Computation and sampling for Schubert specializations

Abstract

We present computational results on principal specializations $\mathfrak{S}_w(1^n)$ of Schubert polynomials, which count reduced pipe dreams and reduced bumpless pipe dreams (RBPD). We find the first counterexample, at $n=17$, to the Merzon-Smirnov conjecture (arXiv:1410.6857) that the maximum of $\mathfrak{S}_w(1^n)$ over $S_n$ is attained at a layered permutation. The simulations suggest that $\lim_{n \to \infty} \log(\max_{w\in S_n}\mathfrak{S}_w(1^n))/n^2$ equals the maximal layered permutations' constant from Morales-Pak-Panova (arXiv:1805.04341). We also explore the random permutation drawn from the distribution proportional to $\mathfrak{S}_w(1^n)$, revealing permuton-like asymptotics similar to those for Grothendieck polynomials by Morales-Panova-Petrov-Yeliussizov (arXiv:2407.21653). We implement and compare three recurrences for $\mathfrak{S}_w(1^n)$: the descent formula (Macdonald), transition formula (Lascoux--Schutzenberger), and cotransition formula (Knutson). For sampling uniformly random RBPDs (whose count is $\sum_{w\in S_n} \mathfrak{S}_w(1^n)$), we show that reducedness breaks the sublattice property of the ASM lattice, preventing monotone CFTP and causing false coalescence. We develop an efficient MCMC sampler with macroscopic "droop" updates for connectivity and fast mixing. Our code computes $\mathfrak{S}_w(1^n)$ up to $n\sim 20$ and samples random RBPDs up to $n\sim 60$ on a personal computer ($n\sim 100$ on a cluster).

Figures (17)

• Figure 1: Pipe dreams for $n=4$. Left: A reduced pipe dream for $w = 1432$ where each pair of pipes crosses at most once. Right: A non-reduced (forbidden) pipe dream where pipes 3 and 4 cross twice. Color is added merely as a visual aid to highlight the pipes' paths.
• Figure 2: Top: The six tile types used in bumpless pipe dreams. Bottom: A reduced bumpless pipe dream for $n=4$ corresponding to the permutation $w=2143$.
• Figure 3: The simulated density histogram for $n=100$ for the Schubert (left) and Grothendieck (right) random permutations, displaying striking similarities in behavior. The permuton limit for the Grothendieck case is proven in GrothendieckShenanigans2024, while the Schubert case is open.
• Figure 4: Rothe diagram $D(w)$ for $w=(3,1,4,2)$. Dots mark positions $(i,w(i))$; shaded cells form $D(w)$. (Here $|D(w)| = 3 = \ell(w)$.)
• Figure 5: Height functions on $n=4$ BPDs. Values of $h_D(i,j)$ appear at dual-lattice vertices; six-vertex lines are the level lines of $h_D$.

Theorems & Definitions (47)

• Conjecture 1: merzon2016determinantal
• Theorem 1
• proof
• Remark 1: Typical vs. maximal
• Theorem 2: Descent formula
• proof : Proof of \ref{['thm:descent']}
• Theorem 3: Transition formula LascouxSchutzenberger1985LR
• Theorem 4: Cotransition formula Knutson2019cotransition
• Remark 2
• Remark 3