TORSION UNITS OF INTEGRAL GROUP RINGS AND SCHEME RINGS
A Thesis
Submitted to the Faculty of Graduate Studies and Research
In Partial Fulfillment of the Requirements
For the Degree of
Doctor of Philosophy
In
Mathematics
University of Regina
By
Gurmail Singh
Regina, Saskatchewan
August, 2015
c Copyright 2015: Gurmail Singh
UNIVERSITY OF REGINA
FACULTY OF GRADUATE STUDIES AND RESEARCH
SUPERVISORY AND EXAMINING COMMITTEE
Gurmail Singh, candidate for the degree of Doctor of Philosophy in Mathematics, has presented a thesis titled, Torsion Units of Integral Group Rings and Scheme Rings, in an oral examination held on August 26, 2015. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material.
External Examiner: *Dr. Yuanlin Li, Brock University
Co-Supervisor: Dr. Allen Herman, Department of Mathematics & Statistics
Co-Supervisor: Dr. Shaun Fallat, Department of Mathematics & Statistics
Committee Member: Dr. Fernando Szechtman, Department of Mathematics & Statistics
Committee Member: Dr. Karen Meagher, Department of Mathematics & Statistics
Committee Member: **Dr. Robert Hilderman, Department of Computer Science
Chair of Defense: Dr. Christopher Yost, Department of Biology
*via SKYPE **Not present at defense Abstract
We study torsion units of algebras over the ring of integers Z with nice bases.
These include integral group rings, integral adjacency algebras of association schemes and integral C-algebras.
Torsion units of group rings have been studied extensively since the 1960’s.
Much of the attention has been devoted to the Zassenhaus conjecture for normal- ized torsion units of ZG, which says that they should be rationally conjugate (i.e. in QG) to elements of the group G. In recent years several new restrictions on inte- gral partial augmentations for torsion units of ZG have been introduced that have improved the effectiveness of the Luthar-Passi method for checking the Zassen- haus conjecture for specific finite groups G. We have implemented a computer program that constructs units of QG that have integral partial augmentations that are relevant to the Zassenhaus conjecture. Indeed, any unit of ZG with these par- tial augmentations would be a counterexample to the conjecture. In all but three
i exceptions among groups of order less than 160, we have constructed units of
QG with these partial augmentations that satisfy a condition which implies they cannot be rationally conjugate to an element of ZG. Currently our package has computational difficulties with the Luthar-Passi method for some of the groups of order 160.
As C-algebras are generalization of groups, it is natural to ask about torsion units of C-algebras. We establish some basic results about torsion units of C- algebras analogous to what happens for torsion units of group rings. These results can be immediately applied to give new results for Schur rings, Hecke algebras, adjacency algebras of association schemes and fusion rings. We also investigate the possibility for a conjecture analogous to the Zassenhaus conjecture in the C- algebra setting.
ii Acknowledgment
I am grateful to my supervisors Professor Allen W. Herman and Professor Shaun
M. Fallat for all of the their support, understanding, patience, and knowledge they have provided me over the course of my study. Without their supervision and mentorship this would not have been possible.
I am thankful to Dr. Yuanlin Li, my external examiner. I also wish to thank
Dr. Karen Meagher and Dr. Fernando Szechtman for their advice and careful reading of the manuscript. Their suggestions have been valuable and helpful for my thesis.
Finally, the financial support of my supervisors’s NSERC grants, Department of Mathematics and Statistics, and the Faculty of Graduate Studies and Research during my PhD program allowed me to focus solely on my PhD program during these past years.
iii To my family
iv Contents
Acknowledgment iii
Dedication iv
1 Introduction 1
2 Background 5
2.1 Group rings ...... 5
2.2 Generalized C-algebras and table algebras ...... 9
2.3 Association schemes and scheme rings ...... 12
2.4 Representations and characters of semisimple algebras ...... 18
2.4.1 Semisimple algebras ...... 18
2.4.2 Representation theory of semisimple algebras ...... 21
2.4.3 Representation theory of groups ...... 24
v 3 Basic Tools 28
3.1 Torsion units of ZG and partial augmentations ...... 29
3.2 Luthar-Passi method ...... 30
3.3 The standard feasible trace of a C-algebra ...... 36
4 Normalized Torsion Units of Integral Group Rings 45
4.1 Partial augmentations and rational conjugacy ...... 46
4.2 Computer implementation of the Luthar-Passi method ...... 49
4.3 Computer construction of torsion units with prescribed partial aug-
mentations ...... 55
4.4 Partially central torsion units of QG ...... 65
5 Torsion units of C-algebras 69
5.1 Torsion units of RB ...... 70
5.2 Torsion units for integral C-algebras with a
standard character ...... 75
5.3 Lagrange’s theorem for normalized torsion
units of Z¯ B ...... 82
5.4 Applications to Schur rings and Hecke algebras ...... 90
5.4.1 Schur rings ...... 90
vi 5.4.2 Integral Hecke algebras ...... 92
6 Future Work 93
6.1 Categorical aspects ...... 94
6.2 When all units of ZB are trivial ...... 96
6.3 Normalized automorphisms of QB ...... 98
vii Chapter 1
Introduction
Let R be a commutative ring with identity. A group ring RG is a ring as well as a free R-module whose basis is a multiplicative group G. When the ring R is replaced with a field K then KG is called a group algebra. The trivial units of a group ring RG are scalar multiples of a single group element in G by a unit of the ring R. We will investigate a conjecture that would characterize the nontrivial torsion units of the integral group ring ZG. To get a broader perspective of what is really going on, we formulate and study the analogous conjecture in the general settings of the integral C-algebras and integral adjacency algebras of association schemes (a.k.a scheme rings). Along the way we establish several basic results to set up the necessary machinery in these new settings.
1 In this dissertation we present new results on torsion units of group rings,
C-algebras, and scheme rings. Using the Luthar-Passi method and our new con- struction for partially central units we show how we have verified the Zassenhaus conjecture for all but three groups of order up to 159. In the case of C-algebras we establish some fundamental results for torsion units of integral C-algebras with s- tandard character, and we prove a Lagrange-type theorem for integral C-algebras that applies directly to integral scheme rings.
Our initial motivation came from the fact that the Zassenhaus conjecture for integral group rings was open for fairly small groups, and so there was an opportu- nity to raise the lower bound or to find a counterexample . The algebraic study of association schemes, table algebras, and C-algebras has seen a number of signifi- cant advances in the last five years, due to the realization of close connections with
finite group theory. As the bases for C-algebras and finite association schemes are close to finite groups, we felt we could formulate and investigate versions of the
Zassenhaus conjecture in these settings.
The purpose of this dissertation is to shed light upon various properties of torsion units for both group rings, C-algebras and scheme rings, to verify Zassen- haus’s conjecture for group rings in some new cases and to move a step toward establishing the Zassenhaus conjecture for a broader class of rings.
2 The dissertation is organized in the following manner. In Chapter 2, we de-
fine the algebraic structures such as group rings, adjacency algebras and torsion units of integral C-algebras. We give a brief review of particular known results for torsion units of integral group rings. In Section 2.1, we define integral group rings, their normalized torsion units, and we give the statement of the Zassenhaus conjecture for the normalized torsion units of integral group rings. In Section 2.2, we define generalized C-algebras and table algebras. In Section 2.3, we define as- sociation schemes and scheme rings and demonstrate how an association scheme is a generalization of a group. In Section 2.4, we define semisimple algebras and state some background results for semisimple algebras that will be used. Then we define representations and characters for algebras, and give a similar treatment for groups. In Chapter 3, we state all the preliminary results that will be used in
Chapters 4 and 5 to prove the results. In Section 3.1, we state the results used in the Luthar-Passi method. In Section 3.2, we demonstrate the Luthar-Passi method by applying it to the group A4. In Section 3.3, we introduce standard feasible of a C-algebra that will be needed in Chapters 5. In Chapter 4, we examine the extent to which rational conjugacy of units in QG is determined by partial aug- mentations. Also we show that partially central units are not conjugate in QG to elements of ZG and we construct partially central units of QG for a group of order
3 48. This demonstrates the method we used to verify the Zassenhaus conjecture for all but three groups of orders up to 159. In Chapter 5, we prove a generalization of the Berman-Higman Lemma for integral C-algebras and using this we prove a
Lagrange-type theorem for integral C-algebras. We apply these results directly to several familiar settings, including that of integral scheme rings. We try to move a step toward generalizing the Zassenhaus conjecture to integral C-algebras by proving conjugacy of finite subgroups of units of KB in LB implies their conju- gacy in KB, where K and L are infinite subfields of C with K ⊆ L and B is the distinguished basis for the C-algebra.
4 Chapter 2
Background
In this chapter we provide background material on group rings, scheme rings, and
C-algebras which we will need in Chapter 4 and Chapter 5 to prove the main theorems. Basic definitions and concepts of groups, rings, fields, and algebras that are commonly encountered in undergraduate mathematics courses will be assumed. These can be found in [11].
2.1 Group rings
Definition 2.1.1. Let R be a commutative ring with identity 1 , 0 and let G be a
finite group. A group ring RG of G over R is a free module over R with basis G,
5 i.e. the set of all formal finite sums
X αgg, where g ∈ G, αg ∈ R.
The addition in RG is given by:
X X X αgg + βgg = (αg + βg)g, the multiplication in RG by:
X X X ( αgg)( βhh) = (αgβg−1t)t, where t = gh, g h g, t and scalar multiplication by:
X X a( αgg) = aαgg, for all a ∈ R. g∈G g∈G
If we take ring R = Z to be the ring of integers, then the group ring ZG is called the integral group ring of G. When we take R = Q to be the field of rational numbers the group ring QG is called the rational group algebra of G. If we take
R = C to be the field of complex numbers, the group ring CG is called the complex group algebra of G. In general when we take R to be any field then RG is called a group algebra over R.
If x and g are elements of a group G, we write x ∼ g when x and g are conjugate in G. Let K(G) be a complete set of representatives for the conjugacy
6 P P classes of G. When u = ugg ∈ ZG, ε(u) = ug denotes the augmentation g∈G g∈G P of u, and εx(u) = ug denotes the partial augmentation of u with respect to x∼g x ∈ G. We note that augmentation is an algebra homomorphism, so the kernel of this homomorphism is called the augmentation ideal.
A unit in a ring R with identity is any element x ∈ R which has a two-sided multiplicative inverse y in R. Thus x is a unit of R if and only if for some y ∈ R, xy = yx = 1. If R is a ring with identity, then U(R) denotes the group of units of R. A unit of a ring R with finite multiplicative order is called a torsion unit.
The subset of R consisting of torsion units is denoted by U(R)tor. The subgroup of
U(R) consisting of units with augmentation 1 is denoted by V(R). This is the set of normalized units of R and V(R)tor denotes the set of normalized torsion units of P R. If u = ugg is a unit in a group ring RG with ug ∈ R, then the support of u is g∈G the set supp(u) = {g ∈ G : ug , 0}. A unit u of a group ring RG is a trivial unit if u = ugg for some ug ∈ U(R) and a unique element g in the support of u, where
U(R) denotes the group of units of R.
