{"id":"https://openalex.org/W7223035533","doi":"https://doi.org/10.48550/arxiv.2610.25036","title":"Schur-Positve Symmetric Functions from Products: Log-Majorization, Total-Positivity, and PSD Matrices","display_name":"Schur-Positve Symmetric Functions from Products: Log-Majorization, Total-Positivity, and PSD Matrices","publication_year":2026,"publication_date":"2026-10-18","ids":{"openalex":"https://openalex.org/W7223035533","doi":"https://doi.org/10.48550/arxiv.2610.25036"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.25036","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.25036","pdf_url":"https://arxiv.org/pdf/2610.25036","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.25036","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5038115890","display_name":"Isaiah Siegl","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Siegl, Isaiah","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5038115890"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/symmetric-functions","display_name":"symmetric functions","score":0.8759999871253967},{"id":"https://openalex.org/keywords/chromatic-symmetric-functions","display_name":"chromatic symmetric functions","score":0.9139999747276306},{"id":"https://openalex.org/keywords/total-positivity","display_name":"total positivity","score":0.9850000143051147},{"id":"https://openalex.org/keywords/positive-semidefinite-matrices","display_name":"positive semidefinite matrices","score":0.9269999861717224},{"id":"https://openalex.org/keywords/independence-polynomial","display_name":"independence polynomial","score":0.6779999732971191},{"id":"https://openalex.org/keywords/incomparability-graphs","display_name":"incomparability graphs","score":0.9639999866485596},{"id":"https://openalex.org/keywords/directed-graphs","display_name":"directed graphs","score":0.6110000014305115},{"id":"https://openalex.org/keywords/graph-laplacian","display_name":"graph Laplacian","score":0.4129999876022339},{"id":"https://openalex.org/keywords/chromatic-polynomial","display_name":"chromatic polynomial","score":0.3499999940395355}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8149999976158142},{"id":"https://openalex.org/C3746008","wikidata":"https://www.wikidata.org/wiki/Q981351","display_name":"Symmetric function","level":2,"score":0.7307000160217285},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5781999826431274},{"id":"https://openalex.org/C196956537","wikidata":"https://www.wikidata.org/wiki/Q202021","display_name":"Chromatic scale","level":2,"score":0.5343000292778015},{"id":"https://openalex.org/C126385604","wikidata":"https://www.wikidata.org/wiki/Q1087609","display_name":"Chromatic polynomial","level":3,"score":0.499099999666214},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4803999960422516},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.46389999985694885},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.45399999618530273},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4271000027656555},{"id":"https://openalex.org/C171436396","wikidata":"https://www.wikidata.org/wiki/Q5156499","display_name":"Complete homogeneous symmetric polynomial","level":5,"score":0.4131999909877777},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4027999937534332},{"id":"https://openalex.org/C64338288","wikidata":"https://www.wikidata.org/wiki/Q2235316","display_name":"Elementary symmetric polynomial","level":4,"score":0.36820000410079956},{"id":"https://openalex.org/C129010115","wikidata":"https://www.wikidata.org/wiki/Q1298683","display_name":"Power sum symmetric polynomial","level":5,"score":0.366100013256073},{"id":"https://openalex.org/C52153879","wikidata":"https://www.wikidata.org/wiki/Q930499","display_name":"Symmetric polynomial","level":4,"score":0.36309999227523804},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.3441999852657318},{"id":"https://openalex.org/C21596040","wikidata":"https://www.wikidata.org/wiki/Q2896771","display_name":"Unimodality","level":2,"score":0.33970001339912415},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.33820000290870667},{"id":"https://openalex.org/C170412648","wikidata":"https://www.wikidata.org/wiki/Q4736415","display_name":"Alternating polynomial","level":4,"score":0.33640000224113464},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3353999853134155},{"id":"https://openalex.org/C204710682","wikidata":"https://www.wikidata.org/wiki/Q3036983","display_name":"Integral graph","level":5,"score":0.33480000495910645},{"id":"https://openalex.org/C78136718","wikidata":"https://www.wikidata.org/wiki/Q7600073","display_name":"Stanley symmetric