research article on the torsion units of integral adjacency...

Research ArticleOn the Torsion Units of Integral Adjacency Algebras ofFinite Association Schemes

Allen Herman and Gurmail Singh

Department of Mathematics and Statistics University of Regina Regina SK Canada S4S 0A2

Correspondence should be addressed to Allen Herman allenhermanureginaca

Received 26 August 2014 Accepted 26 November 2014 Published 16 December 2014

Academic Editor Burkhard Kulshammer

Copyright copy 2014 A Herman and G Singh This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Torsion units of group rings have been studied extensively since the 1960s As association schemes are generalization of groupsit is natural to ask about torsion units of association scheme rings In this paper we establish some results about torsion units ofassociation scheme rings analogous to basic results for torsion units of group rings

1 Introduction

In this paper we will consider torsion units of rings generatedby finite association schemes which we now define Let119883 bea finite set of size 119899 gt 0 Let 119878 be a partition of 119883 times 119883 suchthat every relation in 119878 is nonempty For a relation 119904 isin 119878 therecorresponds an adjacency matrix denoted by 120590

119904 which is the

119899 times 119899 (0 1)-matrix whose (119894 119895) entries are 1 if (119894 119895) isin 119904 and 0otherwise (119883 119878) is an association scheme if

(i) 119878 is a partition of119883times119883 consisting of nonempty sets(ii) 119878 contains the identity relation 1

119883= (119909 119909) 119909 isin 119883

(iii) for all 119904 in 119878 the adjoint relation 119904lowast = (119910 119909) isin 119883times119883

(119909 119910) isin 119904 also belongs to 119878(iv) for all 119904 119905 and 119906 in 119878 there exists a nonnegative integer

structure constant 119886119904119905119906

such that 120590119904120590119905= sum119906isin119878

119886119904119905119906120590119906

A finite association scheme (119883 119878) is said to have order 119899 = |119883|

and rank 119903 = |119878| For notation and background on associationschemes see [1]

The structure constants of the scheme (119883 119878) make theinteger span of its adjacencymatrices into a naturalZ-algebraZ119878 = oplus

119904isin119878Z120590119904This is known as the integral adjacency algebra

of the scheme (119883 119878) which we will simply refer to as theintegral scheme ring Note that the multiplicative identity ofZ119878 is the 119899 times 119899 identity matrix which is the adjacency matrix1205901119883

= 1205901 Similarly we can define the 119877-algebra 119877119878 for any

commutative ring 119877 with identity which is known as theadjacency algebra of the scheme over 119877

The complex adjacency algebraC119878 is a semisimple algebrawith involution defined by 119906lowast = sum

119904119906119904120590119904lowast This involution is

an antiautomorphismof the algebraC119878Thenatural inclusionC119878 997893rarr 119872

119899(C) is the standard representation ofC119878 (or (119883 119878))

Its character 120588 satisfies 120588(1205901) = 119899 = |119883| and 120588(120590

119904) = 0 for all

1 = 119904 isin 119878 Clearly the degree of the standard representation is119899 = |119883|

It is easy to show using the definition of a scheme thatthe structure constant 119886

1199041199051= 0 if and only if 119905 = 119904

lowast Wewrite 119899

119904instead of 119886

119904119904lowast1and call 119899

119904the valency of 119904 The linear

extension of the valency map defines a degree one algebrarepresentation C119878 rarr C by

119906 = sum

119904isin119878

119906119904120590119904997888rarr 119899119906= sum

119904

119906119904119899119904 (1)

We say that 119904 isin 119878 is a thin element of 119878 when 119899119904= 1 The thin

radicalO120599(119878) of 119878 is the subset consisting of the thin elements

of 119878 It follows from the fact that the valency map is a ringhomomorphism that 120590

119905 119905 isin O

120599(119878) is a group

If 119877 is a ring with identity then 119880(119877) denotes the groupof units of 119877 and 119880(119877)

tor denotes its subset consisting oftorsion units (ie units with finite multiplicative order) Thesubgroup of 119880(Z119878) consisting of units with valency 1 isdenoted by119881(Z119878) Its subset119881(Z119878)tor consists of normalizedtorsion units

Hindawi Publishing CorporationAlgebraVolume 2014 Article ID 842378 5 pageshttpdxdoiorg1011552014842378

2 Algebra

The results of Section 2 show that 119881(Z119878)tor is often equalto the thin radical of 119878when 119878 is a commutative finite schemeIn particular this holds for symmetric association schemes orif the valency of any element of 119878 is divisible by a prime 119901 InSection 3 we establish a ldquoLagrange-typerdquo theorem for finitesubgroups of119881(Z119878)tor by showing that the order of any finitesubgroup of 119881(Z119878)tor divides the order of 119878 and is boundedby the rank of 119878 In Section 4 this result is directly applied toSchur rings and Hecke algebras

Throughout the paper 120577119896will denote a complex primitive

119896th root of unity for a given positive integer 119896 When 119906 isin C119878we will consistently use the notation 119906 = sum

119904119906119904120590119904with 119906

119904isin C

for all 119904 isin 119878

2 The Support of Normalized TorsionUnits of Z119878

Our first lemma is an analogue of Berman-Higmanrsquos propo-sition on torsion units of group rings (see [2 3])

Lemma 1 Let (119883 119878) be a finite association scheme Suppose119906 isin 119881(Z119878)

