On the Torsion Units of Integral Adjacency Algebras of Finite Association Schemes

Torsion units of group rings have been studied extensively since the 1960s. As association schemes are generalization of groups, it is natural to ask about torsion units of association scheme rings. In this paper we establish some results about torsion units of association scheme rings analogous to basic results for torsion units of group rings. 1. Introduction In this paper we will consider torsion units of rings generated by finite association schemes, which we now define. Let be a finite set of size . Let be a partition of such that every relation in is nonempty. For a relation , there corresponds an adjacency matrix, denoted by , which is the ？？-matrix whose entries are 1 if and 0 otherwise. is an association scheme if(i)is a partition of consisting of nonempty sets,(ii) contains the identity relation ,(iii)for all in the adjoint relation also belongs to ,(iv)for all , , and in there exists a nonnegative integer structure constant such that . A finite association scheme is said to have order and rank . For notation and background on association schemes, see [1]. The structure constants of the scheme make the integer span of its adjacency matrices into a natural -algebra . This is known as the integral adjacency algebra of the scheme , which we will simply refer to as the integral scheme ring. Note that the multiplicative identity of is the identity matrix, which is the adjacency matrix . Similarly we can define the -algebra for any commutative ring with identity, which is known as the adjacency algebra of the scheme over . The complex adjacency algebra is a semisimple algebra with involution defined by . This involution is an antiautomorphism of the algebra . The natural inclusion is the standard representation of (or ). Its character satisfies and for all . Clearly the degree of the standard representation is . It is easy to show using the definition of a scheme that the structure constant if and only if . We write instead of and call the valency of . The linear extension of the valency map defines a degree one algebra representation by We say that is a thin element of when . The thin radical of is the subset consisting of the thin elements of . It follows from the fact that the valency map is a ring homomorphism that is a group. If is a ring with identity, then denotes the group of units of and denotes its subset consisting of torsion units (i.e., units with finite multiplicative order). The subgroup of consisting of units with valency is denoted by . Its subset consists of normalized torsion units. The results of Section 2 show that is often