Fourier inversion formula and Plancherel formula for finite groups
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The following problem is from an exercise sheet from a lecture, but it is a bit off-topic, so I would appreaciate some help/hints on it.
I've seen some similar posts, but either with different notation or already assuming a similar formula as the first one below.
Let $G$ be a finite group and $phi , psi : G to mathbb{C}$ and $(V, rho_V) $ be a complex representation of $G$. Let $ operatorname{Irr}(G) = { S_1, ldots, S_t } $ be the irreducible representations of $G$. Then the Fourier transform is defined as $hat{phi} (rho_V) := sum_{g in G} phi (g) g$.
I want to show the following two formulas:
$$ phi(g) = frac{1}{left| G right|} sum_{i= 1}^t dim(S_i) operatorname{Tr}(rho_{S_i} (g^{-1}) circ hat{phi}(rho_{S_i}))$$
$$ sum_{g in G} phi(g^{-1})psi(g) = frac{1}{left| G right|} sum_{i= 1}^t dim(S_i) operatorname{Tr}(hat{phi}(rho_{S_i}) circ hat{psi}(rho_{S_i}))$$
group-theory representation-theory
edited Jan 5 at 22:36
Bernard
123k741117
asked Jan 5 at 22:32
MPB94MPB94
27017
$endgroup$
add a comment |
$begingroup$
The following problem is from an exercise sheet from a lecture, but it is a bit off-topic, so I would appreaciate some help/hints on it.
I've seen some similar posts, but either with different notation or already assuming a similar formula as the first one below.
Let $G$ be a finite group and $phi , psi : G to mathbb{C}$ and $(V, rho_V) $ be a complex representation of $G$. Let $ operatorname{Irr}(G) = { S_1, ldots, S_t } $ be the irreducible representations of $G$. Then the Fourier transform is defined as $hat{phi} (rho_V) := sum_{g in G} phi (g) g$.
I want to show the following two formulas:
$$ phi(g) = frac{1}{left| G right|} sum_{i= 1}^t dim(S_i) operatorname{Tr}(rho_{S_i} (g^{-1}) circ hat{phi}(rho_{S_i}))$$
$$ sum_{g in G} phi(g^{-1})psi(g) = frac{1}{left| G right|} sum_{i= 1}^t dim(S_i) operatorname{Tr}(hat{phi}(rho_{S_i}) circ hat{psi}(rho_{S_i}))$$
group-theory representation-theory
edited Jan 5 at 22:36
Bernard
123k741117
asked Jan 5 at 22:32
MPB94MPB94
27017
$endgroup$
add a comment |
$begingroup$
The following problem is from an exercise sheet from a lecture, but it is a bit off-topic, so I would appreaciate some help/hints on it.
I've seen some similar posts, but either with different notation or already assuming a similar formula as the first one below.
Let $G$ be a finite group and $phi , psi : G to mathbb{C}$ and $(V, rho_V) $ be a complex representation of $G$. Let $ operatorname{Irr}(G) = { S_1, ldots, S_t } $ be the irreducible representations of $G$. Then the Fourier transform is defined as $hat{phi} (rho_V) := sum_{g in G} phi (g) g$.
I want to show the following two formulas:
$$ phi(g) = frac{1}{left| G right|} sum_{i= 1}^t dim(S_i) operatorname{Tr}(rho_{S_i} (g^{-1}) circ hat{phi}(rho_{S_i}))$$
$$ sum_{g in G} phi(g^{-1})psi(g) = frac{1}{left| G right|} sum_{i= 1}^t dim(S_i) operatorname{Tr}(hat{phi}(rho_{S_i}) circ hat{psi}(rho_{S_i}))$$
group-theory representation-theory
edited Jan 5 at 22:36
Bernard
123k741117
asked Jan 5 at 22:32
MPB94MPB94
27017
$endgroup$
The following problem is from an exercise sheet from a lecture, but it is a bit off-topic, so I would appreaciate some help/hints on it.
I've seen some similar posts, but either with different notation or already assuming a similar formula as the first one below.
Let $G$ be a finite group and $phi , psi : G to mathbb{C}$ and $(V, rho_V) $ be a complex representation of $G$. Let $ operatorname{Irr}(G) = { S_1, ldots, S_t } $ be the irreducible representations of $G$. Then the Fourier transform is defined as $hat{phi} (rho_V) := sum_{g in G} phi (g) g$.
I want to show the following two formulas:
$$ phi(g) = frac{1}{left| G right|} sum_{i= 1}^t dim(S_i) operatorname{Tr}(rho_{S_i} (g^{-1}) circ hat{phi}(rho_{S_i}))$$
$$ sum_{g in G} phi(g^{-1})psi(g) = frac{1}{left| G right|} sum_{i= 1}^t dim(S_i) operatorname{Tr}(hat{phi}(rho_{S_i}) circ hat{psi}(rho_{S_i}))$$
group-theory representation-theory
group-theory representation-theory
edited Jan 5 at 22:36
Bernard
123k741117
asked Jan 5 at 22:32
MPB94MPB94
27017
edited Jan 5 at 22:36
Bernard
123k741117
asked Jan 5 at 22:32
MPB94MPB94
27017
edited Jan 5 at 22:36
Bernard
123k741117
edited Jan 5 at 22:36
Bernard
123k741117
edited Jan 5 at 22:36
Bernard
123k741117
123k741117
asked Jan 5 at 22:32
MPB94MPB94
27017
asked Jan 5 at 22:32
MPB94MPB94
27017
asked Jan 5 at 22:32
MPB94MPB94
27017
27017
add a comment |
add a comment |
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