119905119900119903 Then 1199061

= 0 rArr 119906 = 1205901

Proof Let Γ C119878 rarr 119872119899(C) be the standard representation

of (119883 119878) of degree 119899 = |119883| Let 120588 be the standard characterso

120588 (120590119904) =

119899 if 119904 = 1119883

0 otherwise(2)

Γ(119906) is diagonalizable since Γ(119906)119896 = 119868 for some integer 119896If specΓ(119906) denotes the list of eigenvalues of Γ(119906) (includingmultiplicities) then specΓ(119906) = 120577

119891119894

119896119899

119894=1consists of 119896th roots

of 1 Now 120588(119906) = sum119899

119894=1120577119891119894

119896= 1199061sdot 119899Then |119906

1|119899 = |sum

119899

119894=1120577119891119894

119896| le 119899

and |1199061|119899 = 119899 hArr all 120577119891119894

119896rsquos are equal to 1205771198911

119896 Thus 119906

1isin minus1 0 1

and 1199061

= 0 rArr Γ(119906) = 1205771198911

119896sdot 119868 rArr 119906 = 120577

1198911

1198961205901= 11990611205901 As

119906 isin 119881(Z119878)tor 1199061

= minus1 Therefore 119906 = 1205901

Let (119883 119878) be an association scheme and let 119877 be acommutative ring with identity Let 119906 = sum

119904isin119878119906119904120590119904isin 119877119878 then

119904 in 119878 belongs to the support of 119906 (briefly supp(119906)) if and onlyif 119906119904

= 0 We will say that 119906 isin 119880(119877119878) is a trivial unit if 119906is a unit of 119877119878 for which 119906 = 119906

119904120590119904for some 119906

119904isin 119880(119877) and

a unique element 119904 in the support of 119906 which is necessarilya thin element Trivial units of Z119878 are permutation matriceswith possibly negative sign in the standard representation

Proposition 2 Let 119906 be a unit ofZ119878 with 119906119906lowast = 1205901 Then 119906 is

a trivial unit

Proof Consider 119906119906lowast = 1205901rArr 1 = (119906119906

lowast

)1= sum119904119906119904119906119904119899119904=

sum119904|119906119904|2

119899119904 Since 119906 isin Z119878 it follows that 119906

119904= 0 except for

exactly one 119904 isin 119878 with 119899119904= 1 and 119906

119904= plusmn1

Proposition 3 Let 119906 isin 119881(Z119878)119905119900119903 If 119904 isin O

120599(119878) cap supp (119906) and

120590119904commutes with 119906 then 119906 = 120590

119904is a trivial unit of ZO

120599(119878)

Proof Let 119906 isin 119881(Z119878)tor Let 119904 isin O

120599(119878) cap supp(119906) for which

120590119904commutes with 119906 Then 120590

119904is a unit of Z119878 and 119906

119904= 0 Let

1199061015840

= 120590minus1

119904119906 Since120590

119904commuteswith1199061199061015840 has finite order Since

1199061015840

1= 119906119904

= 0 we must have 1199061015840 = 1205901by Lemma 1 Therefore

119906 = 120590119904

The center of the finite association scheme (119883 119878) isdefined to be 119885(119878) = 119905 isin 119878 120590

119905120590119904= 120590119904120590119905 for all 119904 isin 119878

The scheme (119883 119878) is a commutative scheme if 119885(119878) = 119878 Thenext two corollaries are immediate from Proposition 3

Corollary 4 Let (119883 119878) be a finite association scheme Suppose119906 isin 119881(Z119878)

119905119900119903 is a nontrivial unit If 119904 isin supp (119906) then either119899119904ge 2 or 119904 notin 119885(119878)

Corollary 5 Let (119883 119878) be a finite commutative associationscheme Suppose 119906 isin 119881(Z119878)

119905119900119903 is a nontrivial unit If 119904 isin

supp (119906) then 119899119904ge 2

If 119866 is a finite group then it is well known that centraltorsion units ofZ119866 are trivial [4Theorem 21]We are able toextend this result to finite association schemeswhose nonthinelements have valencies divisible by a single prime

Theorem 6 Suppose (119883 119878) is a finite association schemeSuppose there is a prime integer 119901 that divides 119899

119904for every 119904 isin 119878

with 119899119904gt 1 Then every normalized central torsion unit of Z119878

is a trivial unit

Proof Let 119906 = sum119904isin119878

119906119904120590119904isin 119881(Z119878)

tor be a central element ofZ119878 with multiplicative order 119896 Suppose 119906 is not trivial ByProposition 3 every 119904 isin supp(119906) has 119899

119904gt 1 Our assumption

then implies that 119901 divides 119899119904 for every 119904 isin supp(119906)

Then 1 = (119906119896

)1

isin spanZ(12059011990411205901199042 sdot sdot sdot 120590119904119896)1 119904119894

isin

supp(119906) for 1 le 119894 le 119896 If (12059011990411205901199042sdot sdot sdot 120590119904119896)1

= 0 then itis divisible by (120590

1199041120590119904lowast

1

)1

= 1198991199041 hence divisible by 119901 This

contradicts 1 = (119906119896

)1 hence the result follows

A finite association scheme (119883 119878) is 119901-valenced for someprime integer119901 if 119899

119904is a power of119901 for all 119904 isin 119878We know that

for a finite abelian group 119866 every torsion unit of the integralgroup ringZ119866 is a trivial unit The next corollary generalizesthis result to 119901-valenced commutative schemes

