# Yang-Baxter representations of the infinite symmetric group

July 01, 2017

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 Yang-Baxter representations of the Temperley-Lieb algebra at parameter
$q=2$, which can be completely classified in terms of their rank and dimension.

Keywords:

Yang-Baxter equation