The study of torsion units of integral group rings has centered upon a funda- mental conjecture made by Hans Zassenhaus in 1966.
Conjecture 2.1.2 (Zassenhaus [32]). Let G be a finite group. Then every element of V(ZG)tor is rationally conjugate to an element of the group G. That is, if u ∈
7 V(ZG)tor, then there exists b ∈ U(QG) and g ∈ G such that u = b−1gb.
Zassenhaus made several stronger conjectures at the time, all of which have now been disproven. This one, however, remains open. The Zassenhaus conjec- ture (abbr. ZC) has been shown to hold in the following general situations:
(i) Abelian groups ([32, Corollary (1.6)]).
(ii) Nilpotent groups (Weiss [35]).
(iii) Groups G with a normal Sylow p-subgroup P for which G/P is abelian
(Hertweck [20]).
(iv) Cyclic-by-abelian groups (Caicedo-Margolis-Del Rio´ [7]).
(v) All groups of order up to 71. (Hofert-Kimmerle [24]).
(vi) G = A o X, where A and X abelian groups, | X |= m < p for every prime
dividing | A |, and m is prime (Marciniak-Ritter-Sehgal-Weiss [28]).
(vii) Frobenius groups (Hertweck [23]).
8 2.2 Generalized C-algebras and table algebras
In this dissertation we shall consider torsion units of generalized integral C-algebras and generalized integral table algebras (see [1]). These are finite-dimensional al- gebras, possibly non-commutative, that have a basis with special properties, one of which is having integral structure constants. Over the complex numbers these algebras share many properties in common with the complex group algebra. The
C in C-algebra stands for “character algebra”.
Definition 2.2.1. A (generalized) C-algebra is a triple (A, B, δ), where A is a
finite-dimensional algebra over C with an R-linear and C-conjugate linear in- volution ∗ : A → A, B = {b0, b1,..., bd} is a distinguished basis, δ : A → C is an algebra homomorphism called the degree map that satisfies the following properties:
(i) 1A ∈ B (throughout we set b0 to be the multiplicative identity in A),
∗ (ii) (bi) = bi∗ ∈ B, for all bi ∈ B, for a transposition ∗ : {0, 1,..., d} →
{0, 1,..., d}
(iii) multiplication in A defines real structure constants in the basis B, i.e. for
9 all bi, b j ∈ B, we have
X bib j = λi jkbk, for some λi jk ∈ R,
bk∈B
∗ (iv) for all bi, b j ∈ B, λi j0 , 0 ⇐⇒ j = i ,
(v) for all bi ∈ B, λii∗0 = λi∗i0 > 0, and
(vi) δ(bi) = λii∗0 for all bi ∈ B.
P ∗ P If u = i uibi ∈ A with ui ∈ C for all i ∈ {0, 1,..., d}, then u = i u¯ibi∗ . The involution ∗ on A is understood by definition to be an antiautomorphism of A; i.e.
(uv)∗ = v∗u∗, for all u, v ∈ A. Being an algebra homomorphism, the degree map satisfies δ(uv) = δ(u)δ(v), for all u, v ∈ A. Note also that δ(u∗) = δ(u), for all u ∈ A.
The real numbers λi jk are the structure constants relative to the basis B. The
C-algebra basis B given in the definition is considered to be standardized because it satisfies δ(bi) = λii∗0, for all bi ∈ B.A C-algebra (A, B, δ) with standardized
+ P | | basis B is said to have order n = δ(B ):= bi∈B δ(bi) and rank r = B = d + 1.
The prototype example of a C-algebra is the complex group algebra CG.
We will have use for a few refinements of this definition. A rational (or inte- gral) C-algebra is a C-algebra whose structure constants in the basis B lie in Q (or
10 respectively, in Z). A table algebra is a C-algebra whose structure constants are nonnegative. An algebra with a distinguished basis satisfying conditions 1 to 5 of the C-algebra definition are called reality-based algebras (RBAs). RBAs may or may not have a degree map. If K is an algebraic number field whose ring of algebraic integers is R, then we will say that a C-algebra (A, B, δ) is R-integral when all of its structure constants in the basis B lie in the intersection of R with
R. If B is the distinguished basis of a C-algebra (or table algebra), then we shall say that B is a C-algebra basis (or table algebra basis).
The set of linear elements of a standardized C-algebra basis B is L(B) = {bi ∈
∗ ∗ ∗ 2 B : bibi = λii 0b0}. Note that b0b0 = b0 = b0, so b0 ∈ L(B). Furthermore, if bi ∈ B
∗ 2 then applying the degree map to bibi = λii∗0b0 gives δ(bi) = δ(bi). Since δ(bi) > 0
∗ −1 we must have λii∗0 = δ(bi) = 1, and so bi = (bi) . We then can conclude that
L(B) is a finite subgroup of normalized units of ZB.
C-algebras and table algebras have been studied in the commutative situation in various forms (see Blau’s recent survey [4]). Noncommutative table algebras were considered in [1] under the name “generalized table algebras”. The alge- bras considered here are almost the same, except we do not assume the structure constants are all nonnegative.
11 2.3 Association schemes and scheme rings
We shall only consider finite association schemes, which we now define.
Definition 2.3.1. Let X be a finite set of size n > 0. Let S be a partition of X × X such that every relation in S is non-empty. 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. The pair (X, S ) is an association scheme if:
(i) S is a partition of X × X consisting of nonempty sets,
(ii) S contains the identity relation 1X := {(x, x): x ∈ X},
(iii) for all s in S , the adjoint relation s∗ := {(y, x) ∈ X × X :(x, y) ∈ s} also
belongs to S , and
(iv) for all s, t ∈ S , there exist nonnegative integer structure constants astu, for P all u ∈ S , such that σsσt = astuσu. u∈S
A finite association scheme (X, S ), or scheme for short, is said to have order n = |X| and rank r = |S |. Since S is a partition of X × X, the sum of all adjacency matrices is the all ones matrix, denoted by J. For notation and background on association schemes see [38].
12 Examples of association schemes include several familiar algebraic structures.
(i) Finite groups. Let G be a finite group of order n. Let Gτ = {gτ : g ∈ G} ⊆
G × G, where (x, y) ∈ gτ ⇐⇒ xg = y. Then (G, Gτ) is an association
scheme of order n and rank n.
(ii) Schur Rings. Let F be a Schur ring partition of a group G of order n, with
|F | = r. For all U ∈ F , define Uτ ⊂ G × G by (x, y) ∈ Uτ ⇐⇒ xg = y,
for some g ∈ U. Let F τ = {Uτ : U ∈ F }. Then (G, F τ) is an association
scheme of order n and rank r.
(iii) Hecke Algebras. Suppose H is a subgroup of a group G with index n. Let
G/H be the set of left cosets of H in G. Let G//H = {gH : g ∈ G} ⊂
G/H × G/H, where (xH, yH) ∈ gH ⇐⇒ y ∈ xHgH. |G//H| = r is the
number of double cosets of H in G. Then (G/H, G//H) is an association
scheme of order n and rank r.
Our first example above shows that an association scheme is a generalization of group. But now we give an example of an association scheme that is not a group.
Example 2.3.2. Let X = {a, b, c, d, e, f } be a set. Let S = {s0, s1, s2, s3} be the
13 relations:
s0 = {(a, a), (b, b), (c, c), (d, d), (e, e), ( f, f )},
s1 = {(a, b), (b, a), (c, d), (d, c), (e, f ), ( f, e)},
s2 = {(a, c), (a, d), (b, c), (b, d), (c, e), (c, f ), (d, e), (d, f ), (e, a), (e, b), ( f, a), ( f, b)},
s3 = {(a, e), (a, f ), (b, e), (b, f ), (c, a), (c, b), (d, a), (d, b), (e, c), (e, d), ( f, c), ( f, d)}.
Then (X, S ) is an association scheme of rank 4 and order 6. The adjacency matrices are:
1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 σ0 = , σ1 = , 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 σ2 = , σ3 = . 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 and the strucuture constants are given by the following equations:
2 2 σ1 = σ0, σ2 = 2σ3,
σ1σ2 = σ2σ1 = σ2, σ2σ3 = σ3σ2 = 2σ0 + 2σ1,
2 σ1σ3 = σ3σ1 = σ3, σ3 = 2σ2.
14 Note that the collection of adjacency matrices is not a group. The adjacency ma- trices σ2 and σ3 only have rank 3 and so are not invertible.
Definition 2.3.3. Let (X, S ) be an association scheme and let R be a commutative ring with identity 1 , 0. The adjacency algebra of the scheme (X, S ) over R is a ring as well as a free module over R with basis {σs : s ∈ S }, i.e. the set of all formal finite sums: X γsσs, where s ∈ S, γs ∈ R. s∈S
The addition in RS is given by:
X X X γsσs + µsσs = (γs + µs)σs. s∈S s∈S s∈S the multiplication in RS is given by:
X X γµ = γsµtσsσt = γsµtastuσu, s,t s,t,u and scalar multiplication by:
X X a( γsσs) = aγsσs, a ∈ R. s∈S s∈S
So the structure constants of the scheme (X, S ) make the integer span of its adjacency matrices into a natural Z-algebra ZS := ⊕s∈S Zσs. This is known as the integral adjacency algebra of the scheme (X, S ), which we will simply refer to
15 as the integral scheme ring. Similarly, we can define rational adjacency algebra
(complex adjacency algebra) to be the rational span of adjacency matrices (resp. complex span of adjacency matrices). Adjacency algebras (integral scheme rings) are examples of C-algebras (resp. integral C-algebras). Note that the multiplica- tive identity of ZS is the n × n identity matrix, which is the adjacency matrix
σ1X := σ1.
It is easy to show using the definition of a scheme that the structure constant
∗ ast1 , 0 if and only if t = s . We write ns instead of ass∗1 and call ns the va- lency of s. The linear extension of the valency map defines a degree one algebra representation CS → C by
X X u = usσs → nu = usns. s∈S s∈S So the valency map for scheme rings generalizes the augmentation map for group rings.
We say that s ∈ S is a thin element of S when ns = 1. The thin radical Oϑ(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. For the same reason, the valency of a unit u ∈ U(CS ) has to be a nonzero element of C,
−1 and thus nu u is a unit of valency 1. The subgroup of U(ZS ) consisting of units of valency 1 is denoted by V(ZS ).
16 The next result introduces a natural involution on the adjacency algebras of
finite association schemes.
Proposition 2.3.4. Let (X, S ) be an association scheme. The complex scheme algebra CS has an involution given as follows: let γ, µ ∈ CS and a ∈ C, for
P ∗ P − γ = γsσs, set γ = γsσs∗ , where denotes the complex conjugate. Then s∈S s∈S
(i) (γ + µ)∗ = γ∗ + µ∗,
(ii) (γµ)∗ = µ∗γ∗,
(iii) (γ∗)∗ = γ, and
(iv) (aγ)∗ = aγ∗.