Corollary 7 If (119883 119878) is a finite 119901-valenced commutativeassociation scheme then every normalized torsion unit of Z119878is a trivial unit

Proof Since (119883 119878) is a commutative association scheme theadjacency algebraZ119878 is a commutative ringTherefore everyunit of Z119878 is central ByTheorem 6 every 119906 isin 119881(Z119878)

tor mustbe a trivial unit that is 119906 = 120590

119904 for some 119904 isin 119878with 119899

119904= 1

An association scheme (119883 119878) is symmetric if all of theadjacency matrices 120590

119904for 119904 isin 119878 are symmetric matrices or

equivalently 119904lowast = 119904 for all 119904 isin 119878 It is easy to show thatsymmetric association schemes are commutative

Theorem 8 Let (119883 119878) be a finite symmetric associationscheme If 119906 isin 119881(Z119878)

119905119900119903 then 119906 = 120590119904 for some 119904 isin 119878 and

1205902

119904= 1205901 In particular torsion units ofZ119878 are trivial with order

2 at most

Algebra 3

Proof Suppose 119906 isin 119881(Z119878) has multiplicative order 119896 Sinceevery element of Z119878 is a symmetric matrix the eigenvaluesof 119906 are totally real algebraic integers Since 119906 has finitemultiplicative order the eigenvalues of 119906 must also be rootsof unity Therefore the only possibilities for eigenvalues of 119906are plusmn1 and the order of 119906 can only be 1 or 2

Suppose 119906 isin 119881(Z119878) is a nontrivial torsion unit whoseorder is 2 Then 119906 = 120590

119904 forall119904 isin 119878 but 1199062 = 120590

1 Also 1199062 =

(sum1199041199062

119904119899119904)1205901 By Corollary 5 119899

119904ge 2 for all 119904 isin supp(119906) so it

follows that (1199062)1ge 2 a contradiction

Theorem 8 has the following immediate consequence

Corollary 9 Let (119883 119878) be a symmetric association scheme If119879 is a finite subgroup of 119881(Z119878) then119879 is an elementary abelian2-group

3 Lagrangersquos Theorem for NormalizedTorsion Units of Z119878

The next proposition extends a result concerning idempo-tents of group algebras over fields of characteristic 0 toadjacency algebras of finite association schemes over fields ofcharacteristic 0

Proposition 10 Let 119870 be a field of characteristic 0 and let(119883 119878) be a finite association scheme of order 119899 Let 119890 =

sum119904isin119878

119890119904120590119904

= 0 1 be a nontrivial idempotent of 119870119878 Then 1198901=

119898119899 isin Q 0 lt 1198901lt 1 where 119899 = |119883| and 119898 is the rank of 119890 as

the matrix in the standard representation

Proof Let Γ be the standard representation and let 120588 be thestandard character of 119870119878 As 119890 is an idempotent we knowthat spec(Γ(119890)) = [1

(119898)

0(119899minus119898)

] as a multiset where 119898 =

rank(Γ(119890)) Thus

120588 (119890) = sum

119904isin119878

119890119904120588 (120590119904) = 1198901sdot 119899 = 119898 (3)

Therefore 1198901= 119898119899 isin 0 1119899 (119899 minus 1)119899 1 and we

have 1198901= 0 = 0119899 hArr 119890 = 0 and 119890

1= 1 = 119899119899 hArr 119890 = 120590

1

Corollary 11 Let (119883 119878) be a finite association scheme Thenthe only idempotents of Z119878 are 0 and 120590

1

Proof Let 119890 isin Z119878 be an idempotent Then 1198901

isin Z ByProposition 10 this implies 119890

1= 0 or 1 and by considering

the rank of Γ(119890) in these respective cases we have 119890 = 0 or1205901

Here we give a glance on the fact that the associationscheme concept generalizes the group concept For moredetails see [1 Section 55] Let (119883 119878) be a finite associationscheme of order 119899 for which every relation in 119878 is thinthat is a thin association scheme Then using the valencymap it follows that 120590

119904 119904 isin 119878 is a group of 119899 distinct

permutation matrices Conversely let 119866 be a group For each119892 in 119866 let 119892

120591 denote the set of all pairs (119890 119891) isin 119866 times

119866 satisfying 119890119892 = 119891 Let 119866120591 denote the set of all sets

119892120591 with 119892 in 119866 Then (119866 119866

120591

) becomes a thin association

scheme So there is correspondence between thin associationschemes and groups called the group correspondence In thiscorrespondence the augmentation map of the integral groupring Z119866 agrees with the valency map of the integral schemering Z[119866120591]

If 119906 = sum119892isin119866

119906(119892)119892 isin Z119866 then augmentation of 119906 issum119892isin119866

119906(119892) isin Z We know any finite subgroup 119867 sube 119881(Z119866)

is a linearly independent set (cf [5 Lemma (371)]) One canaskwhat happens in the case of scheme ringsThenext lemmagives an answer to this question

Lemma 12 Let (119883 119878) be a finite association scheme Thenany finite group of units of valency 1 in Z119878 is a set of linearlyindependent elements

Proof Let 119879 = 1199061= 1205901 1199062 119906

ℓ be a finite group of units

contained in 119881(Z119878) Suppose 11988811199061198941+ sdot sdot sdot + 119888

