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

Authors: Allen Herman; Gurmail Singh
Journal: Algebra
 (2014)

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## 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.


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## Body

## 1. Introduction

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 σs, which is the n×n  (0,1)-matrix whose (i,j) entries are 1 if (i,j)∈s and 0 otherwise. (X,S) is anassociation scheme if(i)
Sis a partition of X×X consisting of nonempty sets,(ii)
S contains the identity relation 1X≔x,x:x∈X,(iii)
for alls in S the adjoint relation s*≔{(y,x)∈X×X:(x,y)∈s} also belongs to S,(iv)
for alls, t, and u in S there exists a nonnegative integer structure constant astu such that σsσt=∑u∈Sastuσu. A finite association scheme (X,S) is said to have order n=X and rank r=S. For notation and background on association schemes, see [1].The structure constants of the scheme(X,S) make the integer span of its adjacency matrices into a natural Z-algebra ZS≔⊕s∈SZσs. This is known as theintegral adjacency algebra of the scheme (X,S), which we will simply refer to as theintegral scheme ring. Note that the multiplicative identity of ZS is the n×n identity matrix, which is the adjacency matrix σ1X≔σ1. Similarly we can define the R-algebra RS for any commutative ring R with identity, which is known as the adjacency algebra of the scheme over R.The complex adjacency algebraCS is a semisimple algebra with involution defined by u*=∑s‍us¯σs*. This involution is an antiautomorphism of the algebra CS. The natural inclusion CS↪Mn(C) is thestandard representation of CS (or (X,S)). Its character ρ satisfies ρ(σ1)=n=X and ρ(σs)=0 for all 1≠s∈S. Clearly the degree of the standard representation is n=X.It is easy to show using the definition of a scheme that the structure constantast1≠0 if and only if t=s*. We write ns instead of ass*1 and call ns thevalency of s. The linear extension of the valency map defines a degree one algebra representation CS→C by
(1)u=∑s∈S‍usσs⟶nu=∑s‍usns.
We say that s∈S is athin element of S when ns=1. Thethin radicalOϑ(S) of S is the subset consisting of the thin elements of S. It follows from the fact that the valency map is a ring homomorphism that {σt:t∈Oϑ(S)} is a group.IfR is a ring with identity, then U(R) denotes the group of units of R and U(R)tor denotes its subset consisting of torsion units (i.e., units with finite multiplicative order). The subgroup of U(ZS) consisting of units with valency 1 is denoted by V(ZS). Its subset V(ZS)tor consists of normalized torsion units.The results of Section2 show that V(ZS)tor is often equal to the thin radical of S when S is a commutative finite scheme. In particular this holds for symmetric association schemes or if the valency of any element of S is divisible by a prime p. In Section 3 we establish a “Lagrange-type’’ theorem for finite subgroups of V(ZS)tor, by showing that the order of any finite subgroup of V(ZS)tor divides the order of S and is bounded by the rank of S. In Section 4 this result is directly applied to Schur rings and Hecke algebras.Throughout the paperζk will denote a complex primitive kth root of unity for a given positive integer k. When u∈CS, we will consistently use the notation u=∑s‍usσs with us∈C for all s∈S.