function","level":5,"score":0.3172000050544739},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.31360000371932983},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.3125999867916107},{"id":"https://openalex.org/C156903151","wikidata":"https://www.wikidata.org/wiki/Q17102788","display_name":"Ring of symmetric functions","level":4,"score":0.3043000102043152},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.2957000136375427},{"id":"https://openalex.org/C128622974","wikidata":"https://www.wikidata.org/wiki/Q849512","display_name":"Symmetric group","level":2,"score":0.27889999747276306},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.2603999972343445},{"id":"https://openalex.org/C35651441","wikidata":"https://www.wikidata.org/wiki/Q625303","display_name":"Independence (probability theory)","level":2,"score":0.2531000077724457},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.2502000033855438}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.25036","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.25036","pdf_url":"https://arxiv.org/pdf/2610.25036","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.25036","doi":"https://doi.org/10.48550/arxiv.2610.25036","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.25036","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.25036","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.25036","pdf_url":"https://arxiv.org/pdf/2610.25036","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7223035533.pdf","grobid_xml":"https://content.openalex.org/works/W7223035533.grobid-xml"},"referenced_works_count":8,"referenced_works":["https://openalex.org/W1495032532","https://openalex.org/W2295392691","https://openalex.org/W4403582395","https://openalex.org/W7158236919","https://openalex.org/W7162641198","https://openalex.org/W7170268882","https://openalex.org/W7171809228","https://openalex.org/W7171878194"],"related_works":["https://openalex.org/W2785315198","https://openalex.org/W2557587033","https://openalex.org/W2949110295","https://openalex.org/W2969899287","https://openalex.org/W4213305752","https://openalex.org/W2807897965","https://openalex.org/W4406808066","https://openalex.org/W1909522092","https://openalex.org/W1815115176","https://openalex.org/W4402954251"],"abstract_inverted_index":{"For":[0],"a":[1,19,68,79,186,208],"polynomial":[2,61],"$F(t)":[3,175],"=":[4,30,111,176,180],"1":[5],"+":[6,9,13,178],"a_1":[7],"t":[8],"a_2":[10],"t^2":[11],"+\\dots":[12],"a_d":[14],"t^d$":[15],"with":[16],"coefficients":[17],"in":[18,47,121],"$\\mathbb{Q}$-algebra":[20],"$\\mathcal{R}$,":[21],"we":[22,41,73,91,113,129,169,200],"consider":[23],"Schur-positivity":[24,46,83,93,120,171],"of":[25,37,49,62,67,82,84,94,97,104,123,167,172,188,207],"the":[26,53,58,63,85,107,202],"symmetric":[27,87,99,124,204],"function":[28,88,125,205],"$\u03a9_F(\\mathbf{x})":[29],"\\prod_{i":[31],"\\geq":[32,138,149],"1}":[33],"F(x_i)$.":[34],"Using":[35,164],"work":[36,103],"Lindstr\u00f6m":[38],"and":[39,116,145,155,212],"Gessel--Viennot,":[40],"give":[42,78],"sufficient":[43,117],"conditions":[44,118],"for":[45,119,131,140,151,185,197],"terms":[48,122],"directed":[50],"networks.":[51],"In":[52,106,127],"case":[54,108],"when":[55,109,174],"$F(t)$":[56],"is":[57],"multivariate":[59],"independence":[60],"incomparability":[64],"graph":[65,210],"$inc(P)$":[66],"natural":[69],"unit":[70],"interval":[71],"order,":[72],"apply":[74],"these":[75,156],"ideas":[76],"to":[77,160],"new":[80],"proof":[81],"chromatic":[86,98],"$X_{inc(P)}(\\mathbf{x})$.":[89],"Additionally,":[90],"prove":[92,114,170],"certain":[95],"sums":[96],"functions,":[100],"refining":[101],"recent":[102],"Amdeberhan--Shareshian--Stanley.":[105],"$\\mathcal{R}":[110],"\\mathbb{Q}[z_1,z_2,\\dots,z_n]$,":[112],"necessary":[115],"evaluations.":[126],"particular,":[128],"show":[130,213],"$\\mathbf{u},":[132],"\\mathbf{v}":[133],"\\in":[134,192],"\\mathbb{R}_{\\geq":[135],"0}^n$,":[136],"$s_\u03bb(\\mathbf{u})":[137],"s_\u03bb(\\mathbf{v})$":[139],"all":[141,152],"partitions":[142,153],"$\u03bb$":[143],"if":[144,147],"only":[146],"$m_\u03bb(\\mathbf{u})":[148],"m_\u03bb(\\mathbf{v})$":[150],"$\u03bb$,":[154],"inequalities":[157],"are":[158],"equivalent":[159],"$\\mathbf{u}$":[161],"log-majorizing":[162],"$\\mathbf{v}$.":[163],"representation":[165],"theory":[166],"$GL(\\mathbb{C}^n)$,":[168],"$\u03a9_F(\\mathbf{x})$":[173],"\\det(I":[177],"t\\sum_{i":[179],"1}^m":[181],"z_i":[182],"\\mathbf{v}_i":[183],"\\mathbf{v}_i^*)$":[184],"collection":[187],"vectors":[189],"$\\mathbf{v}_1,\\mathbf{v}_2,\\dots,":[190],"\\mathbf{v}_m":[191],"\\mathbb{C}^n$.":[193],"As":[194],"an":[195],"application":[196],"this":[198],"result,":[199],"introduce":[201],"Kirchhoff":[203],"$K_G(\\mathbf{x})$":[206],"loopless":[209],"$G$":[211],"that":[214],"it":[215,216],"always":[217],"Schur-positive.":[218]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-21T00:00:00"}
