Four-weight Spin Models and Jones Pairs

We introduce and discuss Jones pairs. These provide a generalization and a new approach to the four-weight spin models of Bannai and Bannai. We show that each four-weight spin model determines a “dual” pair of association schemes. 1. Jones Pairs The space of k × k matrices acts on itself in three distinct ways: if C ∈ Mk(F), we can define endomorphisms XC , ∆C and YC by XC(M) = CM, ∆C(M) = C ◦M, YC(M) = MC . If A and B are k × k matrices, we say (A,B) is a one-sided Jones pair if XA and ∆B are invertible and (1.1) XA∆BXA = ∆BXA∆B . We call this the braid relation for reasons that will become clear as we proceed. We note that XA is invertible if and only if A is, and ∆B is invertible if and only if the Schur inverse B(−) is defined. (Recall that if B and C are matrices of the same order, then their Schur product B ◦ C is defined by the condition (B ◦ C)i,j = Bi,jCi,j and B ◦ B(−) = J .) We will see that each one-sided Jones pair determines a representation of the braid groupB3, and this is one of the reasons we are interested in Jones pairs. The pair (I, J) forms a trivial but useful example. A one-sided Jones pair (A,B) is invertible if A(−) and B−1 both exist. We observe that XA and YA commute and that (1.2) YA∆BYA = ∆BYA∆B if and only if (A,B ) is a one-sided Jones pair. A pair of matrices (A,B) is a Jones pair if both (A,B) and (A,B ) are one-sided Jones pairs. From a Jones pair we obtain a family of representations of the braid groups Br, for all r. In this paper we describe the basic theory of Jones pairs. We find that if A(−) exists, then (A,A(−)) is a Jones pair if and only if A is a spin model in the sense of Received by the editors November 9, 2001. 2000 Mathematics Subject Classification. Primary 05E30; Secondary 20F36. Support from a National Sciences and Engineering Council of Canada operating grant is gratefully acknowledged by the second author. c ©2003 American Mathematical Society