## 2. The Support of Normalized Torsion Units ofZS

Our first lemma is an analogue of Berman-Higman’s proposition on torsion units of group rings (see [2, 3]).Lemma 1.
Let(X,S) be a finite association scheme. Suppose u∈V(ZS)tor. Then u1≠0⇒u=σ1.Proof.
LetΓ:CS→Mn(C) be the standard representation of (X,S) of degree n=X. Let ρ be the standard character; so
(2)ρσs=nifs=1X0otherwise.
Γ
(
u
) is diagonalizable since Γ(u)k=I for some integer k. If specΓ(u) denotes the list of eigenvalues of Γ(u) (including multiplicities), then specΓ(u)=ζkfii=1n consists of kth roots of 1. Now ρ(u)=∑i=1nζkfi=u1·n. Then u1n=∑i=1nζkfi≤n, and u1n=n⇔ all ζkfi’s are equal to ζkf1. Thus u1∈{-1,0,1}, and u1≠0⇒Γ(u)=ζkf1·I⇒u=ζkf1σ1=u1σ1. As u∈V(ZS)tor, u1≠-1. Therefore u=σ1.Let(X,S) be an association scheme and let R be a commutative ring with identity. Let u=∑s∈Susσs∈RS; then s in S belongs to thesupport of u (briefly supp⁡(u)) if and only if us≠0. We will say that u∈U(RS) is atrivial unit if u is a unit of RS for which u=usσs for some us∈U(R) and a unique element s in the support of u, which is necessarily a thin element. Trivial units of ZS are permutation matrices with possibly negative sign in the standard representation.Proposition 2.
Letu be a unit of ZS with uu*=σ1. Then u is a trivial unit.Proof.
Consideruu*=σ1⇒1=(uu*)1=∑s‍usus¯ns=∑s‍us2ns. Since u∈ZS, it follows that us=0 except for exactly one s∈S with ns=1 and us=±1.Proposition 3.
Letu∈V(ZS)tor. If s∈Oϑ(S)∩supp(u) and σs commutes with u, then u=σs is a trivial unit of ZOϑ(S).Proof.
Letu∈V(ZS)tor. Let s∈Oϑ(S)∩supp⁡(u) for which σs commutes with u. Then σs is a unit of ZS and us≠0. Let u′=σs-1u. Since σs commutes with u, u′ has finite order. Since u1′=us≠0, we must have u′=σ1 by Lemma 1. Therefore u=σs.Thecenter of the finite association scheme (X,S) is defined to be Z(S)={t∈S:σtσs=σsσt,foralls∈S}. The scheme (X,S) is acommutative scheme if Z(S)=S. The next two corollaries are immediate from Proposition 3.Corollary 4.
Let(X,S) be a finite association scheme. Suppose u∈V(ZS)tor is a nontrivial unit. If s∈supp(u), then either ns≥2 or s∉Z(S).Corollary 5.
Let(X,S) be a finite commutative association scheme. Suppose u∈V(ZS)tor is a nontrivial unit. If s∈supp(u), then ns≥2.IfG is a finite group, then it is well known that central torsion units of ZG 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(X,S) is a finite association scheme. Suppose there is a prime integer p that divides ns for every s∈S with ns>1. Then every normalized central torsion unit of ZS is a trivial unit.Proof.
Letu=∑s∈Susσs∈V(ZS)tor be a central element of ZS with multiplicative order k. Suppose u is not trivial. By Proposition 3, every s∈supp⁡(u) has ns>1. Our assumption then implies that p divides ns, for every s∈supp⁡(u).
Then1=(uk)1∈span⁡Z{(σs1σs2⋯σsk)1:si∈supp⁡(u),for1≤i≤k}. If (σs1σs2⋯σsk)1≠0, then it is divisible by (σs1σs1*)1=ns1, hence divisible by p. This contradicts 1=(uk)1; hence the result follows.A finite association scheme(X,S) is p-valenced for some prime integer p if ns is a power of p for all s∈S. We know that, for a finite abelian group G, every torsion unit of the integral group ring ZG is a trivial unit. The next corollary generalizes this result to p-valenced commutative schemes.Corollary 7.
If(X,S) is a finite p-valenced commutative association scheme, then every normalized torsion unit of ZS is a trivial unit.Proof.
Since(X,S) is a commutative association scheme, the adjacency algebra ZS is a commutative ring. Therefore, every unit of ZS is central. By Theorem 6, every u∈V(ZS)tor must be a trivial unit; that is, u=σs, for some s∈S with ns=1.An association scheme(X,S) issymmetric if all of the adjacency matrices σs for s∈S are symmetric matrices, or, equivalently, s*=s for all s∈S. It is easy to show that symmetric association schemes are commutative.Theorem 8.
Let(X,S) be a finite symmetric association scheme. If u∈V(ZS)tor, then u=σs, for some s∈S and σs2=σ1. In particular, torsion units of ZS are trivial with order 2 at most.Proof.
Supposeu∈V(ZS) has multiplicative order k. Since every element of ZS is a symmetric matrix, the eigenvalues of u are totally real algebraic integers. Since u has finite multiplicative order, the eigenvalues of u must also be roots of unity. Therefore, the only possibilities for eigenvalues of u are ±1, and the order of u can only be 1 or 2.
Supposeu∈V(ZS) is a nontrivial torsion unit whose order is 2. Then u≠σs, ∀s∈S, but u2=σ1. Also, u2=(∑s‍us2ns)σ1. By Corollary 5, ns≥2 for all s∈supp⁡(u), so it follows that (u2)1≥2, a contradiction.Theorem8 has the following immediate consequence.Corollary 9.
Let(X,S) be a symmetric association scheme. If T is a finite subgroup of   V(ZS), then T is an elementary abelian 2-group.


