Big data approach to Kazhdan-Lusztig polynomials
Abel Lacabanne, Daniel Tubbenhauer, Pedro Vaz
TL;DR
This work investigates the structure of Kazhdan-Lusztig polynomials for the symmetric group $\mathrm{S}_n$ using big-data techniques, pushing computations up to $n=11$ and treating polynomial coefficients with vectorization, visualization, and topological data analysis to reveal distributional patterns. It establishes empirical and conjectural results on nonzero density, the number and growth of KL polynomials, unimodality prevalence, root distributions (including real-root frequency and Perron–Frobenius behavior), and topology-based visualizations via ballmapper, linking representation-theoretic polynomials to data-science methodologies. The authors propose several conjectures (e.g., density $\mathrm{den}_n\sim n^{-(2+a)}$ and exponential growth of growth quantities) and outline future directions, including extensions to other KL-type polynomials and deeper connections to canonical bases and webs. The work provides a data-driven perspective on long-standing combinatorial objects, suggesting new conjectures, offering partial proofs/heuristics, and highlighting the potential of topological techniques for uncovering structure in algebraic invariants with broad applicability.
Abstract
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.