[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-135466-en":3,"doc-seo-135466-105":31,"detail-sidebar-cat-0-en-105":92},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},135466,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","NONCOMMUTATIVE CHROMATIC QUASI-SYMMETRIC FUNCTIONS, MACDONALD POLYNOMIALS, AND THE YANG-BAXTER EQUATION","Chromatic quasi-symmetric functions associated with Shareshian-Wachs are lifted to WQSym, enabling a study of alphabet transformations in a noncommuting setting. The work proposes a noncommutative analogue of Macdonald polynomials compatible with a noncommutative Haglund–Wilson formula, and introduces a multi-t version. For rectangular partitions at q = 0, the commutative images are conjectured to match multi-t Hall–Littlewood functions, with Macdonald polynomials realizable via equivariant traces of Yang–Baxter elements in Hecke algebras.","arXiv :2502 .09072v2 [math .CO] 4 Nov 2025  \nNONCOMMUTATIVE CHROMATIC QUASI-SYMMETRIC FUNCTIONS, MACDONALD POLYNOMIALS, AND THE YANG-BAXTER EQUATION  \nJEAN-CHRISTOPHE NOVELLI AND JEAN-YVES THIBON  \nAbstract. As shown in our paper [JCTA 177 (2021), Paper No. 105305], the chromatic quasi-symmetric function of Shareshian-Wachs can be lifted to WQSym, the algebra of quasi-symmetric functions in noncommuting variables. We investigate here its behaviour with respect to classical transformations of alphabets and propose a noncommutative analogue of Macdonald polynomials compatible with a noncommutative version of the Haglund-Wilson formula. We also introduce a multi-t version of these noncommutative analogues. For rectangular partitions, their commutative images at q = 0 appear to coincide with the multi-t Hall-Littlewood functions introduced in [Lett. Math. Phys. 35 (1995), 359–374] . This leads us to conjecture that for rectangular partitions, multi-t Macdonald polynomials are obtained as equivariant traces of certain Yang-Baxter elements of Hecke algebras. We also conjecture that all (ordinary) Macdonald polynomials can be obtained in this way. We conclude with some remarks relating various aspects of quasi-symmetric chromatic functions to calculations in Hecke algebras. In particular, we show that all modular relations are given by the product formula of the Kazhdan-Lusztig basis.  \n1. Introduction  \nThis paper is a continuation of [35] . In this reference, we obtained a noncommutative analogue of the Carlsson-Mellit identity [7] relating unicellular LLT polynomialsand the quasi-symmetric chromatic polynomials [37] of Dyck graphs  \n(1) XG (t, X) = (t − 1) −n LLTG (t,(t − 1)X) .  \nWe first showed how to deduce this relation from a morphism from the Guay-Paquet Hopf algebra to QSym, and then obtained a noncommutative version by extending this morphism to WQSym.  \nThis method can be extended to other transformations of alphabets. In Section 3, we describe the image of the noncommutative chromatic quasi-symmetric function XG by the extension to WQSym of the ω-involution of QSym, and the image of the noncommutative unicellular LLT-polynomials LLTG (A;t) by the transformation A 7→ A (q − 1) . In Section 4, we propose a noncommutative lift of the Macdonald polynomials Hµ to WQSym. The rationale for this definition is the Haglund-Wilson formula expressing the J-functions in terms of chromatic quasi-symmetric functions:  \n2020 Mathematics Subject Classification. 05E05, 20C30, 60C05 .  \nKey words and phrases. Noncommutative symmetric functions, Quasi-symmetric functions, LLT polynomials, Macdonald polynomials, Yang-Baxter equation, Hecke algebras, chromatic polynomials.  \n2 J.-C. NOVELLI AND J.-Y. THIBON  \nif we replace in this formula the ordinary XH by their noncommutative version, we obtain a multiplicity-free sum of terms qi tm Mu where i, m are HHL statistics inv and maj of the packed word u interpreted as the row-reading of a filling of the integer partition µ associated with u. This definition is then extended to a multi-t-analogue. Next, we relate Macdonald polynomials to the Yang-Baxter bases of Hecke algebras, and propose a conjectural expression of the Macdonald polynomials as equivariant traces of intertwiners. We also relate the e-positivity conjecture to properties of the Yang-Baxter basis. Finally, we conclude by showing that all modular relations derive from the product formula of the Kazhdan-Lusztig basis of the Hecke algebra.  \nAcknowledgements. This research has been partially supported by the project CARPLO of the Agence Nationale de la Recherche (ANR-20-CE40-0007) .  \n2. General setup  \nRecall that the Guay-Paquet Hopf algebra G is based on finite simple undirected graphs with vertices labelled by the integers from 1 to n = |V(G)| . The product is the shifted concatenation: G · H = G ∪ H[n] where H [n] is H with labels shifted by the number n of vertices of G.  \nThe parameter t arises in the coproduct. If G is a graph ","cbCaijdn2Tz7aelp","https://ap.wps.com/l/cbCaijdn2Tz7aelp","pdf",479889,4,1,26,"English","en",105,"# 1. Introduction\n# 2. General setup","[{\"question\":\"What is the main object the paper studies and how is it lifted to WQSym?\",\"answer\":\"It studies the chromatic quasi-symmetric function of Shareshian-Wachs and lifts it to WQSym, the algebra of quasi-symmetric functions in noncommuting variables. This lifting allows the authors to analyze behaviour under classical alphabet transformations in a noncommutative framework.\"},{\"question\":\"What noncommutative version of Macdonald polynomials is proposed?\",\"answer\":\"The paper proposes a noncommutative analogue of Macdonald polynomials Hμ in WQSym. The definition is designed to be compatible with a noncommutative analogue of the Haglund–Wilson formula, obtained by replacing ordinary chromatic quasi-symmetric functions with their noncommutative counterparts.\"},{\"question\":\"What conjectural connection links multi-t Macdonald polynomials to Yang–Baxter elements?\",\"answer\":\"For rectangular partitions, the authors conjecture that multi-t Macdonald polynomials arise as equivariant traces of certain Yang–Baxter elements of Hecke algebras. They further conjecture that all ordinary Macdonald polynomials can be obtained in the same way.\"}]","NONCOMMUTATIVE CHROMATIC QUASI-SYMMETRIC FUNCTIONS, MACDONALD POLYNOMIALS, AND THE YANG-BAXTER EQUATION | PDF",1787312869,66,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":29},"noncommutative-chromatic-quasi-symmetric-functions-macdonald-polynomials-and-the-yang-baxter-equation","",{"@graph":37,"@context":86},[38,54,69],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":20},"https://docshare.wps.com/document/noncommutative-chromatic-quasi-symmetric-functions-macdonald-polynomials-and-the-yang-baxter-equation/135466/",{"url":53,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":42,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-09-03","2026-08-21",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What is the main object the paper studies and how is it lifted to WQSym?","Question",{"text":76,"@type":77},"It studies the chromatic quasi-symmetric function of Shareshian-Wachs and lifts it to WQSym, the algebra of quasi-symmetric functions in noncommuting variables. This lifting allows the authors to analyze behaviour under classical alphabet transformations in a noncommutative framework.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What noncommutative version of Macdonald polynomials is proposed?",{"text":81,"@type":77},"The paper proposes a noncommutative analogue of Macdonald polynomials Hμ in WQSym. The definition is designed to be compatible with a noncommutative analogue of the Haglund–Wilson formula, obtained by replacing ordinary chromatic quasi-symmetric functions with their noncommutative counterparts.",{"name":83,"@type":74,"acceptedAnswer":84},"What conjectural connection links multi-t Macdonald polynomials to Yang–Baxter elements?",{"text":85,"@type":77},"For rectangular partitions, the authors conjecture that multi-t Macdonald polynomials arise as equivariant traces of certain Yang–Baxter elements of Hecke algebras. They further conjecture that all ordinary Macdonald polynomials can be obtained in the same way.","https://schema.org",{"og:url":53,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":107,"slug":139},19,"General","general"]