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

Abstract

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 equal to the thin radical of when is a commutative finite scheme. In particular this holds for symmetric association schemes or if the valency of any element of is divisible by a prime . In Section 3 we establish a “Lagrange-type’’ theorem for finite subgroups of , by showing that the order of any finite subgroup of divides the order of and is bounded by the rank of . In Section 4 this result is directly applied to Schur rings and Hecke algebras.

Throughout the paper will denote a complex primitive th root of unity for a given positive integer . When , we will consistently use the notation with for all .

2. The Support of Normalized Torsion Units of

Our first lemma is an analogue of Berman-Higman’s proposition on torsion units of group rings (see [2, 3]).

Lemma 1. Let be a finite association scheme. Suppose . Then .

Proof. Let be the standard representation of of degree . Let be the standard character; so
is diagonalizable since for some integer . If denotes the list of eigenvalues of (including multiplicities), then consists of th roots of 1. Now . Then , and all ’s are equal to . Thus , and . As , . Therefore .

Let be an association scheme and let be a commutative ring with identity. Let ; then in belongs to the support of (briefly ) if and only if . We will say that is a trivial unit if is a unit of for which for some and a unique element in the support of , which is necessarily a thin element. Trivial units of are permutation matrices with possibly negative sign in the standard representation.

Proposition 2. Let be a unit of with . Then is a trivial unit.

Proof. Consider . Since , it follows that except for exactly one with and .

Proposition 3. Let . If and commutes with , then is a trivial unit of .

Proof. Let . Let for which commutes with . Then is a unit of and . Let . Since commutes with , has finite order. Since , we must have by Lemma 1. Therefore .

The center of the finite association scheme is defined to be . The scheme is a commutative scheme if . The next two corollaries are immediate from Proposition 3.

Corollary 4. Let be a finite association scheme. Suppose is a nontrivial unit. If , then either or .

Corollary 5. Let be a finite commutative association scheme. Suppose is a nontrivial unit. If , then .

If is a finite group, then it is well known that central torsion units of are trivial [4, Theorem 2.1]. We are able to extend this result to finite association schemes whose nonthin elements have valencies divisible by a single prime.

Theorem 6. Suppose is a finite association scheme. Suppose there is a prime integer that divides for every with . Then every normalized central torsion unit of is a trivial unit.

Proof. Let be a central element of with multiplicative order . Suppose is not trivial. By Proposition 3, every has . Our assumption then implies that divides , for every .
Then . If , then it is divisible by , hence divisible by . This contradicts ; hence the result follows.

A finite association scheme is -valenced for some prime integer if is a power of for all . We know that, for a finite abelian group , every torsion unit of the integral group ring is a trivial unit. The next corollary generalizes this result to -valenced commutative schemes.

Corollary 7. If is a finite -valenced commutative association scheme, then every normalized torsion unit of is a trivial unit.

Proof. Since is a commutative association scheme, the adjacency algebra is a commutative ring. Therefore, every unit of is central. By Theorem 6, every must be a trivial unit; that is, , for some with .

An association scheme is symmetric if all of the adjacency matrices for are symmetric matrices, or, equivalently, for all . It is easy to show that symmetric association schemes are commutative.

Theorem 8. Let be a finite symmetric association scheme. If , then , for some and . In particular, torsion units of are trivial with order at most.

Proof. Suppose has multiplicative order . Since every element of is a symmetric matrix, the eigenvalues of are totally real algebraic integers. Since has finite multiplicative order, the eigenvalues of must also be roots of unity. Therefore, the only possibilities for eigenvalues of are , and the order of can only be or .
Suppose is a nontrivial torsion unit whose order is . Then , , but . Also, . By Corollary 5, for all , so it follows that , a contradiction.

Theorem 8 has the following immediate consequence.

Corollary 9. Let be a symmetric association scheme. If is a finite subgroup of   , then is an elementary abelian -group.

3. Lagrange’s Theorem for Normalized Torsion Units of

The next proposition extends a result concerning idempotents of group algebras over fields of characteristic to adjacency algebras of finite association schemes over fields of characteristic .

Proposition 10. Let be a field of characteristic 0 and let be a finite association scheme of order . Let be a nontrivial idempotent of . Then , , where and is the rank of as the matrix in the standard representation.

Proof. Let be the standard representation and let be the standard character of . As is an idempotent, we know that as a multiset, where . Thus
Therefore, , and we have and .

Corollary 11. Let be a finite association scheme. Then the only idempotents of are 0 and .

Proof. Let be an idempotent. Then . By Proposition 10, this implies or , and by considering the rank of in these respective cases we have or .

Here we give a glance on the fact that the association scheme concept generalizes the group concept. For more details, see [1, Section 5.5]. Let be a finite association scheme of order for which every relation in is thin, that is, a thin association scheme. Then using the valency map it follows that is a group of distinct permutation matrices. Conversely, let be a group. For each in , let denote the set of all pairs satisfying . Let denote the set of all sets with in . Then becomes a thin association scheme. So there is correspondence between thin association schemes and groups, called the group correspondence. In this correspondence, the augmentation map of the integral group ring agrees with the valency map of the integral scheme ring .

If , then augmentation of is . We know any finite subgroup is a linearly independent set (cf. [5, Lemma (37.1)]). One can ask what happens in the case of scheme rings. The next lemma gives an answer to this question.

