[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84418-en":3,"doc-seo-84418-105":29,"detail-sidebar-cat-0-en-105":95},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84418,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Big Data Approach to Kazhdan–Lusztig Polynomials","Investigates the structure of Kazhdan–Lusztig (KL) polynomials for the symmetric group by applying big-data style computational methods. The study uses exploratory data analysis and topological data analysis to analyze KL polynomials for symmetric groups of up to 11 strands, aiming to expose statistical patterns that are difficult to access via purely combinatorial or geometric arguments. Results and conjectural structures are organized around densities, growth extremes, unimodality, roots, and prime divisors, including a construction termed the KL ballmapper.","arXiv :2412 .01283v3 [math .RT] 13 Jul 2026  \nBIG DATA APPROACH TO KAZHDAN–LUSZTIG POLYNOMIALS  \nABEL LACABANNE, DANIEL TUBBENHAUER AND PEDRO VAZ  \nAbstract . 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.  \nContents  \n1. Introduction 1  \n2. The data and figures 4  \n3. Preliminaries in a nutshell and notation 14  \n4. Density A: Percentage of nonzero KL polynomials 16  \n5. Density B: Number of KL polynomials 17  \n6. Extremes: Growth of KL polynomials 18  \n7. Structure A: Unimodality of KL polynomials 20  \n8. Structure B: Roots of KL polynomials: all roots 21  \n9. Structure C: Roots of KL polynomials: Perron–Frobenius root 22  \n10. Structure D: Prime divisors of KL polynomials 23  \n11. Structure E: The KL ballmapper 24  \nReferences 25  \n1. Introduction  \nWe investigate a data set of Kazhdan–Lusztig (KL) polynomials using techniques traditionally employed in data science.  \n1A. Background and ideas. The Kazhdan–Lusztig (KL) polynomials are fundamental yet enigmatic objects in combinatorial representation theory. First introduced in [KL79] for arbitrary Coxeter groups, this paper focuses on the type A Coxeter group, the symmetric group Sn on {1, ... , n} . In this context, KL polynomials represent entries in a nonnegatively graded change-of-basis matrix between simple and Verma modules of sln. As a result, these polynomials are either zero or belong to 1 + qZ≥0[q], where q denotes the grading variable.  \nAlthough a main focus of research in combinatorics, geometry, and representation theory alike, these polynomials still hold many mysteries. The starting point of this work is the observation that KL polynomials exhibit patterns akin to statistical distributions. By analyzing these distributions, we aim to uncover structures that remain elusive through traditional methods, such as combinatorial or geometric approaches. Our methodology involves systematic data processing and analysis, often referred to as “big data,” utilizing techniques such as data visualization, exploratory data analysis (EDA), and topological data analysis (TDA) . Our approach is inspired by work in knot theory as, for example, [LHS22 , DGS24 , TZ25] and differs from the deep learning approaches to KL theory as in e.g. [DVB+ 21 , Wil24] .  \nUsing these methods, we will discuss several conjectures about KL polynomials and their distribution, and for some of them, we provide a possible approach for proving these conjectures. Additionally, the data science approach offers a different perspective, revealing patterns that, while possibly beyond formal verification, are still worth highlighting.  \nRemark 1A.1 . A critical aspect of this study is the scale of the data: KL polynomials are indexed by pairs of elements in Sn , resulting in (n!)2 values. Although computable, their computation for S11 , with one permutation fixed as trivial, required approximately 60 days using the program in [War11] on the servers of Laboratoirede Mathématiques Blaise Pascal, Université Clermont Auvergne. For S 12 , we anticipate significantly higher  \ncomputational costs, at least 12 times longer, although this is a very conservative lower bound that completely 2020 Mathematics Subject Classification. Primary: 05E10, 62R07, secondary: 20C08, 68P05 .  \nKey words and phrases. Visualization, exploratory data analysis, topological data analysis, conjecturing, Kazhdan–Lusztig polynomials.  \n2 A. LACABANNE, D. TUBBENHAUER AND P. VAZ  \nexcludes potential memory limitations. One particularly notable KL polynomial, with potentially the largest value at q = 1 (when normalized by dividing by the group size) ever computed, see Example 6.7 below, is for  \nS 13 and required approximately 60 days of computation on the same server. ✸  \nNotation 1A.2 . In this paper, we have conjectures and specul","cbCaioJZ3JtpKyso","https://ap.wps.com/l/cbCaioJZ3JtpKyso","pdf",3569449,1,27,"English","en",105,"# Introduction\n# The data and figures\n# Preliminaries in a nutshell and notation\n# Density A: Percentage of nonzero KL polynomials\n# Density B: Number of KL polynomials\n# Extremes: Growth of KL polynomials\n# Structure A: Unimodality of KL polynomials\n# Structure B: Roots of KL polynomials: all roots\n# Structure C: Roots of KL polynomials: Perron–Frobenius root\n# Structure D: Prime divisors of KL polynomials\n# Structure E: The KL ballmapper\n# References","[{\"question\":\"What is the main goal of the study on Kazhdan–Lusztig polynomials?\",\"answer\":\"To uncover structural patterns in Kazhdan–Lusztig polynomials for symmetric groups using computational methods from big data, including exploratory and topological data analysis.\"},{\"question\":\"Which symmetric groups and KL polynomials are analyzed?\",\"answer\":\"KL polynomials for symmetric groups of up to 11 strands are investigated, with the paper focusing on the indexed polynomial data arising from pairs of elements in Sn.\"},{\"question\":\"What kinds of structural questions are addressed?\",\"answer\":\"The work studies densities of nonzero polynomials and total counts, growth extremes, unimodality, and root-related structure (including the Perron–Frobenius root), as well as prime divisors and the KL ballmapper.\"},{\"question\":\"How does the paper position its approach relative to traditional methods and deep learning?\",\"answer\":\"It aims to reveal distributional structures that may be hard to access through traditional combinatorial or geometric techniques, and it explicitly contrasts its methodology with deep learning approaches to KL theory.\"}]",1784195498,68,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":90,"head_meta":92,"extra_data":94,"updated_unix":27},"big-data-approach-to-kazhdanlusztig-polynomials","",{"@graph":35,"@context":89},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/big-data-approach-to-kazhdanlusztig-polynomials/84418/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main goal of the study on Kazhdan–Lusztig polynomials?","Question",{"text":75,"@type":76},"To uncover structural patterns in Kazhdan–Lusztig polynomials for symmetric groups using computational methods from big data, including exploratory and topological data analysis.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which symmetric groups and KL polynomials are analyzed?",{"text":80,"@type":76},"KL polynomials for symmetric groups of up to 11 strands are investigated, with the paper focusing on the indexed polynomial data arising from pairs of elements in Sn.",{"name":82,"@type":73,"acceptedAnswer":83},"What kinds of structural questions are addressed?",{"text":84,"@type":76},"The work studies densities of nonzero polynomials and total counts, growth extremes, unimodality, and root-related structure (including the Perron–Frobenius root), as well as prime divisors and the KL ballmapper.",{"name":86,"@type":73,"acceptedAnswer":87},"How does the paper position its approach relative to traditional methods and deep learning?",{"text":88,"@type":76},"It aims to reveal distributional structures that may be hard to access through traditional combinatorial or geometric techniques, and it explicitly contrasts its methodology with deep learning approaches to KL theory.","https://schema.org",{"og:url":51,"og:type":91,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":93,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":96},[97,101,105,109,114,119,124,127,132,135,139],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Story & 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