Matrix written in terms of Kronecker delta

Question

What would be the matrix form of $P$ defined in equation (1.11) of this published work, where (I have chosen $n=2$, for simplicity)

$$P_{kl}=\begin{cases} \delta_{k,2l-1} ~~~ k \le 2\\\\ \delta_{2+k,2l} ~~~ l\le 2 \end{cases}$$

Answer 1

From the context of the paper (namely the text below Equation (1.2) that defines $\mathbf R$ and the text below Equation (1.4) defining $\mathbf S$), it seems that the permutation matrix $\mathbf P$ being discussed for which $\mathbf P \mathbf R = \mathbf S$ is the commutation matrix $\mathbf P = \mathbf K^{(2,n)}$.

Some examples:

$$n = 2: \mathbf P = \left[\begin{array}{cc|cc}1 & 0 & 0 & 0\\0 & 0 & 1 & 0\\ \hline0 & 1 & 0 & 0\\0 & 0 & 0 & 1\end{array}\right]\\ n=3: \mathbf P = \left[\begin{array}{cc|cc|cc} 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ \hline 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]\\ n=4: \mathbf P = \left[\begin{array}{cc|cc|cc|cc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ \hline 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]. $$ With that, I suspect that the intended description of $\mathbf P$ was as follows. $$ P_{kl} = \begin{cases} \delta_{k,2l-1} & k \leq n,\\ \delta_{k-n,2l-2} & k > n. \end{cases} $$