Proof. For (i)
∗ X X X ∗ ∗ (γ + µ) = (γs + µs)σs∗ = γsσs∗ + µsσs∗ = γ + µ . s∈S s∈S s∈S
For (ii)
∗ P ∗ P ∗ (γµ) = ( γsµtσsσt) = ( γsµtastuσu) s,t s,t,u P P = γsµtastuσu∗ = γsµtat∗ s∗u∗ σu∗ s,t,u s,t,u
P ∗ ∗ = ( µtσt∗ )(γsσs∗ ) = µ γ . For (iii)
∗ X ∗ ∗ X γ = γsσs∗ =⇒ (γ ) = γsσs = γ. s∈S s∈S
17 For (iv)
∗ X ∗ X ∗ X ∗ ∗ (aγ) = (a γsσs) = ( aγsσs) = aγsσs∗ =⇒ (aγ) = aγ . s∈S s∈S s∈S
It now follows that CS is a generalized integral C-algebra for any finite as- sociation scheme (X, S ). Its involution is given in Proposition 2.3.4, the set of adjacency matrices is its C-algebra basis, and the valency map is its degree map.
2.4 Representations and characters of semisimple al-
gebras
2.4.1 Semisimple algebras
Here we collect some basic theory of semisimple algebras.
Theorem 2.4.1 (Krull-Schmidt-Azumaya [10]). Every finite-dimensional algebra over a field K has a decomposition A = A1 ⊕ · · · ⊕ Ah in which the Ai are inde- composable two-sided ideals of A. The list of algebras occurring in any such indecomposable decomposition of A is uniquely determined up to algebra iso- morphism.
18 Definition 2.4.2. An algebra A is a simple algebra over a field if A , 0 and the only ideals of A are 0 and A.
Every finite-dimensional simple algebra over a field is isomorphic to a full matrix algebra over a division algebra over the same field; cf. [10, Theorem 3.28].
Definition 2.4.3. Let K be a field. A finite-dimensional K-algebra A is semisimple if its indecomposable two-sided ideals are isomorphic to simple K-algebras (i.e. full matrix rings over K-division algebras).
Definition 2.4.4. Let A be a semisimple algebra over a field K with decomposition of the following form
m A = ⊕ j=1 Mn j (K),
where Mn j (K) is full matrix ring of n j × n j matrices over K. Then A is called a split semisimple algebra.
Definition 2.4.5. Let R be a ring with identity. Then the Jacobson radical J(R) of a ring R is the intersection of all maximal left ideals of R.
Lemma 2.4.6 (Nakayama’s Lemma). Let R be a ring with identity, and let J(R) be the Jacobson radical of R. If M is any finitely generated R-module and J(R)M =
M, then M = 0.
19 Proof. Suppose M , 0 and let {m1, m2,..., mn} be a minimal generating set of M over R. Since M = J(R)M, we have
mn = rlml + r2m2 + ... + rnmn, for some r1, r2,..., rn ∈ J(R).
Thus (1 − rn)mn = rlml + ... + rn−1mn−1. It is enough to prove that 1 − rn is a unit.
Suppose 1 − rn is not a unit. Let I be a maximal ideal of R that contains 1 − rn.
Since 1 < I and rn < I we have that rn is not in J(R). Hence 1 − rn is a unit.
Theorem 2.4.7. If A is a finite-dimensional algebra over a field K, then J(A) is a nilpotent ideal; i.e. there exists a positive integer k such that J(A)k = 0.
Proof. Since A is finite-dimensional and J(A)2 ⊆ J(A), there exists a positive integer m such that J(A)m = J(A)m+i for all positive integers i, in particular
J(A)m = J(A)m+1. Hence J(A)m = J(A)J(A)m, and so by Nakayama’s Lemma,
J(A)m = 0; cf. [11].
Proposition 2.4.8 (Theorem 5.18 [10]). A finite-dimensional K-algebra A is semisim- ple if and only if its Jacobson radical is 0.
In the next theorem we shall reproduce [1, Theorem 3.11] in the setting of reality-based algebras.
20 Theorem 2.4.9. Let (A, B) be an RBA. Then A is a semisimple algebra.
Proof. Let J be the Jacobson radical of A. Since ∗ is an antiautomorphism of A,
J∗ = J. Assume that J , {0}. Since J is nilpotent, there exists a minimal m ∈ N, m ≥ 2 such that Jm = {0}. Set I = Jm−1. By the choice of m, I , {0} and I2 = {0}.
∗ ∗ P Since J = J, I = I. Take a non-zero x ∈ I, where x ∈ b∈B xbb, for xb ∈ C. Then
∗ P ∗ ∗ 2 x ∈ b∈B x¯bb ∈ I. Further, xx ∈ I = {0}. On the other hand, the coefficient of 1
∗ P in the product xx is equal to b∈B λbb∗0 xb x¯b > 0, contradiction.
Since C-algebras are just RBAs with a degree map, it is immediate from The- orem 2.4.9 that C-algebras are semisimple algebras. As table algebras and the adjacency algebras of finite association schemes are C-algebras, these are also semisimple.
2.4.2 Representation theory of semisimple algebras
Definition 2.4.10. Let K be a field. Let A be a K-algebra. An algebra represen- tation is a K-algebra homomorphism X : A → Mn(K).
Definition 2.4.11. Let A be an algebra with basis B = {b0, b1,..., bd}. Let {λi jk :
0 ≤ i, j, k ≤ d} be the structure constants relative to the basis B. The representa-
21 tion of A defined by the C-linear extension of
bi 7→ (λi jk)k, j is called the left regular representation of A.
In the next example we give the left regular representation of a C-algebra.
Example 2.4.12 (Example 4.3 [29]). Let (A, B, δ) be the integral table algebra whose distinguished basis B = {1 := b0, b1, b2} satisfies
2 b1 = 2b0 + b1,
2 b2 = 25b0 + 25b1 + 22b2,
b1b2 = b2b1 = 2b2. The images of the basis elements in the left regular representation of A are: 1 0 0 0 2 0 0 0 25 b0 = 0 1 0 , b1 = 1 1 0 , b2 = 0 0 25 . 0 0 1 0 0 2 1 2 22
The subalgebra of M3(C) generated by these basis matrices is isomorphic to A.
Note that the entries of these basis matrices are precisely the structure constants relative to the original basis B.
Definition 2.4.13. An algebra representation X of A is irreducible if its image
X(A) is a simple algebra.
22 Definition 2.4.14. Let A be an algebra over a field K. Two representations X1, X2 :
A −→ Mn(K) are equivalent if there is an invertible matrix P ∈ Mn(K) such that
−1 X1(x) = PX2(x)P , for all x ∈ A.
Definition 2.4.15. Let X : A −→ Mn(K) be a representation of A. The map
χ : A −→ K given by χ(x) = tr(X(x)), for all x ∈ A is called the character of X.
Also we say that the representation X affords the character χ, and n is the degree of the representation. Obviously, χ(1) = n, where 1 is the unity element of A.
Definition 2.4.16. Let A be an algebra over a field K. Let X be a representation of
A. If X is an irreducible representation affording the character χ, then χ is called an irreducible character. The set of all irreducible characters of A is denoted by
Irr(A).
Theorem 2.4.17. The equivalence classes of irreducible representations of a semisim- ple algebra are distinguished by the distinct irreducible characters.
Proof. Let A be a semisimple algebra over a field K. Let X1 and X2 be equivalent representations of A of degree n affording the characters χ1 and χ2, respectively.
−1 Then there exists an invertible matrix P ∈ Mn(K) such that X1(x) = PX2(x)P ,
−1 −1 for all x ∈ A. Since tr(PX2(x)P ) = tr(X2(x)PP ) = tr(X2(x)) we have that
χ1(x) = χ2(x), for all x ∈ A. Conversely, let X1 and X2 be two non-equivalent irre-
23 ducible representations affording the same character χ. Therefore X1 and X2 cor- respond to two different simple A-modules. By [10, Theorem (3.41)], coordinate functions for both representations must be linearly independent. For any non-zero x ∈ A, entries of the matrices X1(x) and X2(x) are linearly independent. Therefore, tr(X1(x)) − tr(X2(x)) , 0, but on the other hand, tr(X1(x)) = χ(x) = tr(X2(x)) for all x in A. So we reached a contradiction.
Let A be a semisimple algebra over a field K and let χ1, χ2, . . . , χm be the ir- reducible characters of A. If X1 and X2 are representations of A with characters
χ1 and χ2, respectively, then X1 ⊗ X2 affords χ1χ2. Since the product of two char- acters is a character, C[Irr(A)] is known as a ring of virtual characters. A virtual character is a character exactly when it lies in N[Irr(A)].
2.4.3 Representation theory of groups
Definition 2.4.18. Let G be a finite group. A representation of G is an algebra representation of its complex group algebra CG; i.e. an algebra homomorphism
X : CG −→ Mn(C).
P P If u ∈ ugg ∈ CG then X(u) = ugX(g). The characters of representations
24 of CG are referred to as the characters of the group G. Especially important are the characters of irreducible representations of CG, which carry a great deal of information about the group G. The set of distinct irreducible characters of the group G is denoted by Irr(G).
The character table of a group of G is an array, in which the rows are indexed by the distinct irreducible characters starting with the principal character, the character that affords the trivial representation, and the columns by the conjugacy classes of G, starting with the class consisting of the identity element. The value at the (i, j)-position in the character table is the value of i-th character at j-th conjugacy class.
Example 2.4.19. Let G = S 3 = {1, (12), (13), (23), (123), (321)} be the symmetric group on 3 elements. The irreducible representations of G are: the trivial char- acter, the alternating character, and the 2-dimensional representation of S 3 for which 0 1 0 −1 7→ 7→ (1, 2) , (1, 2, 3) 1 0 1 −1 This defines an irreducible degree 2 representation because its image on G is isomorphic to S 3 and the vector space the 6 matrices generate is 4-dimensional.
The character table for symmetric group S 3 is:
25 C1 C2 C3
χ1 1 1 1
χ2 1 −1 1
χ3 2 0 −1
Since C-algebras and adjacency algebras of association schemes are semisim- ple algebras with distinguished bases, they also have a character theory.
We finish this section with a proof of Kronecker’s theorem; cf. [36, Theorem
4.5.4], which will be needed later on in Chapter 5.
Theorem 2.4.20 (Kronecker). Let α , 0 be an algebraic integer. If α is not a root of unity, then at least one number Galois conjugate to α has absolute value strictly grater than 1.
Proof. Let {α1, . . . , αn} be the set of all numbers Galois conjugate to α. Suppose on the contrary that |αi| ≤ 1, i = 1,..., n, and for each k ≥ 0 consider the polynomial
k k n n−1 fk(x) = (x − α1) ··· (x − αn) = x + ak,n−1 x + ··· + ak,0.
