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.