I want to find the answer of $$\int dU \ U^m X \ U^{\dagger m}$$

Where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $U(n)$ and $dU$ is a normalized Haar measure. $X$ is a given self-adjoint matrix.

I used Schur lemma and found that the answer is of the form

(Edit: the correct form is ) $$pX+(1-p)tr(X)\frac{I}{n}$$

with $p\in[0,1]$, but if this is true then by comparing above expressions and using the results obtained in my last question, $p$ can be found as $$p=\frac{m-1}{n^2-1}$$ which is greater than 1 for $m>n^2$.

I don't know what is going wrong here. Does anyone have an idea about my solution or another way for solving this integral?

I want to find the answer of $$\int dU \ U^m X \ U^{\dagger m}$$

Where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $U(n)$ and $dU$ is a normalized Haar measure. $X$ is a given self-adjoint matrix.

I used Schur lemma and found that the answer is of the form

(Edit: the correct form is ) $$pX+(1-p)tr(X)\frac{I}{n}$$

with $p\in[0,1]$, but if this is true then by comparing above expressions and using the results obtained in my last question, $p$ can be found as $$p=\frac{m-1}{n^2-1}$$ which is greater than 1 for $m>n^2$.

I don't know what is going wrong here. Does anyone have an idea about my solution or another way for solving this integral?

I want to find the answer of $$\int dU \ U^m X \ U^{\dagger m}$$

Where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $U(n)$ and $dU$ is a normalized Haar measure. $X$ is a given self-adjoint matrix.

I used Schur lemma and found that the answer is of the form $$pX+(1-p)\frac{I}{n}$$

with $p\in[0,1]$, but if this is true then by comparing above expressions and using the results obtained in my last question, $p$ can be found as $$p=\frac{m-1}{n^2-1}$$ which is greater than 1 for $m>n^2$.

I don't know what is going wrong here. Does anyone have an idea about my solution or another way for solving this integral?

I want to find the answer of $$\int dU \ U^m X \ U^{\dagger m}$$

Where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $U(n)$ and $dU$ is a normalized Haar measure. $X$ is a given self-adjoint matrix.

I used Schur lemma and found that the answer is of the form

(Edit: the correct form is ) $$pX+(1-p)tr(X)\frac{I}{n}$$

with $p\in[0,1]$, but if this is true then by comparing above expressions and using the results obtained in my last question, $p$ can be found as $$p=\frac{m-1}{n^2-1}$$ which is greater than 1 for $m>n^2$.

I don't know what is going wrong here. Does anyone have an idea about my solution or another way for solving this integral?