### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 064, 23 pages      arXiv:1801.09635      https://doi.org/10.3842/SIGMA.2018.064
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

### The Functional Method for the Domain-Wall Partition Function

Jules Lamers
Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Göteborg, Sweden

Received January 30, 2018, in final form June 17, 2018; Published online June 26, 2018

Abstract
We review the (algebraic-)functional method devised by Galleas and further developed by Galleas and the author. We first explain the method using the simplest example: the computation of the partition function for the six-vertex model with domain-wall boundary conditions. At the heart of the method lies a linear functional equation for the partition function. After deriving this equation we outline its analysis. The result is a closed expression in the form of a symmetrized sum - or, equivalently, multiple-integral formula - that can be rewritten to recover Izergin's determinant. Special attention is paid to the relation with other approaches. In particular we show that the Korepin-Izergin approach can be recovered within the functional method. We comment on the functional method's range of applicability, and review how it is adapted to the technically more involved example of the elliptic solid-on-solid model with domain walls and a reflecting end. We present a new formula for the partition function of the latter, which was expressed as a determinant by Tsuchiya-Filali-Kitanine. Our result takes the form of a `crossing-symmetrized' sum with $2^L$ terms featuring the elliptic domain-wall partition function, which appears to be new also in the limiting case of the six-vertex model. Further taking the rational limit we recover the expression obtained by Frassek using the boundary perimeter Bethe ansatz.

Key words: six-vertex model; solid-on-solid model; reflecting end; functional equations.

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