{^{g}W}.$

This formula is known as Mackey’s restriction formula.

Proof

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