119898119906119894119898

= 0 is anexpression ofminimal length where 119906

119894119895are elements of119879 and

the coefficients 119888119895isin Z are not all 0 Since 119879 is a group we can

assumewithout loss of generality that 1199061198941= 1205901 Expressing the

119906119894119895for 119895 = 2 119898 as 119906

119894119895= sum119904119906119894119895 119904120590119904 we have by Lemma 1

that 119906119894119895 1

= 0 for 119895 = 2 119898 It follows that

0 = (11988811205901+ 11988821199061198942+ sdot sdot sdot + 119888

119898119906119894119898)1

= 1198881 (4)

contradicting theminimal length assumptionTherefore119879 isa linearly independent set

For a finite group 119866 the order of any finite subgroup 119867

of119881(Z119866) divides the order of119866 [5 Lemma (373)] Our maintheorem shows this also holds for schemes

Theorem 13 Let (119883 119878) be a finite association scheme of order119899 and rank 119903 Then the order of any finite subgroup 119879 of119881(Z119878)divides 119899 and is at most 119903 Symbolically |119879| divides |119883| and|119879| le |119878|

Proof Since Z119878 is a free module with basis 119878 any basis ofZ119878 must have |119878| elements By Lemma 12 119879 is a linearlyindependent subset Therefore |119879| le |119878|

Now let 119890 = sum119905isin119879

(119905|119879|) = |119879| = 1198902 where

sum119905isin119879

119905 = Let Γ be the standard representation and let120588 be the standard character of (119883 119878) Since 119890

2

= 119890 = 0spec(Γ(119890)) = [1

(119898)

0(119899minus119898)

] where119898 is the rank of the matrixΓ(119890) Therefore 120588(119890) = 119898 isin Z+ Also 120588(119890) = (1|119879|)120588() =

(1|119879|)119899()1

= 119899|119879| since the argument of Lemma 12implies ()

1= 1 Therefore 119898 = 119899|119879| hence |119879| divides

119899 = |119883| which proves the theorem

We have been unable to settle the question of whether ornot the order of any finite subgroup of119881(Z119878)must divide theorder of O

120599(119878) Related to this is a possible generalization of

the Zassenhaus conjecture on torsion units to integral schemerings which would be that any normalized torsion unit ofZ119878should be conjugate inQ119878 to some 120590

119904 for an 119904 isin O

120599(119878)

If 119867 is a subgroup of 119881(Z119866) for a finite group 119866 with|119867| = |119866| then Z119866 = Z119867 (cf [5 Lemma (374)]) Thefollowing lemma proves an analogous result for schemes

4 Algebra

Lemma 14 Let (119883 119878) be a finite association scheme with rank119903 If 119879 is a finite subgroup of 119881(Z119878) with |119879| = 119903 then Z119878 =

Z119879

Proof By Lemma 12119879 is linearly independent and thusQ119878 =

Q119879 It follows that Z119878 supe Z119879 and 119898Z119878 sub Z119879 for somepositive integer119898

Let 119879 = 1199051= 1205901 1199052 119905

119903 and let 119904 isin 119878 Then

119898120590119904=

119903

sum

119894=1

119888119894119905119894 for some 119888

119894isin Z (5)

We wish to show that each 119888119894is a multiple of 119898 For each 119895 isin

1 119903 we have

119898120590119904119905minus1

119895= 1198881198951205901+sum

119894 =119895

119888119894(119905119894119905minus1

119895) (6)

Since by Lemma 1 (119905119894119905minus1

119895)1= 0 for 119894 = 119895 the coefficient of 120590

1

on the right hand side is 119888119895whereas on the left hand side it is a

multiple of119898 It follows that119898 | 119888119895for 119895 = 1 119903 Therefore

120590119904isin Z119879 for all 119904 isin 119878 and hence Z119878 = Z119879

While thin association schemes give immediate exampleswhere the conclusion of the preceding theorem holds we areuncertain as to whether Z119878 can possess a finite subgroup ofnormalized units of order |119878|when (119883 119878) is not thinThe nextexample shows that it is certainly possible for the adjacencyalgebra Q119878 to be ring isomorphic to a group algebra when(119883 119878) is not thin

Example 15 Let (119883 119878) be the fifth association schemeof order27 in Hanaki and Miyamotorsquos classification of small asso-ciation schemes [6] This is a commutative nonsymmetricscheme of order 27 and rank 3 We have 119878 = 1

119883 119904 119904lowast

where 119899

119904= 119899119904lowast = 13 and the structure constants of 119878 are

determined by 1205902119904= 6120590119904+7120590119904lowast 120590119904120590119904lowast = 13120590

1+6120590119904+6120590119904lowast and

1205902

119904lowast = 7120590

119904+ 6120590119904lowast

Analysis of the character table of 119878 (see [6]) shows thatQ119878 cong Q119862

3 where 119862

3is a cyclic group of order 3 Let 120594

1be

the irreducible character of C119878 corresponding to the valencymap and let 120595 120595 be the other two irreducible characters ofC119878 Let 119890

1205941 119890120595 119890120595 be the centrally primitive idempotents

of C119878 the character formula for which can be found in [1Lemma 916] An element V ofC119878with order 3 and valency 1is given by

V = 1198901205941+ 1205773119890120595+ 1205772

3119890120595 (7)

and since V is fixed by complex conjugation V isin Q119878 Usingthe character formula for centrally primitive idempotents ofC119878 we find that

