Each row in a (row-)stochastic matrix is a discrete probability distribution, and as such lives on a simplex. The space of stochastic matrices is the Cartesian product of these simplices. Is there another description for this space?
For example, a $2\times2$ stochastic matrix could be parameterized as $$ \begin{bmatrix} a & 1-a\\ b & 1-b\\ \end{bmatrix} $$
with $a,b \in [0,1]$. To me, that space looks like a unit square. Each point on the unit square is associated with a point $(a,b)$ which can then be associated with a stochastic matrix.
There's another possible shape I can identify with the space, which is a 3-simplex with 4 points:
$ \begin{bmatrix} 1 & 0\\ 1 & 0\\ \end{bmatrix}, \begin{bmatrix} 0 & 1\\ 1 & 0\\ \end{bmatrix}, \begin{bmatrix} 1 & 0\\ 0 & 1\\ \end{bmatrix}, \begin{bmatrix} 0 & 1\\ 0 & 1\\ \end{bmatrix} $
Any convex combination of those four points produces a valid stochastic matrix. However, there are distinct convex combinations that produce the same matrix. So while I think I can squint and interpret the space unit square as a simplex (by adding the two diagonals and "raising" one diagonal off the plane), it doesn't seem like the most appropriate characterization.
Is there a special word for "Cartesian product of simplices"? By the way, here's what I think is a depiction of the Cartesian product of two 2-simplices
