= Matrix-valued Gowers U2 quantity
{title2=$\|f\|_{U^2}^4=\mathbb E_{xy^{-1}zw^{-1}=e}\operatorname{tr}(f(x)f(y)^*f(z)f(w)^*)$}
= Matrix-valued U2 quantity
{synonym}
For $f:G\to M_n(\mathbb C)$ on a <finite group>, use uniform <expectation> over the solutions of $xy^{-1}zw^{-1}=e$ and the ordinary, unnormalized <trace> to define the displayed fourth-order quantity. If $T_\rho$ is its <matrix Fourier block>, this quantity equals $\sum_\rho d_\rho\operatorname{Tr}((T_\rho T_\rho^*)^2)$, so it is real and nonnegative. For a scalar function on an <abelian group>, it reduces to the usual fourth power of the <Gowers uniformity norm> of order two. The unnormalized <trace> means a unitary matrix-valued function can have fourth-order quantity as large as $n$.
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