V =1

9(minus41205901minus 120590119904+ 2120590119904lowast) (8)

and V2 = Vlowast So if 119879 = 1205901 V V2 then 119879 is a finite subgroup

of normalized units of Q119878 for which Q119878 = Q119879 In this caseZ[19]119878 sube Z119879 ⊊ Z119878

Proposition 16 Let (119883 119878) be the association scheme of order119899 and rank 2 Then

10038161003816100381610038161003816119881 (Z119878)

tor10038161003816100381610038161003816 = 1 if 119899 ge 3

2 if 119899 = 2(9)

Proof Let 119906 be a normalized torsion unit of Z119878 with multi-plicative order 119896 Our Lagrange theorem for schemes impliesthat 119896 divides 119899 and 119896 le 2 So we are done if 119899 is odd Suppose119896 = 2 Since 119878 is symmetric and 119906 isin Z119878 119906 = 119906

lowast Therefore1199062

= 1205901rArr 119906119906

lowast

= 1205901 and so by Proposition 2 119906 = 120590

119904for

some 119904 isin 119878 with 119899119904= 1 Such an element of the scheme of

rank 2 with 119904 = 1119883only exists when 119899 = 2

For symmetric schemes of rank 3 we have already seenthat normalized torsion units must be trivial with order 2Nonsymmetric association schemes of rank 3 such as theone seen in the example above arise naturally from stronglyregular directed graphs

Proposition 17 Let (119883 119878) be a finite association scheme oforder 119899 gt 2 and rank 3 with 119878 = 1

119883 119904 119904lowast

If 119899119904gt 1 then

|119881(Z119878)119905119900119903

| = 1

Proof Suppose 119906 isin 119881(Z119878) is a normalized torsion unit with119906 = 120590

1 By Lemma 1 supp(119906) = 119904 119904

lowast

so 119906 = 119886120590119904+ 119887120590119904lowast for

some 119886 119887 isin Z Since 119899119906= 1 we have 1 = 119886119899

119904+119887119899119904lowast = (119886+119887)119899

119904

which is not possible as 119886 119887 isin Z

4 Applications to Schur Rings andHecke Algebras

Let 119866 be a finite group of order 119899 Let ZF be a Schur ringdefined on the group 119866 This means that F is a partition ofthe set 119866 into nonempty subsets for which we consider thefollowing

(i) 1119866 isin F

(ii) for all 119880 = 1198921 119892119896 isin F 119880lowast = 119892

minus1

1 119892

minus1

119896 isin F

(iii) for all 119880119881119882 isin F there exist nonnegative integers120582119880119881119882

such that

= sum

119882isinF

120582119880119881119882

(10)

where = sum119892isin119880

119892 denotes the sum of the elements of 119880 inthe group ring Z119866

The Schur ringZF is defined to be theZ-span of 119880 isin

F considered as a subring ofZ119866ZF is a freeZ-module ofrank 119903 = |F| By extension of scalars we can consider theSchur ring 119877F for any commutative ring 119877 We will referto a partition of 119866 with the above properties as a Schur ringpartition of 119866 One example of a Schur ring partition is thepartitionF of 119866 into its conjugacy classes in which case thecomplex Schur ring CF is isomorphic to the center of thegroup ring C119866

We claim that the Schur ring ZF is isomorphic to anintegral scheme ring Given the group 119866 and Schur ring

Algebra 5

partition F let F120591 be the images of subsets in F under thegroup correspondence So given 119880 isin F we set

119880120591

= (119909 119910) isin 119866 times 119866 119909119892 = 119910 for some 119892 isin 119880 (11)

Using the properties of the Schur ring partition F it isstraightforward to show that (119866F120591) is an association schemeof order 119899 = |119866| and rank 119903 = |F| Furthermore ZF ≃

Z[F120591] as rings where the isomorphism is produced by therestriction of the regular representation of 119866 to ZF Therestriction of the augmentation map on the group ring toZF corresponds to the valency map of Z[F120591] under thisisomorphism The following corollary is the application ofour Lagrange theorem for scheme rings to this special case

Corollary 18 LetF be a Schur ring partition of a finite group119866 Then the order of any finite subgroup of 119881(ZF) divides |119866|and is at most |F|

Let119867 be a subgroup of a finite group 119866 that has index 119899Let119866119867 be the set of left cosets of119867 in119866 Let 119903 be the numberof distinct double cosets 119867119892119867 of 119867 in 119866 Corresponding toeach double coset119867119892119867 for 119892 isin 119866 let

119892119867

= (119909119867 119910119867) 119910 isin 119909119867119892119867 (12)

Let119866U119867 = 119892119867

119892 isin 119866Then (119866119867119866U119867) is an associationscheme of order 119899 and rank 119903This type of association schemeis known as a Schurian scheme and its rational adjacencyalgebra Q[119866U119867] is ring isomorphic to the ordinary Heckealgebra 119890

119867Q119866119890119867 where 119890

119867= (1|119867|)sum

ℎisin119867ℎ (For details

see [7] and note that the argument given there for this factdoes not require that the field be algebraically closed) Theapplication of our Lagrange theorem for scheme rings in thisspecial case gives the next result

Corollary 19 Let119867 be a subgroup of a finite group119866 that has119899 left cosets and 119903 double cosets Then the order of any finitesubgroup of 119881(Z[119866U119867]) divides 119899 and is at most 119903

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Thework of Allen Herman has been supported by an NSERCDiscovery Grant

References

[1] P-H ZieschangTheory of Association Schemes SpringerMono-graphs in Mathematics Springer Berlin Germany 2005

