# A Weyl invariance constructed from Clebsch-Gordan Coefficients.

Let $V$ and $\tilde{V}$ be irreducible representations of SU(N) with tensor decomposition:

$$V \otimes \tilde{V} = \bigoplus_i U_i$$

\noindent were $U_i$ are also irreps of SU(N). Let $V = \oplus V_\alpha$, $\tilde{V} = \oplus \tilde{V}_\alpha$ and $U_1 = \oplus U_\alpha$ be weight space decompositions of $V$, $\tilde{V}$ and $U_1$. Let $\{v_{\alpha,i}\}$, $\{\tilde{v}_{\beta,i}\}$, $\{u_{\mu,i}\}$ be some orthonormal basis of $V_\alpha$, $\tilde{V}_\beta$ and $U_\mu$, respectively. Denote Clebsch-Gordan coefficients with:

$$\langle u_{\mu,n}|v_{\alpha,i}\otimes \tilde{v}_{\beta,j}\rangle = C^{(\mu,n)}_{(\alpha,i),(\beta,j)}.$$

Note that the CG-coefficients with the above notation are only non zero if $\mu=\alpha+\beta$. I am very interested in the following expression:

$$P_{V\tilde{V}}^{U_1}(\alpha,\beta) = \sum_{i,j,n} |C^{(\alpha+\beta,n)}_{(\alpha,i),(\beta,j)}|^2$$

First, note that $\{v_{\alpha,i}\otimes \tilde{v}_{\beta,j}\}_{i,j}$ spans $V_\alpha\otimes \tilde{V}_\beta$ and $\{u_{\mu,n}\}_n$ spans $U_{\alpha+\beta}$, both spaces being subspaces of $(V \otimes \tilde{V})_{\alpha+\beta}$. Thus $P_{V\tilde{V}}^{U_1}(\alpha,\beta)$ can be viewed as a measure of how similar $V_\alpha\otimes \tilde{V}_\beta$ and $U_{\alpha+\beta}$ are as subspaces of $(V \otimes \tilde{V})_{\alpha+\beta}$. Second and more important, note that the above expression is independent of choice of basis for $V_\alpha$, $\tilde{V}_\beta$ and $U_\mu$, as long as they are orthonormal.

My question is, is P Weyl invariant, and if so, is there a reference in which this is shown?

$$> P_{V\tilde{V}}^{U_1}(W\alpha,W\beta)= P_{V\tilde{V}}^{U_1}(\alpha,\beta),~~~~~~\forall W \in\mathfrak{W}. >$$

Here $\mathfrak{W}$ denotes the Weyl group. My intiution says P should be Weyl invariant. Also, I would think that the above holds not only for SU(N) but for any simply Lie Group.

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