Abstract: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.
| Comments: | 42 pages, 3 figures |
| Subjects: | Combinatorics (math.CO); Algebraic Geometry (math.AG) |
| Cite as: | arXiv:2605.26009 [math.CO] |
| (or arXiv:2605.26009v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.26009 arXiv-issued DOI via DataCite (pending registration) |
From: Jingqi Li [view email]
[v1]
Mon, 25 May 2026 16:27:30 UTC (75 KB)
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