[2] G Higman Units of group rings [PhD thesis] University ofOxford 1940

[3] S D Berman ldquoOn the equation 119909119898

= 1 in an integral groupringrdquo Ukrainskii Matematicheski Zhurnal vol 7 pp 253ndash2611955

[4] J A Cohn and D Livingstone ldquoOn the structure of groupalgebras Irdquo Canadian Journal of Mathematics vol 17 pp 583ndash593 1965

[5] S K Sehgal Units in Integral Group Rings vol 69 of PitmanMonographs and Surveys in Pure and Applied MathematicsLongman Scientific and Technical Harlow UK 1993

[6] A Hanaki and I Miyamoto Classification of Small AssociationSchemes httpmathshinshu-uacjpsimhanakias

[7] A Hanaki and M Hirasaka ldquoTheory of Hecke algebras toassociation schemesrdquo SUT Journal of Mathematics vol 38 no1 pp 61ndash66 2002

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Algebra

The results of Section 2 show that 119881(Z119878)tor is often equalto the thin radical of 119878when 119878 is a commutative finite schemeIn particular this holds for symmetric association schemes orif the valency of any element of 119878 is divisible by a prime 119901 InSection 3 we establish a ldquoLagrange-typerdquo theorem for finitesubgroups of119881(Z119878)tor by showing that the order of any finitesubgroup of 119881(Z119878)tor divides the order of 119878 and is boundedby the rank of 119878 In Section 4 this result is directly applied toSchur rings and Hecke algebras

Throughout the paper 120577119896will denote a complex primitive

119896th root of unity for a given positive integer 119896 When 119906 isin C119878we will consistently use the notation 119906 = sum

119904119906119904120590119904with 119906

119904isin C

for all 119904 isin 119878

2 The Support of Normalized TorsionUnits of Z119878

Our first lemma is an analogue of Berman-Higmanrsquos propo-sition on torsion units of group rings (see [2 3])

Lemma 1 Let (119883 119878) be a finite association scheme Suppose119906 isin 119881(Z119878)

119905119900119903 Then 1199061

= 0 rArr 119906 = 1205901

Proof Let Γ C119878 rarr 119872119899(C) be the standard representation

of (119883 119878) of degree 119899 = |119883| Let 120588 be the standard characterso

120588 (120590119904) =

119899 if 119904 = 1119883

0 otherwise(2)

Γ(119906) is diagonalizable since Γ(119906)119896 = 119868 for some integer 119896If specΓ(119906) denotes the list of eigenvalues of Γ(119906) (includingmultiplicities) then specΓ(119906) = 120577

119891119894

119896119899

119894=1consists of 119896th roots

of 1 Now 120588(119906) = sum119899

119894=1120577119891119894

119896= 1199061sdot 119899Then |119906

1|119899 = |sum

119899

119894=1120577119891119894

119896| le 119899

and |1199061|119899 = 119899 hArr all 120577119891119894

119896rsquos are equal to 1205771198911

119896 Thus 119906

1isin minus1 0 1

and 1199061

= 0 rArr Γ(119906) = 1205771198911

119896sdot 119868 rArr 119906 = 120577

1198911

1198961205901= 11990611205901 As

119906 isin 119881(Z119878)tor 1199061

= minus1 Therefore 119906 = 1205901

Let (119883 119878) be an association scheme and let 119877 be acommutative ring with identity Let 119906 = sum

119904isin119878119906119904120590119904isin 119877119878 then

119904 in 119878 belongs to the support of 119906 (briefly supp(119906)) if and onlyif 119906119904

= 0 We will say that 119906 isin 119880(119877119878) is a trivial unit if 119906is a unit of 119877119878 for which 119906 = 119906

119904120590119904for some 119906

119904isin 119880(119877) and

a unique element 119904 in the support of 119906 which is necessarilya thin element Trivial units of Z119878 are permutation matriceswith possibly negative sign in the standard representation

Proposition 2 Let 119906 be a unit ofZ119878 with 119906119906lowast = 1205901 Then 119906 is

a trivial unit

Proof Consider 119906119906lowast = 1205901rArr 1 = (119906119906

lowast

)1= sum119904119906119904119906119904119899119904=

sum119904|119906119904|2

119899119904 Since 119906 isin Z119878 it follows that 119906

119904= 0 except for

exactly one 119904 isin 119878 with 119899119904= 1 and 119906

119904= plusmn1

Proposition 3 Let 119906 isin 119881(Z119878)119905119900119903 If 119904 isin O

120599(119878) cap supp (119906) and

120590119904commutes with 119906 then 119906 = 120590

119904is a trivial unit of ZO

120599(119878)

Proof Let 119906 isin 119881(Z119878)tor Let 119904 isin O

120599(119878) cap supp(119906) for which

120590119904commutes with 119906 Then 120590

119904is a unit of Z119878 and 119906

119904= 0 Let

1199061015840

= 120590minus1

119904119906 Since120590

119904commuteswith1199061199061015840 has finite order Since

1199061015840

1= 119906119904

= 0 we must have 1199061015840 = 1205901by Lemma 1 Therefore

119906 = 120590119904

The center of the finite association scheme (119883 119878) isdefined to be 119885(119878) = 119905 isin 119878 120590

119905120590119904= 120590119904120590119905 for all 119904 isin 119878

The scheme (119883 119878) is a commutative scheme if 119885(119878) = 119878 Thenext two corollaries are immediate from Proposition 3