Since α is an algebraic integer, it follows that ak,n−1,,..., ak,0 ∈ Z. The con-
n n dition |αi| ≤ 1 for i = 1,..., n imply that |ak,s| ≤ Cs, where Cs are the binomial coefficients. Therefore the coefficients of the polynomials f1, f2,... assume only
26 finitely many values, and hence, among these polynomials, there are only finitely many distinct ones. But then the set of roots of these polynomials is also finite, and all the numbers α, α2, α3,... are in this set. Therefore
αi = α j for some i, j ∈ N and i , j.
Since α , 0, it follows that αi− j = 1.
27 Chapter 3
Basic Tools
Let ZG be the integral group ring of a finite group G. In the first section of this chapter we shall present some results that establish a relationship between torsion units of ZG and partial augmentations. In Section 3.2 we introduce the
Luthar-Passi method, a character-theoretic method that can be used to verify the
ZC for the integral group ring of a specific finite group G. We will demonstrate the Luthar-Passi method by using it to prove the ZC for the integral group ring of the alternating group A4. In the last section of the chapter we develop idempotent formulas and orthogonality relations for C-algebras and association schemes that will be needed for our discussions in Chapters 4 and 5.
28 3.1 Torsion units of ZG and partial augmentations
First, we will record two foundational results for later reference concerning the torsion units of ZG. The next lemma of Berman-Higman shows that if e, the identity of G, lies in the support of a torsion unit of ZG then the unit has to be a trivial unit.
P k Lemma 3.1.1 (Corollary (1.3) [33]). Let u = ugg ∈ ZG. If u = e and ue , 0 then u = uee.
Next we have a Lagrange-type theorem for ZG.
Theorem 3.1.2 (Proposition (1.9) [32]). Let ZG be the integral group ring of a
finite group G. Suppose u ∈ ZG is a normalized torsion unit of order k. Then k is a divisor of the order of the group G.
The following lemma establishes the relation between the order of an element of the support of a torsion unit u of ZG and the partial augmentation of u with respect to that element. This result is deep, it depends on Weiss’ theory of p- permutation lattices.
Lemma 3.1.3 (Theorem 2.7 [28]). Let u be a normalized torsion unit of ZG with order k. Let g ∈ G and let p be a prime. If p divides the order of the element g but not the order of u, then εg(u) = 0.
29 3.2 Luthar-Passi method
The first conjecture of Zassenhaus (ZC) states that every element of V(ZG)tor is rationally conjugate to an element of group G, where G is a finite group. That is, if u ∈ V(ZG)tor, then there exists b ∈ U(QG) and g ∈ G such that u = b−1gb. One computational method for verifying the ZC is the Luthar-Passi method (see [27]).
In this section we will explain and demonstrate how this method works.
Let K(G) be a complete set of representatives for the conjugacy classes of G.
A normalized torsion unit u of QG has trivial partial augmentations if there is a unique x ∈ K(G) for which εx(u) = 1 and εg(u) = 0 for all other g ∈ K(G).
The idea of the Luthar-Passi method for verifying ZC for a group G is based on showing that for all k > 1 dividing |G|, the trivial partial augmentations are the only possible integral solutions of the Luthar-Passi equations for units of order k.
k Let U ∈ Mn(C) be a matrix with U = 1, k > 1. Then U is a diagonalizable matrix whose eigenvalues are the k-th roots of unity. Let ζk be a k-th root of unity.
` Let µ` be the multiplicity of ζk as an eigenvalue of U. Then
r r r(k−1) Tr(U ) = µ01 + µ1ζk + ... + µk−1ζk , ∀1 ≤ r ≤ k.
We get a system of k equations in k unknowns that can be solved by inverting the
30 coefficient matrix, which is of Vandermonde type. The solution is:
1 Xk µ = Tr(Ur)ζ−`r ` = 0, 1,... k − 1. ` k k r=1
Let u be a unit in ZG, uk = 1, k ≥ 1. Let χ be any character of G of degree n and
` let X be the corresponding representation. The multiplicity µ`(u, χ) of ζk as an eigenvalue of X(u) is given by
1 Xk µ (u, χ) = χ(ur)ζ−`r; ` = 0, 1,..., k − 1. (3.1) ` k k r=1
We will refer to these as the Luthar-Passi equations. Collecting together those r which have the same gcd with k we have
1 X X 0 0 µ (u, χ) = χ(udr )ζ−d`r . ` k k d|k 0 k r mod d
k d d d k Since (u ) = 1, χ(u ) is a sum of n ( d )-th roots ξ1, ξ2, . . . , ξn of unity. Therefore
k for (r, d ) = 1,
dr r r d σr χ(u ) = ξ1 + ··· + ξn = (χ(u )) ,
d dr d where σr is the automorphism ζk 7→ ζk of Q(ζk ). It follows that
1 X d −d` µ (u, χ) = Tr d (χ(u )ζ ); ` = 0, 1,..., k − 1. ` k Q(ζ )/Q k d|k
By the Luthar-Passi equations from equation (3.1), we have
r r r(k−1) χ(u ) = µ0(u, χ)1 + µ1(u, χ)ζ + ··· + µk−1(u, χ)ζ , ∀ 1 ≤ r ≤ k, ∀ χ ∈ Irr(G).
31 The equations corresponding to a fixed χ can be written in matrix form as
[χ]1×k = [µ]1×k · Fk×k,
2 3 k where [χ]1×k = [χ(u), χ(u ), χ(u ), . . . , χ(u )] is the χ-vector,
[µ]1×k = [µ0(u, χ), µ1(u, χ), µ2(u, χ), . . . , µk−1(u, χ)]
is the µ-vector and Fk×k is the Fourier matrix 1 1 1 ... 1 2 3 ζk ζk ζk ... 1 2 4 6 Fk×k = ζ ζ ζ ... 1 k k k . . . . . . . . . . k−1 2(k−1) 3(k−1) ζk ζ ζ ... 1 The µ-vector is a vector whose entries are nonnegative integers whose sum is n, the degree of the character. The χ-vector is a matrix whose entries are linear polynomial expressions in the partial augmentations. We can convert [χ]1×k =
[µ]1×k · Fk×k into a set of linear equations in the partial augmentations and we can solve these equations. A solution to the system of Luthar-Passi equations is a list of integer partial augmentations such that for all χ ∈ Irr(G), there is a choice of
µ-vector [µ`(u, χ)]` for which [χ] = [µ] · F.
Example 3.2.1. We will demonstrate the Luthar-Passi method by using it to verify the ZC for ZA4 for torsion units of order 3. The character table of A4 (see [9, §32])
32 is :
g1 g2 g3 g4
χ1 1 1 1 1
2 χ2 1 1 ζ3 ζ3 2 χ3 1 1 ζ3 ζ3
χ4 3 −1 0 0
The orders of conjugacy class representatives g1, g2, g3, g4 (in order) are 1, 2, 3, P and 3. Let u = ugg be a normalized torsion unit in ZA4 of order 3, and let
εi = εgi (u), 1 ≤ i ≤ 4,
be the partial augmentations of u. We have for any character χ of A4 of degree n,
χ(u) = ε1χ(g1) + ε2χ(g2) + ε3χ(g3) + ε4χ(g4).
Since the order of u is 3, by Lemma 3.1.1, ε1 = 0, and by Lemma 3.1.3, ε2 = 0.
Therefore, χ(u) = ε3χ(g3) + ε4χ(g4).
Now we generate the Luthar-Passi equations. For ` = 0, 1, 2, we have
1 P d −d` µ`(u, χ) = 3 TrQ(ζd)/Q(χ(u )ζ3 ) d|3 3
1 −` 3 −3` = [TrQ(ζ )/Q(χ(u)ζ ) + Tr 3 (χ(u )ζ )] 3 3 3 Q(ζ3 )/Q 3
1 −` σ2 −2` = 3 [χ(u)ζ3 + χ(u) ζ3 + χ(1)].
1 Therefore µ (u, χ) = [χ(u)ζ−` + χ(u)σ2 ζ−2`) + χ(1)], for ` = 0, 1, 2. ` 3 3 3
33 ` We know the µ`(u, χ) for 0 ≤ ` ≤ 2 and χ ∈ Irr(G) are the multiplicities of ζ3 as an eigenvalue of X(u), where X is the representation affording the character χ. So all the multiplicities are nonnegative integers and their sum is χ(1), the degree of the representation X.
Now for χ = χ1,
1 1 µ (u, χ ) = [(ε + ε ) + (ε + ε ) + 1] = [2ε + 2ε + 1], 0 1 3 3 4 3 4 3 3 4
1 1 µ (u, χ ) = [(ε + ε )ζ−1 + (ε + ε )ζ−2 + 1] = [−ε − ε + 1], 1 1 3 3 4 3 3 4 3 3 3 4 1 1 µ (u, χ ) = [(ε + ε )ζ−2 + (ε + ε )ζ−1 + 1] = [−ε − ε + 1]. 2 1 3 3 4 3 3 4 3 3 3 4
Since partial augmentations are integers, the only possible µ-vector for χ1 is
[1, 0, 0]. Therefore ε3 + ε4 = 1.
For χ = χ2,
1 1 µ (u, χ ) = [(ε ζ2 + ε ζ ) + (ε ζ + ε ζ2) + 1] = [−ε − ε + 1], 0 2 3 3 3 4 3 3 3 4 3 3 3 4
1 1 µ (u, χ ) = [(ε ζ2 + ε ζ )ζ−1 + (ε ζ + ε ζ2)ζ−2 + 1] = [−ε + 2ε + 1], 1 2 3 3 3 4 3 3 3 3 4 3 3 3 3 4 1 1 µ (u, χ ) = [(ε ζ2 + ε ζ )ζ−2 + (ε ζ + ε ζ2)ζ−1 + 1] = [2ε − ε + 1]. 2 2 3 3 3 4 3 3 3 3 4 3 3 3 3 4
There are three possible µ-vectors for χ2, [1, 0, 0], [0, 1, 0] and [0, 0, 1]. The µ- vector [1, 0, 0] produces no compatible solution. The other two µ-vectors [0, 1, 0] and [0, 0, 1] produce ε3 = 1, ε4 = 0 and ε3 = 0, ε4 = 1 respectively.
34 For χ = χ3,
1 1 µ (u, χ ) = [(ε ζ + ε ζ2) + (ε ζ2 + ε ζ ) + 1] = [−ε − ε + 1]. 0 3 3 3 3 4 3 3 3 4 3 3 3 4 1 1 µ (u, χ ) = [(ε ζ + ε ζ2)ζ−1 + (ε ζ2 + ε ζ )ζ−2 + 1] = [2ε − ε + 1]. 1 3 3 3 3 4 3 3 3 3 4 3 3 3 3 4 1 1 µ (u, χ ) = [(ε ζ + ε ζ2)ζ−2 + (ε ζ2 + ε ζ )ζ−1 + 1] = [−ε + 2ε + 1]. 2 3 3 3 3 4 3 3 3 3 4 3 3 3 3 4
Also for χ3, there are three possible µ-vectors, [1, 0, 0], [0, 1, 0] and [0, 0, 1]. A- gain, the µ-vector [1, 0, 0] produces no compatible solution. The other two µ- vectors [0, 1, 0] and [0, 0, 1] are consistent with ε3 = 0, ε4 = 1 and ε3 = 1, ε4 = 0 respectively.
