Generating Sets of Stochastic Matrices
Frederik vom Ende, Fereshte Shahbeigi
TL;DR
The paper addresses generating sets for the semigroup of column-stochastic matrices $s(n)$ by developing a divisibility framework with indivisible elements and building blocks. It adapts a sign-based divisibility criterion to $s(n)$ and proves that all $2\times2$ stochastic matrices are divisible, while characterizing prime elements in $s(3)$ up to $S_3\times S_3$-conjugation. It then constructs explicit generating sets: for $n=2$ with $G=\{0110\}\cup \mathrm{conv}\{\mathbf{1},1100\}$ yielding $N_G(s(2))=4$, and for $n=3$ with $G=\{010100001\}\cup \mathrm{conv}\{\mathbf{1},101010000,101000010,001100010\}$ giving $N_G(s(3))\le 20$, along with a demonstration that $S_3$ can be generated from a small subset. The work provides a concrete, finite bound on the number of factors needed and offers a strategy that may extend to higher dimensions, while highlighting substantial challenges and open questions for $n\ge 4$ and the corresponding growth of $N_G(S(n))$.
Abstract
This paper introduces the concept of a generating set for stochastic matrices -- a subset of matrices whose repeated composition generates the entire set. Understanding such generating sets requires specifying the "indivisible elements" and "building blocks" within the set, which serve as fundamental components of the generation process. Expanding upon prior studies, we develop a framework that formalizes divisibility in the context of stochastic matrices. We provide a sufficient condition for divisibility that is shown to be necessary in dimension $n=3$, while for $n=2$, all stochastic matrices are shown to be divisible. Using these results, we construct generating sets for dimensions 2 and 3 by specifying the indivisible elements, and, importantly, we give an upper bound for the number of factors required from the generating set to produce the entire semigroup.