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Let $V$ be a $K$-vector space and let $T$ be a $K$-linear map $T: V \otimes V \to V \otimes V$. Denote by $T_i$ the linear operator on $V^{\otimes n} \to V^{\otimes n}$ which acts on the $i$th and $(i+1)$th factors and acts trivially on other factors. I would like to understand the matrix form of $T_i$. For example, let $V$ be $2$ dimensional and $e_1, e_2$ be its standard basis. Suppose that with respect to $e_1\otimes e_1, e_1 \otimes e_2, e_2 \otimes e_1, e_2 \otimes e_2$, $T$ is \begin{align*} \left[ \begin {array}{cccc} \mu_{{1,1}}&\mu_{{1,2}}&\mu_{{1,3}}&\mu_{ {1,4}}\\ \mu_{{2,1}}&\mu_{{2,2}}&\mu_{{2,3}}&\mu_{{2 ,4}}\\ \mu_{{3,1}}&\mu_{{3,2}}&\mu_{{3,3}}&\mu_{{3,4 }}\\ \mu_{{4,1}}&\mu_{{4,2}}&\mu_{{4,3}}&\mu_{{4,4}} \end {array} \right] . \end{align*} Let $n=3$. I obtained that, with respect to $$e_1 \otimes e_1 \otimes e_1, e_1 \otimes e_1 \otimes e_2, e_1 \otimes e_2 \otimes e_1, e_1 \otimes e_2 \otimes e_2, e_2 \otimes e_1 \otimes e_1, e_2 \otimes e_1 \otimes e_2, e_2 \otimes e_2 \otimes e_1, e_2 \otimes e_2 \otimes e_2,$$ \begin{align*} T_1 = \left[ \begin {array}{cccccccc} \mu_{{1,1}}&0&\mu_{{1,2}}&0&\mu_{{1,3 }}&0&\mu_{{1,4}}&0\\ 0&\mu_{{1,1}}&0&\mu_{{1,2}}&0& \mu_{{1,3}}&0&\mu_{{1,4}}\\ \mu_{{2,1}}&0&\mu_{{2,2} }&0&\mu_{{2,3}}&0&\mu_{{2,4}}&0\\ 0&\mu_{{2,1}}&0& \mu_{{2,2}}&0&\mu_{{2,3}}&0&\mu_{{2,4}}\\ \mu_{{3,1} }&0&\mu_{{3,2}}&0&\mu_{{3,3}}&0&\mu_{{3,4}}&0\\ 0& \mu_{{3,1}}&0&\mu_{{3,2}}&0&\mu_{{3,3}}&0&\mu_{{3,4}} \\ \mu_{{4,1}}&0&\mu_{{4,2}}&0&\mu_{{4,3}}&0&\mu_{{4 ,4}}&0\\ 0&\mu_{{4,1}}&0&\mu_{{4,2}}&0&\mu_{{4,3}}&0 &\mu_{{4,4}}\end {array} \right], \end{align*} \begin{align*} T_2 = \left[ \begin {array}{cccccccc} \mu_{{1,1}}&\mu_{{1,2}}&\mu_{{1,3}}& \mu_{{1,4}}&0&0&0&0\\ \mu_{{2,1}}&\mu_{{2,2}}&\mu_{{ 2,3}}&\mu_{{2,4}}&0&0&0&0\\ \mu_{{3,1}}&\mu_{{3,2}}& \mu_{{3,3}}&\mu_{{3,4}}&0&0&0&0\\ \mu_{{4,1}}&\mu_{{ 4,2}}&\mu_{{4,3}}&\mu_{{4,4}}&0&0&0&0\\ 0&0&0&0&\mu_ {{1,1}}&\mu_{{1,2}}&\mu_{{1,3}}&\mu_{{1,4}}\\ 0&0&0&0 &\mu_{{2,1}}&\mu_{{2,2}}&\mu_{{2,3}}&\mu_{{2,4}}\\ 0 &0&0&0&\mu_{{3,1}}&\mu_{{3,2}}&\mu_{{3,3}}&\mu_{{3,4}} \\ 0&0&0&0&\mu_{{4,1}}&\mu_{{4,2}}&\mu_{{4,3}}&\mu_{ {4,4}}\end {array} \right]. \end{align*} Are these computations correct? Thank you very much.

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Your computations indeed appear to be correct. Your results for $T_1,T_2$ can be confirmed using the Kronecker product, since we have $$ T_2 = I_2 \otimes T, \quad T_1 = T \otimes I_2. $$

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