For χ = χ4,
1 1 1 µ (u, χ ) = [3] = 1, µ (u, χ ) = [3] = 1, µ (u, χ ) = [3] = 1. 0 4 3 1 4 3 2 4 3
The only possible µ-vector for χ4 is [1, 1, 1], but it does not put any further restric- tion on the values of ε3 and ε4.
Therefore, there are only two solutions to the Luthar-Passi equations for or- der 3, and both correspond to trivial units of V(ZA4). The spectral information corresponding to (ε3, ε4) = (1, 0) is [[1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1]], and to
(ε3, ε4) = (0, 1) it is [[1, 0, 0], [0, 0, 1], [0, 1, 0], [1, 1, 1]]. The corresponding spec- tral information proves to be very useful in the new algorithm for constructing torsion units.
35 3.3 The standard feasible trace of a C-algebra
Definition 3.3.1. Let (A, B, δ) be a C-algebra with standardized basis B. A “fea- sible trace” is a function ρ : A → C that satisfies the condition ρ(uv) = ρ(vu) for all u, v ∈ A.
Definition 3.3.2. Let (A, B, δ) be a C-algebra with standardized basis B. The
“standard feasible trace” is the function ρ : A → C defined by
X + ρ( uibi) = u0δ(B ). i
This a feasible trace because ρ(uv) = ρ(vu) for all u, v ∈ A.
Proposition 3.3.3 (Proposition 5.1 [13]). If ρ is a feasible trace for a finite dimen- sional C-algebra A, then ρ ∈ C[Irr(A)].
P By Proposition 3.3.3, there are complex numbers mχ such that ρ = χ mχχ, where the sum runs over χ ∈ Irr(A). When these multiplicities mχ are nonnegative integers for every χ ∈ Irr(A), ρ is the character of a representation. In this case we say that ρ is the standard character and any representation affording ρ is called a standard representation.
When we associate the relations of a finite association scheme (X, S ) of or- der n with their adjacency matrices, it gives a natural inclusion CS ,→ Mn(C).
36 This representation of CS is called the standard representation of the association scheme (X, S ). Its character ρ satisfies ρ(σ0) = n = |X| and ρ(σs) = 0 for all s ∈ S with s , 1. Clearly the degree of the standard representation is n = |X|.
Now, let (A, B, δ) be a C-algebra, with B = {b0 = 1A, b1,..., bd}. By Theorem
2.4.9, A is semisimple, but C is algebraically closed. Therefore (A, B, δ) is a split semisimple. So
m A ⊕i=1Ai.
Each Ai is isomorphic to a full matrix algebra over C, of degree ni, say, so
m X 2 ni = r (= d + 1). i=1
Because A is semisimple, any homomorphic image of A is isomorphic to the complement of its kernel. Therefore, each simple component of A is isomorphic to the image of an irreducible representation of A. The distinct simple components of A are in one-to-one correspondence with the equivalence classes of irreducible representations of A, and hence with the irreducible characters of A. We relabel the simple components to have
M A = Aχ. χ∈Irr(A) P The identity 1 of A decomposes as 1 = χ eχ, with eχ ∈ Aχ, and it is clear that eχ is a central idempotent of A that is the identity element for the algebra Aχ. The
37 set {eχ : χ ∈ Irr(A)} is a complete set of pairwise orthogonal idempotents of Z(A).
We call these the centrally primitive idempotents of A.
Our next theorem gives the character formula that expresses the centrally prim- itive idempotents of a C-algebra (A, B, δ) in terms of the distinguished basis B.
Theorem 3.3.4. Let (A, B, δ) be a C-algebra with distinguished basis B = {b0, b1,
..., bd}. The centrally primitive idempotent eχ corresponding to χ ∈ Irr(A) can be expressed as: d m X χ(b ∗ ) e χ i b . χ = + i δ(B ) λ ∗ i=0 ii 0
Proof. Let X1,..., Xm be the inequivalent absolutely irreducible representations of A, and let χ1, . . . , χm be the corresponding characters. The notation will be
chosen so that Xs corresponds to eχs in the sense so that
Xs(eχt ) = δstXs(1), 1 ≤ s, t ≤ m.
This implies that Xs has degree χs(1) = χs(eχs ) = ns.
χ Let i j for i, j ∈ {1, . . . , χ(b0)} and χ ∈ Irr(A) be a full set of matrix units of
A. Then this set forms a basis for the split semisimple algebra A, and so we can choose our representations to satisfy
χs χs Xχt (i j ) = δstEi j ,
38 χs where Ei j is the elementary matrix with entry δi j in the image of Xχs . These
χs representations define C-linear coordinate functions ai j : A → C for which
χs χs χs Xχs (x) = (ai j (x))i j for all x ∈ A, χt ∈ IrrA. Then χt(i j ) = δstδi j, so ρ(i j ) = δi jmχs
χs χt χs and i j kl = δstδ jkil .
χ Since {i j : χ ∈ Irr(A), 1 ≤ i, j ≤ χ(b0)} and {bi : 0 ≤ i ≤ d} are bases for the algebra A, for some αk ∈ C, we have
d χ X i j = αkbk. k=0 d d d χ br∗ X br∗ 1 X X =⇒ i j = αkbk = αkλkr∗lbl . λ ∗ λ ∗ λ ∗ rr 0 k=0 rr 0 rr 0 k=0 l=0 d d χ br∗ 1 X X + =⇒ ρ(i j ) = ρ( αkλkr∗lbl) = αrδ(B ). λ ∗ λ ∗ rr 0 rr 0 k=0 l=0 P Since ρ = ψ mψψ, where the sum runs over ψ ∈ Irr(A), we have
χ br∗ X χ br∗ χ br∗ ρ(i j ) = mψψ(i j ) = mχa ji( ). λ ∗ λ ∗ λ ∗ rr 0 ψ rr 0 rr 0 Therefore
m b ∗ α χ aχ r . r = + ji( ) δ(B ) λrr∗0 Hence d m X b ∗ χ χ aχ k b . i j = + ji( ) k δ(B ) λ ∗ k=0 kk 0 So these primitive idempotents are given by:
d χ m X a (bi∗ ) χ χ j j b . j j = + i δ(B ) λ ∗ i=0 ii 0
39 P χ P P χ These satisfy j j j = eχ and χ j j j = b0. Hence it follows that the centrally primitive idempotents of A are:
d m X χ(b ∗ ) e χ i b . χ = + i δ(B ) λ ∗ i=0 ii 0
Corollary 3.3.5. Let (A, B, δ) be a C-algebra with distinguished basis B = {b0, b1,
..., bd}. One full set of primitive idempotents for the simple component of A cor- responding to χ ∈ Irr(A) can be expressed as:
d χ m X a (bi∗ ) χ χ j j b , j j = + i δ(B ) λ ∗ i=0 ii 0
χ where the maps a j j are as defined in Theorem 3.3.4.
Any other full set of primitive orthogonal idempotents of A will be conjugate in A to this set. P The standard feasible trace ρ = χ mχχ of a C-algebra need not be a character.
Nevertheless, Blau has recently shown that in fact mχ > 0 for all χ ∈ Irr(A); cf.
[3, Proposition 1]. We shall reproduce this result.
Proposition 3.3.6. Let (A, B, δ) be a C-algebra with the distinguished basis B = P {b0, b1, b2,..., bd}. Let ρ = χ mχχ be its standard feasible trace, where the sum runs over χ ∈ Irr(A). Then mχ > 0 for all χ ∈ Irr(A).
40 P ∗ P + ∗ Proof. Let x ∈ xibi ∈ A. Then ρ(xx ) = xi x¯iλii∗0δ(B ) implies ρ(eχeχ) > 0.
∗ ∗ P Thus eχeχ , 0, so eχ = eχ and ρ(eχ) > 0. Since ρ(eχ) = mψψ(eχ) = mχχ(1), this implies mχ > 0.
The next corollary is immediate from the last part of the proof of above propo- sition.
Corollary 3.3.7. Let (A, B, δ) be a C-algebra with the distinguished basis B =
∗ {b0, b1, b2,..., bd}. If e is a centrally primitive idempotent then e = e.
We can often use the Theorem 3.3.4 to calculate multiplicities of irreducible characters in the standard feasible trace of a C-algebra.
Example 3.3.8. Let (A, B, δ) be a C-algebra, where A = M5(C) and let B =
{b0, b1, b2, b3}, where
b1b0 = b1,
2 b1 = 3b0 + b1 + b2,
b1b2 = b1 + 2b3,
b1b3 = 2b2 + b3. When we identify B with the image of its regular representation, we have 0 3 0 0 0 0 3 0 0 0 0 3 1 1 1 0 0 1 0 2 0 0 2 1 b1 = , b2 = , b3 = . 0 1 0 2 1 0 2 0 0 2 0 1 0 0 2 1 0 2 0 1 1 1 1 0
41 To calculate the character table for this algebra, it is suffices to simultaneously diagonalize the three matrices. Since the eigenvalues of matrix b1 are distinct, using standard linear algebra techniques we can find non-zero eigenvectors gen- erating each of its eigenspaces. These eigenvectors form the rows of an invertible matrix Q which diagonalizes b1. The matrix Q in this case is: 1 1 1 1 1 2/3 −1/3 −2/3 Q = . 1 −1/3 −1/3 1 1 −2/3 1/3 −2/3
Since the algebra is commutative it will also diagonalize b2 and b3. As the
−1 representations of A all have degree 1, the entries on the diagonal of Q biQ are precisely the character values of bi for each bi. Therefore, the character table of A is:
b0 b1 b2 b3
χ1 1 3 3 3
χ2 1 2 −1 −2
χ3 1 −1 −1 1
χ4 1 −2 3 −2
We note the first row is the degree map. Therefore, δ(b0) = 1, δ(b1) = 3 =
+ δ(b2) = δ(b3), the order δ(B ) = 10, and the rank of A is 4. By Theorem 3.3.4, we
42 know the idempotent corresponding to χ ∈ Irr(A) is:
d m X χ(b ∗ ) e χ i b . χ = + i δ(B ) λ ∗ i=0 ii 0
Now eχ1 = [mχ1 /10](b0 + b1 + b2 + b3), so
2 2 [(eχ1 ) ]0 = [(mχ1 ) /100](b0 + (3b0) + (3b0) + (3b0))0
2 = [(mχ1 ) /100](10b0)0
2 = (mχ1 ) /10.