## 3. Lagrange’s Theorem for Normalized Torsion Units ofZS

The next proposition extends a result concerning idempotents of group algebras over fields of characteristic0 to adjacency algebras of finite association schemes over fields of characteristic 0.Proposition 10.
LetK be a field of characteristic 0 and let (X,S) be a finite association scheme of order n. Let e=∑s∈Sesσs≠0,1 be a nontrivial idempotent of KS. Then e1=m/n∈Q, 0<e1<1, where n=X and m is the rank of e as the matrix in the standard representation.Proof.
LetΓ be the standard representation and let ρ be the standard character of KS. As e is an idempotent, we know that spec(Γ(e))=[1(m),0(n-m)] as a multiset, where m=rank⁡(Γ(e)). Thus
(3)ρe=∑s∈S‍esρσs=e1·n=m.
Therefore,e1=m/n∈{0,1/n,…,(n-1)/n,1}, and we have e1=0=0/n⇔e=0 and e1=1=n/n⇔e=σ1.Corollary 11.
Let(X,S) be a finite association scheme. Then the only idempotents of ZS are 0 and σ1.Proof.
Lete∈ZS be an idempotent. Then e1∈Z. By Proposition 10, this implies e1=0 or 1, and by considering the rank of Γ(e) in these respective cases we have e=0 or σ1.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 (X,S) be a finite association scheme of order n for which every relation in S is thin, that is, a thin association scheme. Then using the valency map it follows that {σs:s∈S} is a group of n distinct permutation matrices. Conversely, let G be a group. For each g in G, let gτ denote the set of all pairs (e,f)∈G×G satisfying eg=f. Let Gτ denote the set of all sets gτ with g in G. Then (G,Gτ) 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 ZG agrees with the valency map of the integral scheme ring Z[Gτ].Ifu=∑g∈Gu(g)g∈ZG, then augmentation of u is ∑g∈Gu(g)∈Z. We know any finite subgroup H⊆V(ZG) 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(X,S) be a finite association scheme. Then any finite group of units of valency 1 in ZS is a set of linearly independent elements.Proof.
LetT={u1=σ1,u2,…,ul} be a finite group of units contained in V(ZS). Suppose c1ui1+⋯+cmuim=0 is an expression of minimal length, where uij are elements of T and the coefficients cj∈Z are not all 0. Since T is a group, we can assume without loss of generality that ui1=σ1. Expressing the uij for j=2,…,m, as uij=∑s‍uij,sσs, we have by Lemma 1 that uij,1=0 for j=2,…,m. It follows that
(4)0=c1σ1+c2ui2+⋯+cmuim1=c1,
contradicting the minimal length assumption. Therefore, T is a linearly independent set.For a finite groupG, the order of any finite subgroup H of V(ZG) divides the order of G [5, Lemma (37.3)]. Our main theorem shows this also holds for schemes.Theorem 13.
Let(X,S) be a finite association scheme of order n and rank r. Then the order of any finite subgroup T of V(ZS) divides n and is at most r. Symbolically, T divides X and T≤S.Proof.
SinceZS is a free module with basis S, any basis of ZS must have S elements. By Lemma 12, T is a linearly independent subset. Therefore T≤S.
Now lete=∑t∈T(t/T)=T^/T=e2, where ∑t∈Tt=T^. Let Γ be the standard representation and let ρ be the standard character of (X,S). Since e2=e≠0, spec(Γ(e))=[1(m),0(n-m)], where m is the rank of the matrix Γ(e). Therefore, ρ(e)=m∈Z+. Also ρ(e)=(1/T)ρ(T^)=(1/T)n(T^)1=n/T, since the argument of Lemma 12 implies (T^)1=1. Therefore, m=n/T; hence T divides n=X, which proves the theorem.We have been unable to settle the question of whether or not the order of any finite subgroup ofV(ZS) must divide the order of Oϑ(S). 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 ZS should be conjugate in QS to some σs, for an s∈Oϑ(S).IfH is a subgroup of V(ZG) for a finite group G with H=G, then ZG=ZH (cf. [5, Lemma (37.4)]). The following lemma proves an analogous result for schemes.Lemma 14.
Let(X,S) be a finite association scheme with rank r. If T is a finite subgroup of  V(ZS) with T=r, then ZS=ZT.Proof.
By Lemma12, T is linearly independent and thus QS=QT. It follows that ZS⊇ZT and mZS⊂ZT for some positive integer m.
LetT={t1=σ1,t2,…,tr} and let s∈S. Then
(5)mσs=∑i=1r‍citi,forsomeci∈Z.
We wish to show that each ci is a multiple of m. For each j∈{1,…,r}, we have
(6)mσstj-1=cjσ1+∑i≠j‍cititj-1.
Since, by Lemma 1, (titj-1)1=0 for i≠j, the coefficient of σ1 on the right hand side is cj whereas on the left hand side it is a multiple of m. It follows that m∣cj for j=1,…,r. Therefore, σs∈ZT for all s∈S, and hence ZS=ZT.While thin association schemes give immediate examples where the conclusion of the preceding theorem holds, we are uncertain as to whetherZS can possess a finite subgroup of normalized units of order S when (X,S) is not thin. The next example shows that it is certainly possible for the adjacency algebra QS to be ring isomorphic to a group algebra when (X,S) is not thin.Example 15.
Let(X,S) 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 S={1X,s,s*}, where ns=ns*=13, and the structure constants of S are determined by σs2=6σs+7σs*, σsσs*=13σ1+6σs+6σs*, and σs*2=7σs+6σs*.
Analysis of the character table ofS (see [6]) shows that QS≅QC3, where C3 is a cyclic group of order 3. Let χ1 be the irreducible character of CS corresponding to the valency map, and let ψ,ψ¯ be the other two irreducible characters of CS. Let {eχ1,eψ,eψ¯} be the centrally primitive idempotents of CS, the character formula for which can be found in [1, Lemma 9.1.6]. An element v of CS with order 3 and valency 1 is given by
(7)v=eχ1+ζ3eψ+ζ32eψ¯,
and since v is fixed by complex conjugation, v∈QS. Using the character formula for centrally primitive idempotents of CS, we find that
(8)v=19-4σ1-σs+2σs*,
and v2=v*. So if T={σ1,v,v2}, then T is a finite subgroup of normalized units of QS for which QS=QT. In this case Z[1/9]S⊆ZT⊊ZS.Proposition 16.
Let(X,S) be the association scheme of order n and rank 2. Then
(9)VZStor=1ifn≥3,2ifn=2.Proof.
Letu be a normalized torsion unit of ZS with multiplicative order k. Our Lagrange theorem for schemes implies that k divides n and k≤2. So we are done if n is odd. Suppose k=2. Since S is symmetric and u∈ZS, u=u*. Therefore, u2=σ1⇒uu*=σ1, and so, by Proposition 2, u=σs for some s∈S with ns=1. Such an element of the scheme of rank 2 with s≠1X only exists when n=2.For symmetric schemes of rank3, 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.Proposition 17.
Let(X,S) be a finite association scheme of order n>2 and rank 3 with S={1X,s,s*}. If ns>1, then |VZS)tor=1.Proof.
Supposeu∈V(ZS) is a normalized torsion unit with u≠σ1. By Lemma 1, supp(u)={s,s*}, so u=aσs+bσs* for some a,b∈Z. Since nu=1, we have 1=ans+bns*=(a+b)ns, which is not possible as a,b∈Z.


