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For $ A= (A_1,\cdots,A_d)\in {\cal L}(E)^d$ such that $ A_iA_j=A_jA_i$ for all $ i,j$ . Why $ $ \sum_{f\in F(n,d)} A_{f}^*A_{f}=\displaystyle\sum_{|\alpha|=n}\frac{n!}{\alpha!}{A^*}^{\alpha}A^{\alpha}\,?$ $ Note that $ F(n,d)$ denotes the set of all functions from $ \{1,\cdots,n\}$ into $ \{1,\cdots,d\}$ and $ A_f:=A_{f(1)}\cdots A_{f(n)}$ , for $ f\in F(n,d)$ . Also $ \alpha = (\alpha_1, \alpha_2,…,\alpha_d) \inRead more