Corollary 4 Let (119883 119878) be a finite association scheme Suppose119906 isin 119881(Z119878)

119905119900119903 is a nontrivial unit If 119904 isin supp (119906) then either119899119904ge 2 or 119904 notin 119885(119878)

Corollary 5 Let (119883 119878) be a finite commutative associationscheme Suppose 119906 isin 119881(Z119878)

119905119900119903 is a nontrivial unit If 119904 isin

supp (119906) then 119899119904ge 2

If 119866 is a finite group then it is well known that centraltorsion units ofZ119866 are trivial [4Theorem 21]We are able toextend this result to finite association schemeswhose nonthinelements have valencies divisible by a single prime

Theorem 6 Suppose (119883 119878) is a finite association schemeSuppose there is a prime integer 119901 that divides 119899

119904for every 119904 isin 119878

with 119899119904gt 1 Then every normalized central torsion unit of Z119878

is a trivial unit

Proof Let 119906 = sum119904isin119878

119906119904120590119904isin 119881(Z119878)

tor be a central element ofZ119878 with multiplicative order 119896 Suppose 119906 is not trivial ByProposition 3 every 119904 isin supp(119906) has 119899

119904gt 1 Our assumption

then implies that 119901 divides 119899119904 for every 119904 isin supp(119906)

Then 1 = (119906119896

)1

isin spanZ(12059011990411205901199042 sdot sdot sdot 120590119904119896)1 119904119894

isin

supp(119906) for 1 le 119894 le 119896 If (12059011990411205901199042sdot sdot sdot 120590119904119896)1

= 0 then itis divisible by (120590

1199041120590119904lowast

1

)1

= 1198991199041 hence divisible by 119901 This

contradicts 1 = (119906119896

)1 hence the result follows

A finite association scheme (119883 119878) is 119901-valenced for someprime integer119901 if 119899

119904is a power of119901 for all 119904 isin 119878We know that

for a finite abelian group 119866 every torsion unit of the integralgroup ringZ119866 is a trivial unit The next corollary generalizesthis result to 119901-valenced commutative schemes

Corollary 7 If (119883 119878) is a finite 119901-valenced commutativeassociation scheme then every normalized torsion unit of Z119878is a trivial unit

Proof Since (119883 119878) is a commutative association scheme theadjacency algebraZ119878 is a commutative ringTherefore everyunit of Z119878 is central ByTheorem 6 every 119906 isin 119881(Z119878)

tor mustbe a trivial unit that is 119906 = 120590

119904 for some 119904 isin 119878with 119899

119904= 1

An association scheme (119883 119878) is symmetric if all of theadjacency matrices 120590

119904for 119904 isin 119878 are symmetric matrices or

equivalently 119904lowast = 119904 for all 119904 isin 119878 It is easy to show thatsymmetric association schemes are commutative

Theorem 8 Let (119883 119878) be a finite symmetric associationscheme If 119906 isin 119881(Z119878)

119905119900119903 then 119906 = 120590119904 for some 119904 isin 119878 and

1205902

119904= 1205901 In particular torsion units ofZ119878 are trivial with order

2 at most

Algebra 3



119904 forall119904 isin 119878 but 1199062 = 120590

1 Also 1199062 =

(sum1199041199062

119904119899119904)1205901 By Corollary 5 119899








sum119904isin119878

119890119904120590119904





(119898)

0(119899minus119898)



120588 (119890) = sum

119904isin119878

119890119904120588 (120590119904) = 1198901sdot 119899 = 119898 (3)


have 1198901= 0 = 0119899 hArr 119890 = 0 and 119890

1= 1 = 119899119899 hArr 119890 = 120590

1


1










119892120591 with 119892 in 119866 Then (119866 119866

120591



If 119906 = sum119892isin119866





Proof Let 119879 = 1199061= 1205901 1199062 119906



119898119906119894119898





119906119894119895for 119895 = 2 119898 as 119906


that 119906119894119895 1


0 = (11988811205901+ 11988821199061198942+ sdot sdot sdot + 119888

119898119906119894119898)1

= 1198881 (4)






Now let 119890 = sum119905isin119879

(119905|119879|) = |119879| = 1198902 where

sum119905isin119879


2

= 119890 = 0spec(Γ(119890)) = [1

(119898)

0(119899minus119898)


(1|119879|)119899()1







119904 for an 119904 isin O

120599(119878)


4 Algebra


Z119879



Let 119879 = 1199051= 1205901 1199052 119905

119903 and let 119904 isin 119878 Then

119898120590119904=

119903

sum

119894=1

119888119894119905119894 for some 119888

119894isin Z (5)


1 119903 we have

119898120590119904119905minus1

119895= 1198881198951205901+sum

119894 =119895

119888119894(119905119894119905minus1

119895) (6)



1






119883 119904 119904lowast

where 119899



1+6120590119904+6120590119904lowast and

1205902

119904lowast = 7120590

119904+ 6120590119904lowast


3 where 119862


1be




V = 1198901205941+ 1205773119890120595+ 1205772

3119890120595 (7)


V =1





10038161003816100381610038161003816119881 (Z119878)

tor10038161003816100381610038161003816 = 1 if 119899 ge 3

2 if 119899 = 2(9)



