{"pk":65010,"title":"Equivalences of LLT polynomials via lattice paths","subtitle":null,"abstract":"The LLT polynomials \\(\\mathcal{L}_{{{\\beta}/{\\gamma}}} (X;t)\\) are a family of symmetric polynomials indexed by a tuple of (possibly skew-)partitions \\({{\\beta}/{\\gamma}}= (\\beta^{(1)}/\\gamma^{(1)},\\ldots,\\beta^{(k)}/\\gamma^{(k)})\\). It has recently been shown that these polynomials can be seen as the partition function of a certain vertex model whose boundary condition is determined by \\({{\\beta}/{\\gamma}}\\). In this paper we describe an algorithm which gives a bijection between the configurations of the vertex model with boundary condition \\({{\\beta}/{\\gamma}} = (\\beta^{(1)}/\\gamma^{(1)},\\beta^{(2)}/\\gamma^{(2)})\\) and those with boundary condition \\(({{\\beta}/{\\gamma}})_{swap} = (\\beta^{(2)}/\\gamma^{(2)},\\beta^{(1)}/\\gamma^{(1)})\\). We prove a sufficient condition for when this bijection is weight-preserving up to an overall factor of \\(t\\), which in turn implies that the corresponding LLT polynomials are equal up to the same overall factor. Extending these techniques, we are able to systematically determine linear relations within families of LLT polynomials.Mathematics Subject Classifications: 05E05Keywords: LLT polynomials, vertex models","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"LLT polynomials"},{"word":"vertex models"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/0f77k104","frozenauthors":[{"first_name":"David","middle_name":"","last_name":"Keating","name_suffix":"","institution":"Department of Mathematics, University of Illinois, Urbana-Champaign, Urbana, U.S.A.","department":""}],"date_submitted":"2025-09-12T15:07:04+02:00","date_accepted":"2025-09-12T15:07:04+02:00","date_published":"2025-09-15T09:00:00+02:00","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/65010/galley/49820/download/"}]}