Does the trivial representation always induce the permutation representation? Is this true for each field $\mathbb{K}$ or just for representations over $\mathbb{C}$?
does the trivial representation always induce the permutation representation?
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group-theory
representation-theory
finite-groups
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0Thanks!Is it also true, that the dimension of a permutation repr. comes from the index? (I mean the index of $G$ and the subgroup you use for induction) – 2012-04-19
1 Answers
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I think about the induced representation $Ind_H^G 1$ as the space of functions $ f: H\backslash G \rightarrow \mathbb{K},$ and $ g \in G : f(x) \mapsto f(xg).$
Than surely $g$ will permute the cosets $H \backslash G$, hence is a permutation action, and since you can write down a basis of functions supported on a single coset $H \gamma$, the dimesnion equals the cardinality of $H \backslash G$, i.e. the index.