{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/bialternant-formula-for-schur-polynomials","title":"Bialternant formula for Schur polynomials with repeating variables","arxiv_id":"2312.15680","date":"2023-12-25","proceeding":null,"authors":["Luis Angel González-Serrano","Egor A. Maximenko"],"abstract":"We consider polynomials of the form $\\operatorname{s}_\\lambda(y_1^{[\\varkappa_1]},\\ldots,y_n^{[\\varkappa_n]})$, where $\\lambda$ is an integer partition, $\\operatorname{s}_\\lambda$ is the Schur polynomial associated to $\\lambda$, and $y_j^{[\\varkappa_j]}$ denotes $y_j$ repeated $\\varkappa_j$ times. We represent $\\operatorname{s}_\\lambda(y_1^{[\\varkappa_1]},\\ldots,y_n^{[\\varkappa_n]})$ as a quotient whose the denominator is the determinant of the confluent Vandermonde matrix, and the numerator is the determinant of some generalized confluent Vandermonde matrix. We give three algebraic proofs of this formula.","url_abs":"https://arxiv.org/abs/2312.15680v1","url_pdf":"https://arxiv.org/pdf/2312.15680v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"bialternant-formula-for-schur-polynomials","repo_url":"https://github.com/egormaximenko/schur_rep","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}