Yang-Baxter representations of the infinite symmetric group

Lechner, Gandalf ORCID: https://orcid.org/0000-0002-8829-3121, Pennig, Ulrich ORCID: https://orcid.org/0000-0001-5441-6130 and Wood, Simon ORCID: https://orcid.org/0000-0002-8257-0378 2019. Yang-Baxter representations of the infinite symmetric group. Advances in Mathematics 355 , 106769. 10.1016/j.aim.2019.106769

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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∞. 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:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	Elsevier
ISSN:	0001-8708
Date of First Compliant Deposit:	5 August 2019
Date of Acceptance:	5 August 2019
Last Modified:	13 Nov 2024 09:00
URI:	https://orca.cardiff.ac.uk/id/eprint/124722

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