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

In this paper we will consider torsion units of rings generated by finite association schemes, which we now define. LetX be a finite set of size n > 0. Let S be a partition of X × X such that every relation in S is nonempty. For a relation s ∈ S, there corresponds an adjacency matrix, denoted by σ s , which is the n × n (0, 1)-matrix whose (i, j) entries are 1 if (i, j) ∈ s and 0 otherwise. (X, S) is an association scheme if


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  > 0. 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  ×  (0, 1)-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 1  := {(, ) :  ∈ }, (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 Z-algebra Z := ⊕ ∈ Z  .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 Z is the  ×  identity matrix, which is the adjacency matrix  1  :=  1 .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 C is a semisimple algebra with involution defined by  * = ∑      * .This involution is an antiautomorphism of the algebra C.The natural inclusion C →   (C) is the standard representation of C (or (, )).Its character  satisfies ( 1 ) =  = || and (  ) = 0 for all 1 ̸ =  ∈ .Clearly the degree of the standard representation is  = ||.
It is easy to show using the definition of a scheme that the structure constant  1 ̸ = 0 if and only if  =  * .We write   instead of   * 1 and call   the valency of .The linear extension of the valency map defines a degree one algebra representation C → C by We say that  ∈  is a thin element of  when   = 1.The thin radical O  () of  is the subset consisting of the thin elements of .It follows from the fact that the valency map is a ring homomorphism that {  :  ∈ O  ()} is a group.
If  is a ring with identity, then () denotes the group of units of  and () tor denotes its subset consisting of torsion units (i.e., units with finite multiplicative order).The subgroup of (Z) consisting of units with valency 1 is denoted by (Z).Its subset (Z) tor consists of normalized torsion units.

Algebra
The results of Section 2 show that (Z) tor 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 (Z) tor , by showing that the order of any finite subgroup of (Z) tor 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  ∈ C, we will consistently use the notation  = ∑      with   ∈ C for all  ∈ .

The Support of Normalized Torsion Units of Z𝑆
Our first lemma is an analogue of Berman-Higman's proposition on torsion units of group rings (see [2,3]).
Proof.Let Γ : C →   (C) be the standard representation of (, ) of degree  = ||.Let  be the standard character; so Γ() is diagonalizable since Γ()  =  for some integer .If specΓ() denotes the list of eigenvalues of Γ() (including multiplicities), then specΓ() = { Let (, ) be an association scheme and let  be a commutative ring with identity.Let  = ∑ ∈     ∈ ; then  in  belongs to the support of  (briefly supp()) if and only if   ̸ = 0. 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 Z are permutation matrices with possibly negative sign in the standard representation.The center of the finite association scheme (, ) is defined to be () = { ∈  :     =     , for all  ∈ }.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  ∈ (Z)  is a nontrivial unit.If  ∈ supp (), then either   ≥ 2 or  ∉ ().

Corollary 5. Let (𝑋, 𝑆) be a finite commutative association
If  is a finite group, then it is well known that central torsion units of Z 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   > 1.Then every normalized central torsion unit of Z is a trivial unit.
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 Z is a trivial unit.The next corollary generalizes this result to -valenced commutative schemes.

Corollary 7. If (𝑋, 𝑆
) is a finite -valenced commutative association scheme, then every normalized torsion unit of Z is a trivial unit.
Proof.Since (, ) is a commutative association scheme, the adjacency algebra Z is a commutative ring.Therefore, every unit of Z is central.By Theorem 6, every  ∈ (Z) tor must be a trivial unit; that is,  =   , for some  ∈  with   = 1.
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  ∈ (Z)  , then  =   , for some  ∈  and  2  =  1 .In particular, torsion units of Z are trivial with order 2 at most.

Algebra 3
Proof.Suppose  ∈ (Z) has multiplicative order .Since every element of Z 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 ±1, and the order of  can only be 1 or 2.
Corollary 9. Let (, ) be a symmetric association scheme.If  is a finite subgroup of (Z), then  is an elementary abelian 2-group.

