Suppose $(X,h)$ is a compact $n$-dimensional Hermitian manifold, with holonomy group $H$. Now we know,since $X$ is a complex manifold, that $H\subset U(n)$, and that there is a representation of $H$ on the holomorphic tangent space to some point $T_pX\cong \mathbb{C}^n$, which we shall denote as $\rho_{hol}$.
Now the Lie algebra $\mathfrak{sl}(n,\mathbb{C})$ has exactly two $n$-dimensional irreducible representations namely the standard representation (i.e. elements of the algebra acting via normal matrix multiplication on $n$ dimensional vectors), $\rho_{st}$ and the representation dual to this, $\rho_{st}^{*}$. This is exercise 15.21 in Fulton and Harris: "Representation Theory". Hence $SU(n)$, as the maximal compact subgroup of $Sl(n,\mathbb{C})$ will also have only two $n$ dimensional irreps. However, $U(n)$ has a non-trivial one dimensional representation \begin{align} &\rho_{det}: U(n) \rightarrow GL(1,\mathbb{C})\\ &\rho_{det}(g)z = \det(g)z \end{align} So for each $k$, $\rho_{st}\otimes \rho_{det}^{k}$ is an irreducible $n$ dimensional representation of $U(n)$; thus we have infinitely many $n$-dimensional irreps.
If $(X,h)$ has holonomy $U(n)$,and the holonomy representation $\rho_{hol}$ is irreducible, it should be true that either \begin{equation} \rho_{hol} = \rho_{st}\otimes \rho_{det}^{k} \end{equation} or \begin{equation} \rho_{hol} = \rho_{st}^{*}\otimes \rho_{det}^{k} \end{equation} for some $k$ since these are the only irreducible $n$-dimensional representations. So, is it possible to determine which case is true, and what $k$ is?
the $k$ is $0$ and the representation is the standard one.
The point is that by definition the holonomy group IS a subgroup of U(n). In particular the action or representation is "faithfull". So to say that the holonomy is U(n) is to say both that the group as abstract group is U(n) and that the action on the tangente space is the standard representation of U(n).
Notice by the way that the representations with $nk + 1$ different from 1 are not faithfull since there exists multiples of the identity acting as the identity. See comment below.
Best, h.