Lemma 12. Let be a finite association scheme. Then any finite group of units of valency 1 in is a set of linearly independent elements.

Proof. Let be a finite group of units contained in . Suppose is an expression of minimal length, where are elements of and the coefficients are not all 0. Since is a group, we can assume without loss of generality that . Expressing the for , as , we have by Lemma 1 that for . It follows that contradicting the minimal length assumption. Therefore, is a linearly independent set.

For a finite group , the order of any finite subgroup of divides the order of [5, Lemma (37.3)]. Our main theorem shows this also holds for schemes.

Theorem 13. Let be a finite association scheme of order and rank . Then the order of any finite subgroup of divides and is at most . Symbolically, divides and .

Proof. Since is a free module with basis , any basis of must have elements. By Lemma 12, is a linearly independent subset. Therefore .
Now let , where . Let be the standard representation and let be the standard character of . Since , , where is the rank of the matrix . Therefore, . Also , since the argument of Lemma 12 implies . Therefore, ; hence divides , which proves the theorem.

We have been unable to settle the question of whether or not the order of any finite subgroup of must divide the order of . Related to this is a possible generalization of the Zassenhaus conjecture on torsion units to integral scheme rings, which would be that any normalized torsion unit of should be conjugate in to some , for an .

If is a subgroup of for a finite group with , then (cf. [5, Lemma (37.4)]). The following lemma proves an analogous result for schemes.

Lemma 14. Let be a finite association scheme with rank . If is a finite subgroup of   with , then .

Proof. By Lemma 12, is linearly independent and thus . It follows that and for some positive integer .
Let and let . Then We wish to show that each is a multiple of . For each , we have Since, by Lemma 1, for , the coefficient of on the right hand side is whereas on the left hand side it is a multiple of . It follows that for . Therefore, for all , and hence .

While thin association schemes give immediate examples where the conclusion of the preceding theorem holds, we are uncertain as to whether can possess a finite subgroup of normalized units of order when is not thin. The next example shows that it is certainly possible for the adjacency algebra to be ring isomorphic to a group algebra when is not thin.

Example 15. Let be the fifth association scheme of order 27 in Hanaki and Miyamoto’s classification of small association schemes [6]. This is a commutative nonsymmetric scheme of order and rank . We have , where , and the structure constants of are determined by , , and .
Analysis of the character table of (see [6]) shows that , where is a cyclic group of order . Let be the irreducible character of corresponding to the valency map, and let be the other two irreducible characters of . Let be the centrally primitive idempotents of , the character formula for which can be found in [1, Lemma 9.1.6]. An element of with order and valency is given by and since is fixed by complex conjugation, . Using the character formula for centrally primitive idempotents of , we find that and . So if , then is a finite subgroup of normalized units of for which . In this case .

Proposition 16. Let be the association scheme of order and rank . Then

Proof. Let be a normalized torsion unit of with multiplicative order . Our Lagrange theorem for schemes implies that divides and . So we are done if is odd. Suppose . Since is symmetric and , . Therefore, , and so, by Proposition 2, for some with . Such an element of the scheme of rank with only exists when .

For symmetric schemes of rank , we have already seen that normalized torsion units must be trivial with order . Nonsymmetric association schemes of rank , such as the one seen in the example above, arise naturally from strongly regular directed graphs.

Proposition 17. Let be a finite association scheme of order and rank with . If , then .

Proof. Suppose is a normalized torsion unit with . By Lemma 1, supp, so for some . Since , we have , which is not possible as .

4. Applications to Schur Rings and Hecke Algebras

Let be a finite group of order . Let be a Schur ring defined on the group . This means that is a partition of the set into nonempty subsets for which we consider the following:(i),(ii)for all , ,(iii)for all , there exist nonnegative integers such that where denotes the sum of the elements of in the group ring .

The Schur ring is defined to be the -span of , considered as a subring of . is a free -module of rank . By extension of scalars we can consider the Schur ring for any commutative ring . We will refer to a partition of with the above properties as a Schur ring partition of . One example of a Schur ring partition is the partition of into its conjugacy classes, in which case the complex Schur ring is isomorphic to the center of the group ring .

We claim that the Schur ring is isomorphic to an integral scheme ring. Given the group and Schur ring partition , let be the images of subsets in under the group correspondence. So, given , we set Using the properties of the Schur ring partition , it is straightforward to show that is an association scheme of order and rank . Furthermore, as rings, where the isomorphism is produced by the restriction of the regular representation of to . The restriction of the augmentation map on the group ring to corresponds to the valency map of under this isomorphism. The following corollary is the application of our Lagrange theorem for scheme rings to this special case.

Corollary 18. Let be a Schur ring partition of a finite group . Then the order of any finite subgroup of divides and is at most .

Let be a subgroup of a finite group that has index . Let be the set of left cosets of in . Let be the number of distinct double cosets of in . Corresponding to each double coset for , let Let . Then is an association scheme of order and rank . This type of association scheme is known as a Schurian scheme, and its rational adjacency algebra is ring isomorphic to the ordinary Hecke algebra , where . (For details, see [7], and note that the argument given there for this fact does not require that the field be algebraically closed.) The application of our Lagrange theorem for scheme rings in this special case gives the next result.

Corollary 19. Let be a subgroup of a finite group that has left cosets and double cosets. Then the order of any finite subgroup of divides and is at most .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The work of Allen Herman has been supported by an NSERC Discovery Grant.