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  "body": "\n## Statement\n\nGeorge Mackey established a criterion to verify the irreducibility of induced representations. For this we will first need some definitions and some specifications with respect to the notation.\n\nTwo representations $V_1$ and $V_2$ of a group $G$ are called **disjoint**, if they have no irreducible component in common, i.e. if $\\langle V_1,V_2\\rangle_G =0.$\n\nLet $G$ be a group and let $H$ be a subgroup. We define $H_s=sHs^{-1}\\cap H$ for $s \\in G.$ Let $(\\rho, W)$ be a representation of the subgroup $H.$ This defines by restriction a representation $\\text{Res}_{H_s}(\\rho)$ of $H_s.$ We write $\\text{Res}_s(\\rho)$ for $\\text{Res}_{H_s}(\\rho).$ We also define another representation $\\rho^s$ of $H_s$ by $\\rho^s(t)=\\rho(s^{-1}ts).$ These two representations are not to be confused.\n\n**Mackey's irreducibility criterion.** The induced representation $V=\\text{Ind}^G_H(W)$ is irreducible if and only if the following conditions are satisfied:\n\n- $W$ is irreducible\n- For each $s\\in G\\setminus H$ the two representations $\\rho^s$ and $\\text{Res}_s(\\rho)$ of $H_s$ are disjoint. A proof of this theorem may be found in (Serre 1977).\n\nFor the case of $H$ normal, we have $H_s=H$ and $\\text{Res}_s(\\rho)=\\rho$. Thus we obtain the following:\n\n**Corollary.** Let $H$ be a normal subgroup of $G.$ Then $\\text{Ind}^G_H(\\rho)$ is irreducible if and only if $\\rho$ is irreducible and not isomorphic to the conjugates $\\rho^s$ for $s \\notin H.$\n",
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