2 Since (mχ1 ) /10 must be equal to mχ1 /10 and mχ1 > 0 so we get mχ1 = 1.
eχ2 = [mχ2 /10](b0 + (2/3)b1 − (1/3)b2 − (2/3)b3), so
2 2 [(eχ2 ) ]0 = [(mχ2 ) /100](b0 + (4/9)(3b0) + (1/9)(3b0) + (4/9)(3b0))0
2 = [(mχ2 ) /100](4b0)0
2 = (mχ2 ) /25.
2 Since (mχ2 ) /25 must be equal to mχ2 /10 and mχ2 > 0 so we get mχ2 = 5/2.
eχ3 = [mχ3 /10](b0 − (1/3)b1 − (1/3)b2 + (1/3)b3), so
2 2 [(eχ3 ) ]0 = [(mχ3 ) /100](b0 + (1/9)(3b0) + (1/9)(3b0) + (1/9)(3b0))0
2 = [(mχ3 ) /100](2b0)0
2 = (mχ3 ) /50.
2 Since (mχ3 ) /50 must be equal to mχ3 /10 and mχ3 > 0 so we get mχ3 = 5.
43 eχ4 = [mχ4 /10](b0 − (2/3)b1 + (3/3)b2 − (2/3)b3), so
2 2 [(eχ4 ) ]0 = [(mχ4 ) /100](b0 + (4/9)(3b0) + (9/9)(3b0) + (4/9)(3b0))0
2 20 = [(mχ4 ) /100]( 3 b0)0
2 = (mχ4 ) /15.
2 Since (mχ4 ) /15 must be equal to mχ4 /10 and mχ4 > 0 so we get mχ4 = 3/2.
Hence the multiplicities are mχ1 = 1, mχ2 = 5/2, mχ3 = 5, and mχ4 = 3/2.
As these multiplicities are not all integers, this C-algebra has no standard char- acter.
44 Chapter 4
Normalized Torsion Units of
Integral Group Rings
In the first section of this chapter we prove that for a given group G if two torsion units of QG are conjugate in QG then they have equal partial augmentations. The converse holds with some restrictions. In the second and third sections we de- scribe the computer program we have developed to construct torsion units of CG with specific partial augmentations from the spectral information accompanying solutions to the Luthar-Passi equations. In the fourth section we prove a result about the partially central torsion units QG for finite group G.
45 4.1 Partial augmentations and rational conjugacy
Our first theorem examines the extent to which rational conjugacy of units in QG is determined by their partial augmentations.
Theorem 4.1.1. Let G be a finite group.
(i) If u, v ∈ QG, and u ∼ v in QG, then εx(u) = εx(v), for all x ∈ G.
(ii) Suppose u and v are torsion units in QG with the same order k > 1. Further
d d assume that u ∼ v for all divisors d of k with 1 < d ≤ k, and that εx(u) =
εx(v) for all x ∈ G. Then u ∼ v in QG.
Proof. (i). Suppose u ∼ v in QG. Then χ(u) = χ(v) for all χ ∈ Irr(G). Let K(G) be a complete set of representatives for the conjugacy classes of G. Then, for all
χ ∈ Irr(G), we have that
X X χ(u) = χ(v) =⇒ εx(u)χ(x) = εx(v)χ(x). x∈K(G) x∈K(G)
Since the character table of a finite group G is an invertible matrix, it follows that
εx(u) = εx(v), for all x ∈ G.
(ii). Let u1 be a torsion unit of QG with order k. Then u1 produces a specific list
(µ) = (µ`(u1, χ))`,χ of nonnegative integers that are associated with the collection
46 of all the Luthar-Passi equations
1 X d −`d µ`(u1, χ) = TrQ(ζd)/Q(χ(u1)ξ ) k k d|k as ` runs through 0 to k − 1 and χ runs over Irr(G) (see Section 3.2). Our as- sumptions that χ(ud) = χ(vd) for d > 1 dividing k and that u and v have the same partial augmentations mean the right hand side of the collection of Luthar-Passi equations for the unit u matches the right hand side of the Luthar-Passi equations for the unit v. Therefore, the left hand sides also match, so µ`(u, χ) = µ`(v, χ) for all ` and χ.
The k nonnegative integers µ`(u1, χ), ` = 0,..., k − 1 for a fixed χ, each repre-
` sent the multiplicity of ζk as an eigenvalue of X(u1), for any irreducible represen- tation X with character χ. Observe that the full spectrum of X(u1) is determined by these multiplicities. When u1 has finite order, X(u1) is a matrix of finite order, and is therefore diagonalizable. If χ has degree n, then any matrix in GL(n, C) with this same spectrum will be conjugate to X(u1) in GL(n, C). In particular, since
µ`(u, χ) = µ`(v, χ) for all `, X(u) is conjugate to X(v) in GL(n, C). Since χ is an absolutely irreducible character, X(CG) = Mn(C), so in fact we have that X(u) and
X(v) are conjugate in X(CG).
Since this is the case for all irreducible characters χ of G, if we let R be the regular representation of G, then it must be the case that R(u) and R(v) are
47 conjugate in R(CG). Since R is a faithful representation of G, this implies that u and v are conjugate in CG.
Since u and v are torsion units of QG, it follows from [32, Lemma 37.5] that u and v are in fact conjugate in QG (see also Lemma 5.2.11 in the next chapter).
Some recent observations by Hertweck are particularly useful in improving performance of computer verifications of the ZC for small groups using the Luthar-
Passi method. We use two of these in particular. The first of these is Lemma 3.1.3.
From this it follows that εx(u) can be nonzero only when o(x) divides o(u). The second is that if G is a solvable group and u ∈ V(ZG)tor, then G has an element x for which o(x) = o(u) and εx(u) , 0 (see [22]). Furthermore, Lemma 3.1.1 ensures that the coefficient of the identity in any nontrivial torsion unit of ZG is always 0. Another property that is useful for dealing with some nontrivial solu- tions to the Luthar-Passi equations when k has prime power order is a special case of [8, Theorem 4.1]:
Proposition 4.1.2. Let u ∈ V(ZG)tor, and o(u) = pn. Then the following equations hold:
P (i) 0 ≡ ( ug) mod p, for 1 ≤ m < n, o(g)=pm P (ii) 1 ≡ ( ug) mod p. o(g)=pn
48 For further restrictions on partial augmentations see [21].
4.2 Computer implementation of the Luthar-Passi
method
We have written a computer program in GAP [12] that can be used to verify the ZC for the integral group ring of a small finite group G. The first part is a GAP imple- mentation of the Luthar-Passi method. It sets up the Luthar-Passi equations using the character table of the group, and uses a recursive Groebner basis technique to solve them. The output is a list of partial augmentations with their accompanying
µ-vectors. The second part constructs a unit with this list of partial augmentations by finding matrices in the image of each irreducible representation whose spec- trum matches the χ-component of the µ-vector. We continue to use the notation of Section 3.2.
Let K(G) be a complete set of representatives for the conjugacy classes of G.
A normalized torsion unit u of QG has trivial partial augmentations if there is a unique x ∈ K(G) for which εx(u) = 1 and εg(u) = 0 for all other g ∈ K(G). The idea of the Luthar-Passi method for verifying the ZC for G is based on showing that for all k > 1 dividing |G|, the trivial partial augmentations are the only possible
49 integral solutions of the Luthar-Passi equations for units of order k.
We shall explain our GAP program by examining the units of order k = 6 for the group G that is identified as SmallGroup(72,40) in the GAP library of small groups. The group G is a non-split central extension of the form (C3 × C3): D8.
Our notation for the character table and generators of G was produced using GAP
[12]. The character table for group G=SmallGroup(72,40) is:
1a 2a 2b 2c 3a 4a 6a 6b 3b
χ1 1 1 1 1 1 1 1 1 1
χ2 1 −1 −1 1 1 1 −1 −1 1
χ3 1 −1 1 1 1 −1 −1 1 1
χ4 1 1 −1 1 1 −1 1 −1 1
χ5 2 .. −2 2 ... 2
χ6 4 −2 .. 1 . 1 . −2
χ7 4 . −2 . −2 .. 1 1
χ8 4 . 2 . −2 .. −1 1
χ9 4 2 .. 1 . −1 . −2 where 2a, 2b, 2c, 3a, 3b, 4a, 6a, 6b are representatives of conjugacy classes (in or- der) C2, C3, C4, C5, C9, C6, C7, C8 of elements of order 2, 3, 4 and 6.
Step 1: Create the Fourier matrix F = F6×6.
50 Step 2: Create a list of indeterminates xi, one for each conjugacy class of the group. These will represent the various partial augmentations of the unit u in our calculations.
Step 3: Determine the list of cases for powers of u of smaller order. This tells us the conjugacy classes for the group elements ud when d divides k. For our example, we will have 6 cases:
3 2 Case 1 : u ∈ C2 and u ∈ C5.
3 2 Case 2 : u ∈ C2 and u ∈ C9.
3 2 Case 3 : u ∈ C3 and u ∈ C5.
3 2 Case 4 : u ∈ C3 and u ∈ C9.
3 2 Case 5 : u ∈ C4 and u ∈ C5.
3 2 Case 6 : u ∈ C4 and u ∈ C9.
We proceed to find all solutions to the Luthar-Passi equations of order k in each case.
Step 4: For each irreducible character χ of G we repeat the following steps.
Step 4a: Create the χ-vector using the character table of the group, the infor- mation about the particular case, and the list of indeterminates.
3 2 For example, the χ-vector for k = 6, χ5 and Case 1 : u ∈ C2 and u ∈ C5 is:
[2, −2 ∗ x4 + 2 ∗ x5 + 2 ∗ x9, 2, 0, 2, −2 ∗ x4 + 2 ∗ x5 + 2 ∗ x9],
51 where xi is the partial augmentation for the ith conjugacy class in the character table.
Step 4b: Create a list of possible µ vectors for the multiplicities of various powers of ζk that could occur for X(u) when X is a representation of G affording
χ.
In our example, since χ5 has degree 2 and all of its character values are ratio- nal, the possible µ-vectors are:
[2, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0], [1, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 1], [0, 0, 1, 0, 1, 0].
Step 4c, part (i): For each µ-vector in step 4b, find the list of polynomial equations determined by entries of the matrix equation χ ∗ F−1 − µ = 0.
Step 4c, part (ii): Combine the list of polynomial equations with each Groeb- ner basis carried forward from previous χ’s, and solve the homogeneous system formed by these equations by calculating its reduced Groebner basis. When the solution exists, record the reduced Groebner basis and associated µ-vector. Pass this information forward to the next character. All chains of valid choices of µ- vector are carried forward to the next character. Step 4 is concluded once all characters are exhausted.
