[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-290398-105":59,"doc-detail-290398-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","machine-learning-kronecker-coefficients-abstract-machine-learning-prediction-of-zero-kronecker-coefficients","MACHINE-LEARNING KRONECKER COEFFICIENTS - Abstract - Machine-learning Prediction of Zero Kronecker Coefficients","","The Kronecker coefficients quantify the tensor-product decomposition multiplicities of two irreducible symmetric-group representations. Unlike Littlewood–Richardson coefficients for GLN(C), no known combinatorial description exists, and determining whether a coefficient is zero is NP-hard. This work trains standard machine-learning models—Nearest Neighbors, Convolutional Neural Networks, and Gradient Boosting Decision Trees—to perform binary classification of zero versus nonzero Kronecker coefficients with high accuracy (about 0.98).",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/machine-learning-kronecker-coefficients-abstract-machine-learning-prediction-of-zero-kronecker-coefficients/290398/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/machine-learning-kronecker-coefficients-abstract-machine-learning-prediction-of-zero-kronecker-coefficients/290398.png","ImageObject",300,407,{"name":92,"@type":93},"Eliana","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-09-17",true,{"@type":102,"interactionType":103,"userInteractionCount":39},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What are Kronecker coefficients in this paper?","Question",{"text":112,"@type":113},"They are the decomposition multiplicities g^ν_{λ,μ} appearing when tensoring two irreducible representations of the symmetric group S_n.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Why is deciding whether a Kronecker coefficient is zero difficult?",{"text":117,"@type":113},"There is no known combinatorial description, and the problem of deciding whether a given Kronecker coefficient is non-zero is NP-hard.",{"name":119,"@type":110,"acceptedAnswer":120},"Which machine-learning methods and how accurate are they?",{"text":121,"@type":113},"The paper uses Nearest Neighbors, Convolutional Neural Networks, and Gradient Boosting Decision Trees (LightGBM) to classify zero vs nonzero coefficients, achieving accuracy around 0.98.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},290398,1789641614,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":39,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":34,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":143,"preview_page_count":4},4398048949847,"https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267","arXiv :2306 .04734v1 [math .RT] 7 Jun 2023  \nMACHINE-LEARNING KRONECKER COEFFICIENTS  \nKYU-HWAN LEE⋆  \nAbstract. The Kronecker coefficients are the decomposition multiplicities of the tensor product of two irreducible representations of the symmetric group. Unlike the Littlewood–Richardson coefficients, which are the analogues for the general linear group, there is no known combinatorial description of the Kronecker coefficients, and it is an NP-hard problem to decide whether a given Kronecker coefficient is zero or not. In this paper, we show that standard machine-learning algorithms such as Nearest Neighbors, Convolutional Neural Networks and Gradient Boosting Decision Trees may be trained to predict whether a given Kronecker coefficient is zero or not. Our results show that a trained machine can efficiently perform this binary classification with high accuracy (≈ 0 .98) .  \n1. Introduction  \nOne of the major goals in representation theory is to decompose a representation into irreducible representations, and algebraic combinatorics plays a crucial role as it produces a practical way to describe such a decomposition. A prototype can be found in the decomposition of the tensor product of two irreducible representations of the general linear group GLN (C), where the Littlewood–Richardson rule completely describes the decomposition using skew semi-standard tableaux. Therefore it comes with surprise to realize that we do not have a similar success with the symmetric group until now.  \nTo be precise, let Sn be the symmetric group of degree n and consider representations of Sn over C. The irreducible representations Sλ of Sn are parametrized by partitions λ ⊢ n, and the dimension of Sλ is equal to the number of standard tableaux of shape λ . Consider the tensor product of two irreducible representations Sλ and Sµ for λ,µ ⊢ n. Then their tensor product is decomposed into a sum of irreducible representations:  \nSλ⊗ Sµ = Mgνλ,µ Sν (gνλ,µ ∈ Z≥0) .  \nν⊢n  \nThe decomposition multiplicities gνλ,µ are called the Kronecker coefficients.  \nIn stark contrast with the Littlewood–Richardson coefficients for GLN(C), no combinatorial description has been known for gνλ,µ since Murnaghan [Mur] initially posed the question in 1938, and it is still considered as one of the main problems in the combinatorial representation theory. For special cases, various combinatorial formulas are known due to Remmel [Rem], Ballantine–Orellana [BO], Remmel–Whitehead [RW], Blasiak–Mulmuley–Sohoni [BMS] and Blasiak [Bla] . Recently, the coefficients gνλ,µ have also been studied from the viewpoint of computational complexity. In particular, it is shown by Ikenmeyer, Mulmuley and Walter [IMW] that deciding whether a given  \nDate: June 9, 2023 .  \n⋆ This work was partially supported by a grant from the Simons Foundation (\\#712100) .  \nKronecker coefficient is non-zero is NP-hard, and other interesting results are obtained by Pak and Panova [PP1, PP2] .  \nIn this paper we apply machine-learning to the Kronecker coefficients gνλ,µ to show that a machine can be trained to learn these coefficients. Though this seems to be the first time for machinelearning to be applied to the coefficients gνλ,µ , there has been mounting evidence that a machine can be trained to learn mathematical structures. Indeed, starting with superstring theory [He1, He2 , KS, Ru, CHKN], machine-learning has been applied, for example, to group theory [HK], graph theory [HY, Wag], metric geometry [AHO, DLQ], knot invariants [JKP], and to number theory [HLOa, HLOb, HLOc] . Moreover, a new paradigm of AI-aided research in mathematics is emerging [DVB+] .  \nAs the main machine-learning methods for this paper, we utilize Nearest Neighbors, Convolutional Neural Networks and Gradient Boosting Decision Trees (specifically, LightGBM [LGBM]) as they perform better than other standard machine-learning algorithms. For a broad introduction to machine-learning, see [GBC, HTF] . Our datasets of the Kronecker coeffic","cbCail5DZ00xdQoQ","https://ap.wps.com/l/cbCail5DZ00xdQoQ","pdf",538528,"English","# Introduction\n# Methodology\n## Generating Datasets\n# Experimental Setup\n# Results and Observations\n# Conclusion and Future Directions","[{\"question\":\"What are Kronecker coefficients in this paper?\",\"answer\":\"They are the decomposition multiplicities g^ν_{λ,μ} appearing when tensoring two irreducible representations of the symmetric group S_n.\"},{\"question\":\"Why is deciding whether a Kronecker coefficient is zero difficult?\",\"answer\":\"There is no known combinatorial description, and the problem of deciding whether a given Kronecker coefficient is non-zero is NP-hard.\"},{\"question\":\"Which machine-learning methods and how accurate are they?\",\"answer\":\"The paper uses Nearest Neighbors, Convolutional Neural Networks, and Gradient Boosting Decision Trees (LightGBM) to classify zero vs nonzero coefficients, achieving accuracy around 0.98.\"}]","MACHINE-LEARNING KRONECKER COEFFICIENTS - Abstract - Machine-learning Prediction of Zero Kronecker Coefficients | PDF",18]