{"title":"Special Functions for Hyperoctahedral Groups Using Bosonic Lattice Models","authors":"Ben Brubaker, Will Grodzicki, Andrew Schultz","doi":"10.1007/s00026-024-00734-x","DOIUrl":null,"url":null,"abstract":"<div><p>Recent works have sought to realize certain families of orthogonal, symmetric polynomials as partition functions of well-chosen classes of solvable lattice models. Many of these use Boltzmann weights arising from the trigonometric six-vertex model <i>R</i>-matrix (or generalizations or specializations of these weights). In this paper, we seek new variants of bosonic models on lattices designed for Cartan type <i>C</i> root systems, whose partition functions match the zonal spherical function in type <i>C</i>. Under general assumptions, we find that this is possible for all highest weights in rank two and three, but not for higher rank. In ranks two and three, this may be regarded as a new generating function formula for zonal spherical functions (also known as Hall–Littlewood polynomials) in type <i>C</i>.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 2","pages":"491 - 539"},"PeriodicalIF":0.7000,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-024-00734-x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Recent works have sought to realize certain families of orthogonal, symmetric polynomials as partition functions of well-chosen classes of solvable lattice models. Many of these use Boltzmann weights arising from the trigonometric six-vertex model R-matrix (or generalizations or specializations of these weights). In this paper, we seek new variants of bosonic models on lattices designed for Cartan type C root systems, whose partition functions match the zonal spherical function in type C. Under general assumptions, we find that this is possible for all highest weights in rank two and three, but not for higher rank. In ranks two and three, this may be regarded as a new generating function formula for zonal spherical functions (also known as Hall–Littlewood polynomials) in type C.
期刊介绍：
Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board.
The scope of Annals of Combinatorics is covered by the following three tracks:
Algebraic Combinatorics:
Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices
Analytic and Algorithmic Combinatorics:
Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms
Graphs and Matroids:
Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches
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