3 2 We shall demonstrate the full procedure for Case 1 (u ∈ C2 and u ∈ C5) giv- ing the polynomials derived from the Luthar-Passi equations for each character
52 and calculating the Groebner basis at each stage. For χ1, since the unit is nor- malized the µ-vector will be [1, 0, 0, 0, 0, 0]. The six corresponding Luthar-Passi polynomials are:
1/3 ∗ x2 + 1/3 ∗ x3 + 1/3 ∗ x4 + 1/3 ∗ x5 + 1/3 ∗ x7 + 1/3 ∗ x8 + 1/3 ∗ x9 − 1/3,
1/6 ∗ x2 + 1/6 ∗ x3 + 1/6 ∗ x4 + 1/6 ∗ x5 + 1/6 ∗ x7 + 1/6 ∗ x8 + 1/6 ∗ x9 − 1/6,
−1/6 ∗ x2 − 1/6 ∗ x3 − 1/6 ∗ x4 − 1/6 ∗ x5 − 1/6 ∗ x7 − 1/6 ∗ x8 − 1/6 ∗ x9 + 1/6,
−1/3 ∗ x2 − 1/3 ∗ x3 − 1/3 ∗ x4 − 1/3 ∗ x5 − 1/3 ∗ x7 − 1/3 ∗ x8 − 1/3 ∗ x9 + 1/3,
−1/6 ∗ x2 − 1/6 ∗ x3 − 1/6 ∗ x4 − 1/6 ∗ x5 − 1/6 ∗ x7 − 1/6 ∗ x8 − 1/6 ∗ x9 + 1/6,
1/6 ∗ x2 + 1/6 ∗ x3 + 1/6 ∗ x4 + 1/6 ∗ x5 + 1/6 ∗ x7 + 1/6 ∗ x8 + 1/6 ∗ x9 − 1/6.
The reduced Groebner basis is: x2 + x3 + x4 + x5 + x7 + x8 + x9 − 1.
To save space and time we show the list of polynomials that occur in each case from now on without repeating scalar multiples.
For χ2 and µ-vector [0, 0, 0, 1, 0, 0] the Luthar-Passi polynomial is uniqe:
−1/3 ∗ x2 − 1/3 ∗ x3 + 1/3 ∗ x4 + 1/3 ∗ x5 − 1/3 ∗ x7 − 1/3 ∗ x8 + 1/3 ∗ x9 + 1/3.
The reduced Groebner basis from the above polynomial(s) and from the first reduced Groebner basis is: x4 + x5 + x9, x2 + x3 + x7 + x8 − 1.
If we choose the µ-vector [1, 0, 0, 0, 0, 0] instead of [0, 0, 0, 1, 0, 0] then we get the trivial ideal. Hence we cannot use this µ-vector for this case.
53 For χ3 and µ-vector [0, 0, 0, 1, 0, 0] the Luthar-Passi polynomial is:
−1/3 ∗ x2 + 1/3 ∗ x3 + 1/3 ∗ x4 + 1/3 ∗ x5 − 1/3 ∗ x7 + 1/3 ∗ x8 + 1/3 ∗ x9 + 1/3.
The reduced Groebner basis for χ3 corresponding to this sequence of µ-vector choices is: x4 + x5 + x9, x3 + x8, x2 + x7 − 1.
For χ4 and µ-vector [1, 0, 0, 0, 0, 0] the Luthar-Passi polynomial is:
1/3 ∗ x2 − 1/3 ∗ x3 + 1/3 ∗ x4 + 1/3 ∗ x5 + 1/3 ∗ x7 − 1/3 ∗ x8 + 1/3 ∗ x9 − 1/3.
The reduced Groebner basis after χ4 corresponding to this sequence of µ-vector choices is: x4 +x5 +x9, x3 +x8, x2 +x7 −1. Although the polynomials corresponding to χ4 for this choice of µ-vector do not change the Groebner basis, we still need to record and carry forward the µ-vector to the next stage.
For χ5 and µ-vector [1, 0, 0, 1, 0, 0] the Luthar-Passi polynomial is: −2/3∗ x4 +
2/3∗ x5 +2/3∗ x9 and the reduced Groebner basis is: x5 + x9, x4, x3 + x8, x2 + x7 −1.
For χ6 and µ-vector [1, 1, 0, 1, 0, 1] the Luthar-Passi polynomial is: −2/3∗ x2 +
1/3 ∗ x5 + 1/3 ∗ x7 − 2/3 ∗ x9 − 1/3 and the reduced Groebner basis is: x7 − x9 − 1, x5 + x9, x4, x3 + x8, x2 + x9.
For χ7 and µ-vector [0, 1, 1, 0, 1, 1] the Luthar-Passi polynomial is: −2/3∗ x3 −
2/3 ∗ x5 + 1/3 ∗ x8 + 1/3 ∗ x9 and the reduced Groebner basis is: x8 + x9, x7 − x9 −
1, x5 + x9, x4, x3 − x9, x2 + x9..
For χ8 and µ-vector [0, 1, 1, 0, 1, 1] the Luthar-Passi polynomial is: 2/3 ∗ x3 −
54 2/3 ∗ x5 − 1/3 ∗ x8 + 1/3 ∗ x9 and the reduced Groebner basis is: x9, x8, x7 − 1, x5, x4, x3, x2.
For χ9 and µ-vector [1, 0, 1, 1, 1, 0] the Luthar-Passi polynomial is 2/3 ∗ x2 +
1/3∗ x5 −1/3∗ x7 −2/3∗ x9 +1/3 and the reduced Groebner basis is: x9, x8, x7 −1, x5, x4, x3, x2.
Therefore corresponding to Case 1, the chain of µ-vector choices [1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0],
[1, 0, 0, 1, 0, 0], [1, 1, 0, 1, 0, 1], [0, 1, 1, 0, 1, 1], [0, 1, 1, 0, 1, 1], [1, 0, 1, 1, 1, 0]. results in the Groebner basis x9, x8, x7 − 1, x5, x4, x3, x2. This is the Groebner basis for the ideal whose variety consists of the single point (x2, x3, x4, x5, x7, x8, x9) =
(0, 0, 0, 0, 1, 0, 0). In terms of partial augmentations, this means there is only one nonzero partial augmentation in this solution, which is 1 for the 7-th conjugacy class.
4.3 Computer construction of torsion units with pre-
scribed partial augmentations
The GAP program for constructing units that we have developed is explained here using the group G identified as SmallGroup(48,30) in the GAP library. The
55 group G is a non-split central extension of the form C2 : S 4. Our notation for the character table and generators of G was produced using GAP [12]. There are 8 columns in GAP’s character table of G corresponding to elements of order dividing 4: 1a 2a 2b 2c 4a 4b 4c 4d
χ1 1 1 1 1 1 1 1 1
χ2 1 1 1 1 −1 −1 −1 −1
χ3 1 −1 1 −1 −i i −i i
χ4 1 −1 1 −1 i −i i −i
χ5 2 −2 2 −2 0 0 0 0
χ6 2 2 2 2 0 0 0 0
χ7 3 3 −1 −1 −1 −1 1 1
χ8 3 3 −1 −1 1 1 −1 −1
χ9 3 −3 −1 1 −i i i −i
χ10 3 −3 −1 1 i −i −i i
The generators of G in its polycyclic presentation in GAP are: f1 (of order 4), f2
(a central element of order 2), f3 (of order 3), f4 and f5 (both of order 2). The representatives of conjugacy classes indicated by 2a, 2b, 2c, 4a, 4b, 4c, and 4d (in order) are f2, f4, f2 f4, f1, f1 f2, f1 f4, and f1 f2 f4.
56 Let X1,..., Xh be a complete list of representatives of the non-equivalent ir- reducible representations of G. Let χ1, . . . , χh be the irreducible characters of G, and let ei be the centrally primitive idempotent of CG corresponding to χi for i = 1,..., h.
Since our algorithm records the list of µ-vectors for each irreducible character
χ of G associated with a solution to the Luthar-Passi equations, we know the spectrum of Xi(u) for any torsion unit of CG that has the partial augmentations given by that solution. If we can construct a ui ∈ CG for which the spectrum of P Xi(ui) matches the desired spectrum of Xi(u) for each i, then the unit u = i uiei will have the desired spectrum, and hence the desired partial augmentations.
For each irreducible character χi of G, the steps in the procedure are:
Step 1: Construct a representation Xi of G affording χi. For this purpose we use the GAP function IrreducibleRepresentationsDixon.
Step 2: Check if there is a gk ∈ K(G) for which the spectrum of ±Xi(gk) matches that of Xi(u). If so let ui = ±gk.
For example, for the non-trivial partial augmentations [4c, 4a] = [x7, x2] =
[2, −1] of a unit u ∈ ZG of order 4, the µ-vectors corresponding to the irreducible
57 characters of G (in GAP’s ordering) are:
[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [0, 1, 0, 0], [0, 1, 0, 1],
[1, 0, 1, 0], [3, 0, 0, 0], [0, 0, 3, 0], [0, 3, 0, 0], [0, 0, 0, 3].
Let us consider the µ-vector [0, 0, 0, 1] corresponding to irreducible character χ3.
The spectrum corresponding to this µ-vector is [ζ4], so X3(u) = [ζ4]. Since
X3( f1) = ζ4, we can set u3 = f1.
Now consider the µ-vector [0, 3, 0, 0] corresponding to the the irreducible char- acter χ9. The corresponding spectrum is {ζ4, ζ4, ζ4}, that is, ζ4 0 0 X9(u) = 0 ζ4 0 . 0 0 ζ4
But there is no element in group G whose image under X9 is ζ4I, so we have to proceed to step 3.
Step 3: Suppose ui has not been found in Step 2.
Step 3a: Find the field of realization F of the representation Xi. The repre- sentations in GAP are given using their image on the generators of G. So we only need to calculate the field extension of Q generated by the entries of a few matrices.
The field of realization depends on the given representation. For our X9, F is the Gaussian rational field, whereas for X5 it is Q(ζ3).
58 Step 3b: Construct a candidate matrix A of order k from the spectrum of Xi(u).
Use either a diagonal matrix if ζk ∈ F and the factored companion matrix for the spectrum list otherwise. The diagonal matrices are determined directly from the i-th component of the µ-vector. The factored companion matrices are the direct sum of companion matrices corresponding to the irreducible factors of the characteristic polynomial of the diagonal matrix. We don’t use the companion matrix of the characteristic polynomial itself because it may not have order k.
The field of realization for our X is Q(ζ ), but the spectrum from the µ-vector 5 3 0 −1 [0, 1, 0, 1] is {ζ , ζ3}. So we shall use the factored companion matrix . 4 4 1 0
Since the field of realization for X9 is the Gaussian rational field and the spectrum from the µ-vector [0, 3, 0, 0] is {ζ4, ζ4, ζ4}, ζ4 0 0 A = 0 ζ4 0 0 0 ζ4 is the required matrix.
Step 3c: Find a subset B of G for which {Xi(b): b ∈ B} is a Z-basis of Xi(ZG).
First a subset B of G for which Xi(B) is a Q basis of Xi(QG) is found using a recursive algorithm which adds elements of G to the set B one at a time. If X(g) lies in the Q-span of X(B), then g is not added to B, and otherwise, it is added.
