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Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

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research on the structure/properties of permutation matrix/table with $(i,j)th$ entry as $\p...

Is there any research on the structure/properties of permutation matrix/table with $(i,j)th$ entry as $\pi_j\circ \pi_i^{-1}$, where $\{\pi_1,\pi_2,...,\pi_{k!}\}=S_k$? I know if we apply the functio …