Big data approach to Kazhdan-Lusztig polynomials

We investigate the structure of Kazhdan-Lusztig polynomials of the symmetric group by leveraging computational approaches from big data, including exploratory and topological data analysis, applied to the polynomials for symmetric groups of up to 11 strands.

💡 Research Summary

The paper presents a comprehensive computational study of Kazhdan–Lusztig (KL) polynomials for the symmetric groups Sₙ with n up to 11, and a single illustrative example for n = 13. Using the program of Wang (2011) on a high‑performance cluster, the authors compute every KL polynomial P_{v,w}(q) for all pairs of permutations (v,w) in Sₙ, both in the full data set and in the restricted setting where one permutation is the identity. The resulting database, together with high‑resolution figures and source code, is made publicly available.

The authors then apply data‑science techniques—exploratory data analysis (EDA) and topological data analysis (TDA)—to extract statistical and structural information from this massive data set. Their analysis is organized into three main themes: density, extremes, and structure.

Density. Sections 4 and 5 show that the overwhelming majority of KL polynomials are zero; for the full data set the proportion of non‑zero entries exceeds 90 % only for very small n and drops rapidly as n grows. When the identity permutation is fixed, however, almost every non‑zero polynomial is distinct, leading to the striking observation that “almost all non‑zero KL polynomials are different.”

Extremes. Section 6 investigates the size of the largest coefficient and the value at q = 1. Empirical evidence suggests super‑exponential growth in n, far surpassing the linear bound guaranteed by the construction of Pol (1999). An explicit example for S₁₃ exhibits an eight‑digit coefficient, illustrating the rapid escalation.

Structure. Sections 7–11 explore finer features. The vast majority of KL polynomials are unimodal (single peak); bimodal and trimodal cases become exceedingly rare as n increases. Root analysis reveals that about 24 % of the polynomials have real roots, with roughly 44 % of those lying near the unit circle. The leading real root (the Perron–Frobenius or PF root) is positive in almost every case, and violations of the PF property occur in less than 2 % of the data. Using the Ballmapper algorithm from TDA, the authors visualize the high‑dimensional space of KL polynomials, identifying dense “core” regions and sparse outliers.

Based on these observations the authors formulate several conjectures: (1) the proportion of non‑zero KL polynomials decays rapidly with n; (2) maximal coefficients and evaluations at q = 1 grow super‑exponentially; (3) almost all KL polynomials are unimodal and satisfy the PF property; (4) the distribution of roots exhibits fractal‑like patterns. They outline possible proof strategies, suggesting that the graded representation theory of KLR algebras, cellular structures of Hecke algebras, and combinatorics of Coxeter cells could be leveraged.

The paper also discusses limitations—computational cost explodes for n ≥ 12, memory constraints, and the current reliance on experimental evidence rather than rigorous proofs. Future directions include extending the methodology to other Coxeter types, antispherical and spherical KL polynomials, canonical bases of quantum groups, graded representation theory of KLR and Temperley–Lieb algebras, web spaces, and invariants arising from tensor categories and three‑manifold topology.

Acknowledgments note contributions from several experts, the use of ChatGPT for coding and proofreading, and funding sources. Overall, the work demonstrates how big‑data and topological visualization can uncover hidden regularities in Kazhdan–Lusztig theory, providing a new experimental foundation for subsequent theoretical advances.