[math] \newcommand{\mathds}{\mathbb}[/math]

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Write down a complete, simplified proof for the factorization

[[math]] \xymatrix{C(S_{NM}^+)\ar[rr]^{\pi_L}\ar[rd]&&M_{NM}(\mathbb C)\\&C(S_M^+\wr_*G_H)\ar[ur]&} [[/math]]

found above, for [math]L=H\otimes_QK[/math], in the scalar matrix case.