Random Permutations from Bott-Samelson Varieties

Summary · qwen2.5:32b

The study examines probability distributions on \(S_n\) derived from Bott-Samelson varieties over finite fields, showing that for reduced words \(R_1\) and \(R_2\), their associated distributions \(\mathbb{P}_{R_1,q}\) and \(\mathbb{P}_{R_2,q}\) are equal if and only if \(R_1\) and \(R_2\) belong to the same commutation class, indicating that distribution equality implies the words represent the same permutation.

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Exploring the properties of Bott-Samelson varieties and their relation to combinatorial properties of reduced words in a mathematical context.

Excerpt

arXiv:2605.26009v2 Announce Type: replace 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.