On a family of symmetric rational functions

By Alexei Borodin. 2014

125 citations recorded by the source corpus.

This paper is about a family of symmetric rational functions that form a one-parameter generalization of the classical Hall-Littlewood polynomials. We introduce two sets of (skew and non-skew) functions that are akin to P and Q Hall-Littlewood polynomials. We establish (a) a combinatorial formula that represents our functions as partition functions for certain path ensembles in the square grid; (b) symmetrization formulas for non-skew functions; (c) identities of Cauchy and Pieri type; (d) explicit formulas for principal specializations; (e) two types of orthogonality relations for non-skew functions. Our construction is closely related to the half-infinite volume, finite magnon sector limit of the higher spin six-vertex (or XXZ) model, with both sets of functions representing higher spin six-vertex partition functions and/or transfer-matrices for certain domains.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0f429929e91994a2a5362392adbe9f069b0fbf35

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Citations in the corpus

Higher spin six vertex model and symmetric rational functions (2016, methodology): “1 They used duality and coordinate Bethe ansatz to show the integrability of their model; the Bethe ansatz part relied on previous works of Borodin–Corwin–Petrov– Sasamoto [BCPS15b], [BCPS15a], and Borodin [Bor14].”
Stochastic Higher Spin Vertex Models on the Line (2015, methodology): “bols (a;q)n, where n is allowed to be negative are deﬁned as in Mathematicaby: (a;q)n = Qn−1 k=0(1 −aq k), n > 0, 1, n = 0, Q−n−1 k=0 (1 − aq n+k)−1, n < 0. 24 Remark 3.16. To relate to [Bor14], we may rewrite L(J) α (i 1,j 1;i 2,j 2) = (−s) j2(−sv−1)j1−j2q 2j1−j1 2 4 − 2j2−j22 4 (qJq1−j2;q”