Lagrange's Theorem for Normalized Torsion Units of Z𝑆
The next proposition extends a result concerning idempotents of group algebras over fields of characteristic 0 to adjacency algebras of finite association schemes over fields of characteristic 0.
Proposition 10.Let  be a field of characteristic 0 and let (, ) be a finite association scheme of order .Let  = ∑ ∈     ̸ = 0, 1 be a nontrivial idempotent of .Then  1 = / ∈ Q, 0 <  1 < 1, where  = || and  is the rank of  as the matrix in the standard representation.
Corollary 11.Let (, ) be a finite association scheme.Then the only idempotents of Z are 0 and  1 .
Proof.Let  ∈ Z be an idempotent.Then  1 ∈ Z.By Proposition 10, this implies  1 = 0 or 1, and by considering the rank of Γ() in these respective cases we have  = 0 or  1 .
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 Z agrees with the valency map of the integral scheme ring Z[  ].
If  = ∑ ∈ () ∈ Z, then augmentation of  is ∑ ∈ () ∈ Z.We know any finite subgroup  ⊆ (Z) 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 Z is a set of linearly independent elements.Proof.Let  = { 1 =  1 ,  2 , . . .,  ℓ } be a finite group of units contained in (Z).Suppose  1   1 + ⋅ ⋅ ⋅ +      = 0 is an expression of minimal length, where    are elements of  and the coefficients   ∈ Z are not all 0. Since  is a group, we can assume without loss of generality that   1 =  1 .Expressing the    for  = 2, . . ., , as    = ∑     ,   , we have by Lemma 1 that    ,1 = 0 for  = 2, . . ., .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 (Z) divides the order of  [5,Lemma (37.3)].Our main theorem shows this also holds for schemes.We have been unable to settle the question of whether or not the order of any finite subgroup of (Z) must divide the order of O  ().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 Z should be conjugate in Q to some   , for an  ∈ O  ().
If  is a subgroup of (Z) for a finite group  with || = ||, then Z = Z (cf.[5, Lemma (37.4)]).The following lemma proves an analogous result for schemes.Algebra Lemma 14.Let (, ) be a finite association scheme with rank .If  is a finite subgroup of (Z) with || = , then Z = Z.
Proof.By Lemma 12,  is linearly independent and thus Q = Q.It follows that Z ⊇ Z and Z ⊂ Z for some positive integer .

While thin association schemes give immediate examples
where the conclusion of the preceding theorem holds, we are uncertain as to whether Z 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 Q 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 27 and rank 3. We have  = {1  , ,  * }, where   =   * = 13, and the structure constants of  are determined by  2  = 6  + 7  * ,     * = 13 1 + 6  + 6  * , and  2  * = 7  + 6  * .Analysis of the character table of  (see [6]) shows that Q ≅ Q 3 , where  3 is a cyclic group of order 3. Let  1 be the irreducible character of C corresponding to the valency map, and let ,  be the other two irreducible characters of C.Let {  1 ,   ,   } be the centrally primitive idempotents of C, the character formula for which can be found in [1,Lemma 9.1.6].An element V of C with order 3 and valency 1 is given by and since V is fixed by complex conjugation, V ∈ Q.Using the character formula for centrally primitive idempotents of C, we find that and For symmetric schemes of rank 3, we have already seen that normalized torsion units must be trivial with order 2. Nonsymmetric association schemes of rank 3, such as the one seen in the example above, arise naturally from strongly regular directed graphs.

Applications to Schur Rings and Hecke Algebras
Let  be a finite group of order .Let ZF be a Schur ring defined on the group .This means that F is a partition of the set  into nonempty subsets for which we consider the following: ) for all , ,  ∈ F, there exist nonnegative integers   such that where Û = ∑ ∈  denotes the sum of the elements of  in the group ring Z.
The Schur ring ZF is defined to be the Z-span of { Û :  ∈ F}, considered as a subring of Z.ZF is a free Z-module of rank  = |F|.By extension of scalars we can consider the Schur ring F 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 F of  into its conjugacy classes, in which case the complex Schur ring CF is isomorphic to the center of the group ring C.
We claim that the Schur ring ZF is isomorphic to an integral scheme ring.Given the group  and Schur ring partition F, let F  be the images of subsets in F under the group correspondence.So, given  ∈ F, we set   = {(, ) ∈  ×  :  =  for some  ∈ } . (11) Using the properties of the Schur ring partition F, it is straightforward to show that (, F  ) is an association scheme of order  = || and rank  = |F|.Furthermore, ZF ≃ Z[F  ] as rings, where the isomorphism is produced by the restriction of the regular representation of  to ZF.The restriction of the augmentation map on the group ring to ZF corresponds to the valency map of Z[F  ] under this isomorphism.The following corollary is the application of our Lagrange theorem for scheme rings to this special case.
Corollary 18.Let F be a Schur ring partition of a finite group .Then the order of any finite subgroup of (ZF) divides || and is at most |F|.
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   := {(, ) :  ∈ } . (12) Let U := {  :  ∈ }.Then (/, U) is an association scheme of order  and rank .This type of association scheme is known as a Schurian scheme, and its rational adjacency algebra Q[U] is ring isomorphic to the ordinary Hecke algebra   Q  , where   = (1/||) ∑ ℎ∈ ℎ. (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 (Z[U]) divides  and is at most .
then  is a finite subgroup of normalized units of Q for which Q = Q.In this case Z[1/9] ⊆ Z ⊊ Z.Proof.Let  be a normalized torsion unit of Z with multiplicative order .Our Lagrange theorem for schemes implies that  divides  and  ≤ 2. So we are done if  is odd.Suppose  = 2. Since  is symmetric and  ∈ Z,  =  * .Therefore,  2 =  1 ⇒  * =  1 , and so, by Proposition 2,  =   for some  ∈  with   = 1.Such an element of the scheme of rank 2 with  ̸ = 1  only exists when  = 2.