= 1205901rArr 119906119906

lowast


119904for





119883 119904 119904lowast

If 119899119904gt 1 then

|119881(Z119878)119905119900119903

| = 1


1 By Lemma 1 supp(119906) = 119904 119904

lowast

so 119906 = 119886120590119904+ 119887120590119904lowast for


119904+119887119899119904lowast = (119886+119887)119899

119904




(i) 1119866 isin F


minus1

1 119892

minus1

119896 isin F


such that

= sum

119882isinF

120582119880119881119882

(10)






Algebra 5


119880120591






119892119867

= (119909119867 119910119867) 119910 isin 119909119867119892119867 (12)

Let119866U119867 = 119892119867


119867Q119866119890119867 where 119890

119867= (1|119867|)sum






Acknowledgment


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Algebra 3



119904 forall119904 isin 119878 but 1199062 = 120590

1 Also 1199062 =

(sum1199041199062

119904119899119904)1205901 By Corollary 5 119899








sum119904isin119878

119890119904120590119904





(119898)

0(119899minus119898)



120588 (119890) = sum

119904isin119878

119890119904120588 (120590119904) = 1198901sdot 119899 = 119898 (3)


have 1198901= 0 = 0119899 hArr 119890 = 0 and 119890

1= 1 = 119899119899 hArr 119890 = 120590

1


1










119892120591 with 119892 in 119866 Then (119866 119866

120591



If 119906 = sum119892isin119866





Proof Let 119879 = 1199061= 1205901 1199062 119906



119898119906119894119898





119906119894119895for 119895 = 2 119898 as 119906


that 119906119894119895 1


0 = (11988811205901+ 11988821199061198942+ sdot sdot sdot + 119888

119898119906119894119898)1

= 1198881 (4)






Now let 119890 = sum119905isin119879

(119905|119879|) = |119879| = 1198902 where

sum119905isin119879


2

= 119890 = 0spec(Γ(119890)) = [1

(119898)

0(119899minus119898)


(1|119879|)119899()1







119904 for an 119904 isin O

120599(119878)


4 Algebra


Z119879



Let 119879 = 1199051= 1205901 1199052 119905

119903 and let 119904 isin 119878 Then

119898120590119904=

119903

sum

119894=1

119888119894119905119894 for some 119888

119894isin Z (5)


1 119903 we have

119898120590119904119905minus1

119895= 1198881198951205901+sum

119894 =119895

119888119894(119905119894119905minus1

119895) (6)



1






119883 119904 119904lowast

where 119899



1+6120590119904+6120590119904lowast and

1205902

119904lowast = 7120590

119904+ 6120590119904lowast


3 where 119862


1be




V = 1198901205941+ 1205773119890120595+ 1205772

3119890120595 (7)


V =1





10038161003816100381610038161003816119881 (Z119878)

tor10038161003816100381610038161003816 = 1 if 119899 ge 3

2 if 119899 = 2(9)



= 1205901rArr 119906119906

lowast


119904for





119883 119904 119904lowast

If 119899119904gt 1 then

|119881(Z119878)119905119900119903

| = 1


1 By Lemma 1 supp(119906) = 119904 119904

lowast

so 119906 = 119886120590119904+ 119887120590119904lowast for


119904+119887119899119904lowast = (119886+119887)119899

119904




(i) 1119866 isin F


minus1

1 119892

minus1

119896 isin F


such that

= sum

119882isinF

120582119880119881119882

(10)






Algebra 5


119880120591






119892119867

= (119909119867 119910119867) 119910 isin 119909119867119892119867 (12)

Let119866U119867 = 119892119867


119867Q119866119890119867 where 119890

119867= (1|119867|)sum






Acknowledgment


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









4 Algebra


Z119879



Let 119879 = 1199051= 1205901 1199052 119905

119903 and let 119904 isin 119878 Then

119898120590119904=

119903

sum

119894=1

119888119894119905119894 for some 119888

119894isin Z (5)


1 119903 we have

119898120590119904119905minus1

119895= 1198881198951205901+sum

119894 =119895

119888119894(119905119894119905minus1

119895) (6)



1






119883 119904 119904lowast

where 119899



1+6120590119904+6120590119904lowast and

1205902

119904lowast = 7120590

119904+ 6120590119904lowast


3 where 119862


1be




V = 1198901205941+ 1205773119890120595+ 1205772

3119890120595 (7)


V =1





10038161003816100381610038161003816119881 (Z119878)

tor10038161003816100381610038161003816 = 1 if 119899 ge 3

2 if 119899 = 2(9)



= 1205901rArr 119906119906

lowast


119904for





119883 119904 119904lowast

If 119899119904gt 1 then

|119881(Z119878)119905119900119903

| = 1


1 By Lemma 1 supp(119906) = 119904 119904

lowast

so 119906 = 119886120590119904+ 119887120590119904lowast for


119904+119887119899119904lowast = (119886+119887)119899

119904




(i) 1119866 isin F


minus1

1 119892

minus1

119896 isin F


such that

= sum

119882isinF

120582119880119881119882

(10)






Algebra 5


119880120591






119892119867

= (119909119867 119910119867) 119910 isin 119909119867119892119867 (12)

Let119866U119867 = 119892119867


119867Q119866119890119867 where 119890

119867= (1|119867|)sum






Acknowledgment


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Algebra 5


119880120591






119892119867

= (119909119867 119910119867) 119910 isin 119909119867119892119867 (12)

Let119866U119867 = 119892119867


119867Q119866119890119867 where 119890

119867= (1|119867|)sum






Acknowledgment


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article on the torsion units of integral adjacency...

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