The following question is closely related to this one.

Let $\mathrm{U}(n)$ be the group of all (complex) unitary matrices $n\times n$. It is known that all irreducible representations of $\mathrm{U}(n)$ are parameterized by weights $\lambda=(\lambda_1,\ldots,\lambda_n)\in\mathbb{Z}^n$, where $\lambda_1\ge \lambda_1\ge\ldots\ge\lambda_n$; denote this representation by $V_{\lambda}$. It also worths mentioning that if all $\lambda_i\ge 0$, then the corresponding irreducible representation is just $\mathbb{S}^{\lambda}(V)$, where $V\simeq\mathbb{C}^n$ is the standard $n$-dimensional representation. Similarly, if all $\lambda_i\le 0$, then $V_{\lambda}$ is isomorphic to $\mathbb{S}^{\mu}(V)^*$, where $\mu=(-\lambda_n,\ldots,-\lambda_1)$. Here $\mathbb{S}^{\mu}$ is the Schur functor associated to a partition $\mu$.

Now suppose that we have two decreasing sequences of integers: $\lambda=(\lambda_1,\ldots,\lambda_l)$ and $\mu=(\mu_1,\ldots,\mu_k)$, where $\lambda_1\ge\ldots\ge\lambda_l\ge 0\ge\mu_1\ge\ldots\ge\mu_k$ and $k+l\le n$. We are interested in $V_{(\lambda,\mu)}$-isotypic component of the tensor product $V_{\lambda}\otimes V_{\mu}$ which can be regarded as a submodule of $V^{\otimes|\lambda|}\otimes (V^*)^{\otimes|\mu|}$. Here $|\lambda|=\sum_{j=1}^{l}\lambda_j$, $|\mu|=\sum_{j=1}^{k}(-\mu_j)$ and weight $(\lambda,\mu)\in\mathbb{Z}^n$ is defined as $$ (\lambda,\mu)=(\lambda_1,\ldots,\lambda_l,\underbrace{0,\ldots,0}_{n-k-l~\text{zeroes}},\mu_1,\ldots,\mu_k). $$

I think that I saw somewhere the following statement (probably it was in The Classical Groups by H. Weyl but I am not sure and now I can't find this statement there).

**Claim.** The $V_{(\lambda,\mu)}$-isotypic component of $V_{\lambda}\otimes V_{\mu}$ is the intersection of kernels of all tensor contractions $\psi_{p,q}$, where $p=\overline{1,|\lambda|}$ and $q=\overline{1,|\mu|}$, that correspond to the $p$-th contravariant component and to the $q$-th covariant component.

Does anyone know whether this statement is correct or not? Of course I could have forgotten some details, so any similar facts (or corrections to the original statement) would be appreciated.

**Update.** It seems that a similar fact is claimed in A. Kirillov's *Elements of the theory of representations* (see 17.2. Representations of classical compact Lie groups) but the author also refers to H. Weyl's Classical groups without any additional details. Hence, any other references would be appreciated.