Yang-Baxter representations of the infinite symmetric group

Classifies all unitary involutive solutions of the Yang–Baxter equation up to a natural representation-theoretic equivalence, with classes labelled by pairs of Young diagrams and an explicit normal form for each. The proof uses characters of the infinite symmetric group and subfactor techniques.

Abstract

Every unitary involutive solution of the quantum Yang-Baxter equation (“R-matrix”) defines an extremal character and a representation of the infinite symmetric group \(S_\infty\). We give a complete classification of all such Yang-Baxter characters and determine which extremal characters of \(S_\infty\) are of Yang-Baxter form. Calling two involutive R-matrices equivalent if they have the same character and the same dimension, we show that equivalence classes are classified by pairs of Young diagrams, and construct an explicit normal form R-matrix for each class. Using operator-algebraic techniques (subfactors), we prove that two R-matrices are equivalent if and only if they have similar partial traces. Furthermore, we describe the algebraic structure of the equivalence classes of all involutive R-matrices, and discuss several classes of examples. These include unitary Yang-Baxter representations of the Temperley-Lieb algebra at loop parameter \(δ=2\), which can be completely classified in terms of their rank and dimension.

Recommended citation: G. Lechner, U. Pennig and S. Wood, Adv. Math. 355, 106769 (2019)
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