## Bose-Mesner Algebras attached to Jones Pairs

Univ. of  Waterloo

In 1989, Jones used spin models to construct link invariants and braid group representations.
In particular, the Jones polynomial can be obtained from the simplest spin model called the Potts model. In 1995, Bannai and Bannai constructed  four-weight spin models, which are generalization of spin models that give invariants for oriented links.

In an important paper by Jaeger, he discovered the first connection between spin models and Bose-Mesner algebras. In 1995, Jaeger and Nomura showed that  there is a Bose-Mesner algebra attached to every spin model. A similar result for four-weight spin models also holds.
Bose-Mesner algebras, which are equivalent to association schemes, have connections to a vast number of combinatorial objects such as designs, codes and distance regular graphs.
Hence Jaeger's result draws the attention of combinatorialists to spin models.

Let $\schur$ denote the Schur product of two matrices, that is, $(A \schur B)_{ij} = A_{ij} B_{ij}$.
Given a matrix $C$ in $M_n({\Bbb C})$, we define endomorphisms  $X_C$ and $\Delta_C$
on $M_n({\Bbb C})$ by
\begin{equation*}
X_C(M) = CM, \quad \Delta_C(M) = C \schur M.
\end{equation*}
A pair of matrices $(A,B)$ from $M_n({\Bbb C})$ is called a Jones pair if
$X_A$ and $\Delta_B$ are invertible and they give a representation of
braid groups using Jones' construction.
(We save the details for the talk.)

Jones pairs provide a generalization of both spin models and four-weight spin models.  Moreover, Jones pairs also have Bose-Mesner algebras attached to them.
In this talk, we give an introduction of Jones pairs and discuss the Bose-Mesner algebras associated with spin models, four-weight spin models and Jones pairs.
In particular, we describe a family of five Bose-Mesner algebras attached to each Jones pair.

This is joint work with Chris Godsil and Akihiro Munemasa.