Below are examples of the matrix I have in mind: $$ P_{2} = \begin{bmatrix} 1 & & & \\ & & 1 & \\ & 1 & & \\ & & & 1 \end{bmatrix},\quad P_{3} = \begin{bmatrix} 1 & & & & & & & & \\ & & & 1 & & & & & \\ & & & & & & 1 & & \\ & 1 & & & & & & & \\ & & & & 1 & & & & \\ & & & & & & & 1 & \\ & & 1 & & & & & & \\ & & & & & 1 & & & \\ & & & & & & & & 1 \end{bmatrix},\quad\textrm{etc.} $$ I know that it can be written as $$ P_{n} = \sum_{i=1}^{n}\sum_{j=1}^{n} (Q_{ij} \otimes Q_{ij}^{\intercal}), $$ where $Q_{ij}$ is defined to be a matrix with zeros everywhere except for in the $(i,j)$ position. I've seen it come up when computing the variance: $$ \mathrm{Var}(\mathrm{vec}(\mathbf{v} \, \mathbf{v}^{\intercal})) = (I_{n^{2}} + P_{n})(\Sigma \otimes \Sigma) $$ where $\mathbf{v} \sim \mathcal{N}(\mathbf{0},\Sigma)$, and where “$\mathrm{vec}$” is the vectorization operator (the reference I received for this expression was Muirhead, 1982, although I could not verify the source).
More generally, this matrix sometimes arises when I work with vectorization and the Kronecker product, and I was hoping to find out more about it. Any information would be appreciated, but even a name would be useful.
These matrices that have an interesting property-
$$ P_n \cdot \text{vec}(A) = \text{vec}(A^T)$$
I'm not aware of any particular name for them, but that's exactly why they're showing up.
If you write $Q_{ij} = e_i e_j^T$ and then $P_n = \sum_{i,j} (e_i e_j^T) \otimes (e_j e_i^T)$ it becomes a bit clearer - for each index summed over, the $(i,j)$-indexed $n \times n$ block gets a 1 in the $(j, i)$ position.
They are also involutive as you'd expect from a transpose.