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Nested coordinate Bethe ansatz

Gleb Eduardovich Arutyunov

Published Apr 27, 2026• 0 citations

Abstract

This chapter focuses on the diagonalization of a set of commuting scattering operators, or equivalently, transfer matrices, in models involving higher-rank symmetries. The discussion centers on modules more intricate than those of [N – M, M] type, ultimately considering arbitrary irreducible representations of the symmetric group. The core idea is to construct a suitable basis in the representation space that enables the reduction of a higher-dimensional diagonalization problem to a sequence of lower-dimensional ones that are already solved. This reduction is achieved through a recursive scheme in which each step lowers the matrix dimension, enabling its diagonalization–a procedure known as nested Bethe ansatz. The method is framed as a successive dimensional reduction, systematically building on solutions from simpler cases. Special attention is given to Yang’s connection formula and its role in enabling this recursive approach. The chapter extends the framework introduced in Chapter 3, developing a coordinate version of the nested Bethe ansatz and generalizing it to accommodate multiple levels of nesting. This recursive structure reveals deep algebraic insights and plays a pivotal role in the study of integrable systems with rich internal symmetries.

DOI: 10.1017/9781009664837.006

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