Notes for AKT-140205/0:31:49

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In the definition of [math]\displaystyle{ C_A(M) }[/math], the disjoint union is taken over partition of [math]\displaystyle{ A }[/math]; I would like to know why they are all of length [math]\displaystyle{ k }[/math]. Also do the [math]\displaystyle{ A_\alpha }[/math]'s represent clusters?

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