Mackey's criterion

In the mathematical field of representation theory of groups is the criterion of Mackey one of George Mackey -positioned criterion for the irreducibility of induced representations to check finite groups.

Terms and notation

Two representations and a finite group are called disjoint if they have no irreducible component in common, i.e. i.e., if for the scalar product of characters . ${\displaystyle V_{1}}$ ${\displaystyle V_{2}}$ ${\displaystyle G}$ ${\displaystyle \langle V_{1},V_{2}\rangle =0}$

Be a group and be a subgroup. Define for Be a representation of the subgroup This defines by restriction a representation of We write for Also defines another representation of defined by These two representations should not be confused. ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle H_{s}=sHs^{-1}\cap H}$ ${\displaystyle s\in G.}$
${\displaystyle (\rho ,W)}$ ${\displaystyle H.}$ ${\displaystyle {\text{Res}}_{H_{s}}(\rho )}$ ${\displaystyle H_{s}.}$ ${\displaystyle {\text{Res}}_{s}(\rho )}$ ${\displaystyle {\text{Res}}_{H_{s}}(\rho ).}$ ${\displaystyle \rho ^{s}}$ ${\displaystyle H_{s}}$ ${\displaystyle \rho ^{s}(t)=\rho (s^{-1}ts).}$

Furthermore, we denote or the representation of the induced representation of . ${\displaystyle Ind_{H}^{G}(W)}$ ${\displaystyle Ind_{H}^{G}(\rho )}$ ${\displaystyle \rho \colon H\to GL(W)}$ ${\displaystyle G}$

Mackey's Irreducibility Criterion

The induced representation is irreducible if and only if the following conditions are met: ${\displaystyle V={\text{Ind}}_{H}^{G}(W)}$

• ${\displaystyle W}$ is irreducible.
• For each , the two representations and of disjoint. ${\displaystyle s\in G\setminus H}$ ${\displaystyle \rho ^{s}}$ ${\displaystyle {\text{Res}}_{s}(\rho )}$ ${\displaystyle H_{s}}$

A proof of this theorem can be found in.

From the sentence we get the following directly

Corollary

Let be a normal subgroup of Then is irreducible if and only if is irreducible and not isomorphic to the conjugates for . ${\displaystyle H}$ ${\displaystyle G.}$ ${\displaystyle {\text{Ind}}_{H}^{G}(\rho )}$ ${\displaystyle \rho }$ ${\displaystyle \rho ^{s}}$ ${\displaystyle s\notin H}$

proof

If normal , then and and thus the statement follows directly from Mackey's criterion. ${\displaystyle H}$ ${\displaystyle H_{s}=H}$ ${\displaystyle {\text{Res}}_{s}(\rho )=\rho }$ ${\displaystyle \Box }$

literature

1. ^ Jean-Pierre Serre: Linear Representations of Finite Groups. Springer Verlag, New York 1977.