Random Permutations from Bott-Samelson Varieties

Motivated by a recent random pipe dream model, we study a family of probability distributions on $S_n$ arising from Bott--Samelson varieties over finite fields. More precisely, for a word $R$, we consider the Bott--Samelson map $\pi_R:\mathrm{BS}^R\to \mathcal{F}\ell_n$ and define a distribution $\mathbb{P}_{R,q}$ by counting the $\mathbb{F}_q$-points in the inverse images of Schubert cells. For a suitable choice of parameters $p_1=q/(1+q)$ and $p_2=1/q$, this construction recovers a special case of the random pipe dream distribution. The main problem considered in this note is to determine which combinatorial properties of a reduced word are detected by the distribution $\mathbb{P}_{R,q}$. We prove the stronger statement that, for arbitrary reduced words $R_1,R_2$, the equality $\mathbb{P}_{R_1,q}=\mathbb{P}_{R_2,q}$ as functions of $q$ holds if and only if $R_1$ and $R_2$ lie in the same commutation class. In particular, equality of distributions already forces the two words to represent the same permutation. The proof combines the Bott--Samelson interpretation with Demazure products, commutation-class invariants, and Hecke-algebraic arguments.