= Matrix Fourier block
{title2=$T_\rho(A)=\mathbb E_xf(x)A\rho(x)^*$}
For $f:G\to M_n(\mathbb C)$ and a degree-$d_\rho$ <unitary irreducible representation>, the matrix Fourier block is the operator $T_\rho$ on $M_{n\times d_\rho}(\mathbb C)$ defined in the title. Its <singular values> $\lambda_{\rho,j}$ satisfy
$$
\sum_{\rho,j}d_\rho\lambda_{\rho,j}^2=\mathbb E_x\|f(x)\|_{\mathrm{HS}}^2,
\qquad
\sum_{\rho,j}d_\rho\lambda_{\rho,j}^4=\|f\|_{U^2}^4.
$$
Pointwise <operator norm> at most one implies $\|T_\rho\|_{\mathrm{op}}\leq1$. These blocks are operators on a matrix space, rather than simply the entrywise scalar <Fourier transform on a finite group> without any reshaping.
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