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induced representation of the trivial representation

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Statement

Let GG be a finite group and HιGH \overset{\iota}{\hookrightarrow} G a subgroup-inclusion. Then the induced representation in Rep(G) of the 1-dimensional trivial representation 1Rep(H)\mathbf{1} \in Rep(H) is the permutation representation k[G/H]k[G/H] of the coset G-set G/HG/H:

ind H G(1)k[G/H]. \mathrm{ind}_H^G\big( \mathbf{1}\big) \;\simeq\; k[G/H] \,.

This follows directly as a special case of the general formula for induced representations of finite groups (this Example).

It follows that every virtual permutation representation (hence every element of the representation ring R k(G)R_k(G) in the image of the canonical morphism A(G)βR k(G)A(G) \overset{\beta}{\to} R_k(G) from the Burnside ring) is a virtual combination of induced representations of trivial representations.

A generalization of this statement including non-permutation representations is the Brauer induction theorem.

References

See also

Created on January 28, 2019 at 04:45:40. See the history of this page for a list of all contributions to it.