## 4. Applications to Schur Rings and Hecke Algebras

LetG be a finite group of order n. Let ZF be a Schur ring defined on the group G. This means that F is a partition of the set G into nonempty subsets for which we consider the following:(i)
{1G}∈F,(ii)
for allU={g1,…gk}∈F, U*={g1-1,…,gk-1}∈F,(iii)
for allU,V,W∈F, there exist nonnegative integers λUVW such that
(10)U^V^=∑W∈F‍λUVWW^,where U^=∑g∈U‍g denotes the sum of the elements of U in the group ring ZG.The Schur ringZF is defined to be the Z-span of {U^:U∈F}, considered as a subring of ZG. ZF is a free Z-module of rank r=F. By extension of scalars we can consider the Schur ring RF for any commutative ring R. We will refer to a partition of G with the above properties as aSchur ring partition of G. One example of a Schur ring partition is the partition F of G into its conjugacy classes, in which case the complex Schur ring CF is isomorphic to the center of the group ring CG.We claim that the Schur ringZF is isomorphic to an integral scheme ring. Given the group G and Schur ring partition F, let Fτ be the images of subsets in F under the group correspondence. So, given U∈F, we set
(11)Uτ=x,y∈G×G:xg=yforsomeg∈U.
Using the properties of the Schur ring partition F, it is straightforward to show that (G,Fτ) is an association scheme of order n=|G| and rank r=|F|. Furthermore, ZF≃Z[Fτ] as rings, where the isomorphism is produced by the restriction of the regular representation of G to ZF. The restriction of the augmentation map on the group ring to ZF corresponds to the valency map of Z[Fτ] under this isomorphism. The following corollary is the application of our Lagrange theorem for scheme rings to this special case.Corollary 18.
LetF be a Schur ring partition of a finite group G. Then the order of any finite subgroup of V(ZF) divides G and is at most F.LetH be a subgroup of a finite group G that has index n. Let G/H be the set of left cosets of H in G. Let r be the number of distinct double cosets HgH of H in G. Corresponding to each double coset HgH for g∈G, let
(12)gH≔{(xH,yH):y∈xHgH}.
Let G⫽H≔{gH:g∈G}. Then (G/H,G⫽H) is an association scheme of order n and rank r. This type of association scheme is known as aSchurian scheme, and its rational adjacency algebra Q[G⫽H] is ring isomorphic to the ordinary Hecke algebra eHQGeH, where eH=(1/|H|)∑h∈H‍h. (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.
LetH be a subgroup of a finite group G that has n left cosets and r double cosets. Then the order of any finite subgroup of V(Z[G⫽H]) divides n and is at most r.



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