Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 3 / 15F / a

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- Solution a

Solution

For a finite group

G

H, K

, and a complex group representation

V

of

H

, the Mackey restriction formula is

Res_{K}^{G} Ind_{H}^{G} V ≅ g \in K \ G / H ⨁ Ind_{K \cap g H g^{- 1}}^{K} Res_{K \cap g H g^{- 1}}^{g H g^{- 1}} (^{g} V) .

Choose one representative

g

of each double coset

K g H

. The conjugate subgroup representation

^{g} V

is a representation of

g H g^{- 1}

on the same vector space, with

(g h g^{- 1}) v = h v

. Restriction of a representation keeps the vector space and restricts the acting subgroup. induced representation is

C G \otimes_{C H} V

, and analogously for each subgroup of

K

. The displayed isomorphism is an isomorphism of

K

-representations. These definitions specify every summand and its subgroup.

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