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Yang-Baxter representations of the infinite symmetric group

Lechner, Gandalf, Pennig, Ulrich and Wood, Simon 2017. Yang-Baxter representations of the infinite symmetric group. arXiv

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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∞. We give a complete classification of all such Yang-Baxter characters and determine which extremal characters of S∞ 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 Yang-Baxter representations of the Temperley-Lieb algebra at parameter q=2, which can be completely classified in terms of their rank and dimension.

Item Type: Article
Date Type: Published Online
Status: Submitted
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Publisher: Cornell University Press
Date of Acceptance: 1 July 2017
Last Modified: 27 Jan 2019 23:19

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