Then a second recursive algorithm optimizes the choice of elements of G used in
59 B so that every X(g) is a Z-linear combination of the elements {X(b): b ∈ B}.
Let f1, f2, f3, f4, f5 be the generators (as above) of G. The Z-basis of X9(ZG) is generated by the images of the subset
2 2 B = {Identity, f1, f3, f4, f5, f1 f3, f1 f4, f1 f5, f3 , f3 f4, f3 f5, f1 f3 ,
2 2 2 2 f1 f3 f4, f1 f3 f5, f3 f4, f3 f5, f1 f3 f4, f1 f3 f5}.
The images of elements of B under X9 are (in order):
1 0 0 −ζ4 0 −ζ4 0 1 0 −1 1 −1 0 1 0 , 0 −ζ4 0 , 1 0 1 , 0 0 −1 , 0 0 1 0 0 ζ4 1 −1 0 0 −1 0
−1 0 0 −ζ4 0 0 ζ4 0 ζ4 ζ4 −ζ4 0 0 0 1 , −ζ4 0 −ζ4 , 0 0 ζ4 , 0 0 −ζ4 , 0 1 0 ζ4 −ζ4 0 0 −ζ4 0 0 ζ4 0
1 0 1 0 0 −1 0 0 1 0 −ζ4 0 1 0 0 , −1 0 −1 , −1 1 0 , −ζ4 0 0 , −1 1 −1 −1 1 0 −1 0 −1 −ζ4 ζ4 −ζ4
60 ζ4 −ζ4 ζ4 ζ4 0 0 −1 0 −1 −1 1 0 ζ4 0 ζ4 , ζ4 −ζ4 0 , −1 1 −1 , −1 0 0 , −ζ4 ζ4 0 −ζ4 0 −ζ4 1 0 0 1 −1 1
0 0 ζ4 0 0 −ζ4 ζ4 −ζ4 ζ4 , ζ4 0 0 . ζ4 0 0 ζ4 −ζ4 ζ4 P Step 3d: Write A = b∈B abXi(b), ab ∈ F. The coefficients ab are found by sim- ply writing A as a linear combination of the elements X(b) for b in B. This is ac- complished in GAP after designating the specific set B as a basis using one of the options for the Basis command, and then simply asking for Coefficients(B, A)
(see [12]). More meaningful results are sometimes achieved in this step by using a conjugate of A under a permutation matrix.
In our example, the expression for A our implementation produces for X9 is
2 A = (−1/2)X9( f1 f3) + (1/2)X9( f1 f4) + (1/2)X9( f1 f5) + (−1/2)X9( f1 f3 )
2 + (−1/2)X9( f1 f3 f5) + (−1/2)X9( f1 f3 f4). Since A is a scalar matrix this time, it is reasonable, and desirable for consid- erations we will encounter in Section 4.4, to expect a result whose coefficients are constant on conjugacy classes. After manually modifying our choice of the basis
B to favour elements of the class C4a, we obtain
1 A = − X (Cc ), 2 9 4a 61 where Cc4a denotes the sum of elements of the conjugacy class C4a.
It is possible that the GAP command Coefficients(B, A) gives no rational solution. At worst this requires us to switch to an F-basis since there will always be a solution with coefficients in F. P Step 3e: Set ui = b∈B abb. At this step it uses the coefficients that are calcu- lated at step 3d and expresses the ui as a linear combination of elements of B. For example the u9 corresponding to X9 is:
2 u9 = (−1/2) f1 f3 + (1/2) f1 f4 + (1/2) f1 f5 + (−1/2) f1 f3
2 + (−1/2) f1 f3 f5 + (−1/2) f1 f3 f4.
− 1 Alternatively, u9 can be chosen to be 2Cc4a.
We now give the results of carrying out steps 1 to 3 in all cases for the group
G. The nontrivial integer solutions to the Luthar-Passi equations for a unit u of
2 order k = 4 have ε2a = ε2b = ε2c = 0. All of them arise after choosing u to lie in the class 2a. The solutions for the partial augmentations on classes of order 4 come in two patterns:
(ε4a, ε4b, ε4c, ε4d) ∈ {(2, 0, −1, 0), (0, 2, 0, −1), (−1, 0, 2, 0), (0, −1, 0, 2),
(1, −1, 0, 1), (−1, 1, 1, 0), (1, 0, 1, −1), (0, 1, 1, −1)
(0, −1, 1, 1), (−1, 0, 1, 1), (1, 1, 0, −1), (1, 1, −1, 0)}
The lists of multiplicities corresponding to these solutions produce the follow-
62 ing lists of eigenvalues for our candidates for Xχ(u):
χ spec(X(u)) ui
χ1 (1)
χ2 (−1)
χ3 (i), (−i) ← u3 = f1, f1 f2
χ4 (−i), (i)
χ5 (i, −i)
χ6 (1, −1)
χ7 (−1, −1, 1), (−1, 1, 1), (−1, −1, −1), (1, 1, 1) ← u7 = f1, − f1, v, −v
χ8 (−1, 1, 1), (−1, −1, 1), (1, 1, 1), (−1, −1, −1) ← u8 = u7
χ9 (−i, −i, −i), (i, i, i), (−i, −i, i), (−i, i, i) ← u9 = −v, v, − f1, f1
χ10 (i, i, i), (−i, −i, −i), (−i, i, i), (−i, −i, i) ← u9 = u10
− 1 The element v = 2Cc4a is an appropriate scalar multiple of a class sums that will be mapped onto the desired scalar multiples of the identity matrix by an ir-
reducible representation Xi affording the character χi. Let eχi be the centrally primitive idempotent of CG corresponding to χi. The idea is that choosing an P10 appropriate ui ∈ QG for i = 1,..., 10 will result in a unit u = uieχi that has i=1 the desired partial augmentations. It turns out that the choice of u3 will be an ap-
63 propriate choice for u1 through u6, the choice for u7 is appropriate for u8, and the choice for u9 will work for u10. We have given above 2 choices for u3, 4 choices for u7, and 4 choices for u9. These options cover all solutions to the Luthar-Passi equations for order 4 for this group, both trivial and nontrivial.
The torsion units u we construct from the ui will have the form b1e1 + b2e2 + b3e3, where
e1 = eχ1 + eχ2 + eχ3 + eχ4 + eχ5 + eχ6
e2 = eχ7 + eχ8 , and
e3 = eχ9 + eχ10 . are nontrivial orthogonal central idempotents of QG that sum to 1, and b1, b2, and b3 vary among the selections listed above for the ui for i = 3, 7, and 9. The 12 torsion units of order 4 that we get having nontrivial partial augmentations are:
u (ε4a, ε4b, ε4c, ε4d)
f1e1 − ve2 − ve3 (2, 0, −1, 0)
f1e1 + ve2 + ve3 (−1, 0, 2, 0)
f1 f2e1 − ve2 + ve3 (0, 2, 0, −1)
f1 f2e1 + ve2 − ve3 (0, −1, 0, 2)
64 u (ε4a, ε4b, ε4c, ε4d)
f1e1 + f1e2 + ve3 (0, 1, 1, −1)
f1 f2e1 + f1e2 − ve3 (1, 0, −1, 1)
f1e1 − f1e2 − ve3 (1, −1, 0, 1)
f1 f2e1 − f1e2 + ve3 (−1, 1, 1, 0)
f1e1 − ve2 − f1e3 (1, 1, 0, −1)
f1 f2e1 − ve2 + f1e3 (1, 1, −1, 0)
f1e1 + ve2 + f1e3 (0, −1, 1, 1)
f1 f2e1 + ve2 − f1e3 (−1, 0, 1, 1)
A key observation concerning these units is that the ve’s for e ∈ {e2, e3} do not lie in ZGe. We have checked this with GAP by showing ve does not lie in the integral span of Ge.
4.4 Partially central torsion units of QG
The form of the units constructed in Section 4.3 occurs frequently among small group examples. We have discovered a way to show that units of this form do not produce counterexamples to the ZC. Our reasoning for this fact is based on the
65 following lemma:
Lemma 4.4.1. Let e be a central idempotent of QG with e , 0, 1. Suppose v ∈ QG with ve ∈ Z(QG) and ve < ZGe. Then for all t ∈ QG, ve + t(1 − e) is not conjugate in QG to an element of ZG.
Proof. Let t ∈ QG, and let u = ve + t(1 − e). If w is a unit of QG, then uw = ve+tw(1−e). Since tw(1−e) always lies in QG(1−e), and QGe∩QG(1−e) = {0}, it cannot make up the difference between ve and any element of ZGe. This means that uw cannot be an element of ZG, since multiplying it by e does not result in an element of ZGe.
The fact that the ZC holds for units of ZG with order 4 seen in Section 4.3 follows from the Lemma 4.4.1 and Theorem 4.1.1.
In the next example we shall show the limitations of our partially central method by using G:=SmallGroup(72,40) .
Example 4.4.2. Let G be the group SmallGroup(72,40) considered in Section
4.2. The generators of G in its polycyclic presentation in GAP are: f1, f2, f3 (three
3 2 of order 2), f4 and f5 (both of order 3). In Case 5 (u = C4 and u ∈ C5), the
Luthar-Passi method produces a nontrivial solution (ε2, ε4, ε7) = (−1, 1, 1) with corresponding chain of µ-vectors
66 [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0],
[2, 1, 0, 0, 0, 1], [0, 1, 1, 0, 1, 1], [0, 1, 1, 0, 1, 1], [0, 0, 1, 2, 1, 0]. Let 1 be the identity of G. When we use the algorithm of Section 4.3 to construct a unit with these partial augmentations, it produces
u1 = u2 = u3 = u4 = u5 = f3 f5,
2 u6 = (4/3)1 + (−2/3) f4 + (−1) f5 + (−2/3) f4 ,
u7 = f1 f5 = u8, and
2 2 2 u9 = (ζ3 + 2/3 ∗ ζ3 )1 + (−ζ3 + ζ3 ) f1 + (1/3 ∗ ζ3 − 1/3 ∗ ζ3 ) f3
2 2 + (1/3 ∗ ζ3 − 2/3 ∗ ζ3 ) f4 + (1/3 ∗ ζ3 − 1/3 ∗ ζ3 ) f5
2 2 2 + (2/3 ∗ ζ3 − 2/3 ∗ ζ3 ) f3 f5 + (1/3 ∗ ζ3 − 2/3 ∗ ζ3 ) f4 . Therefore
u = f3 f5(eχ1 + eχ2 + eχ3 + eχ4 + eχ5 )
+ u6eχ6 + f1 f5(eχ7 + eχ8 ) + u9eχ9 . The last four µ-vectors [2, 1, 0, 0, 0, 1], [0, 1, 1, 0, 1, 1], [0, 1, 1, 0, 1, 1], [0, 0, 1, 2, 1, 0]
5 2 4 5 2 4 5 2 3 3 4 gives us the spectrum (in order) {1, 1, ζ6, ζ6 }, {ζ6, ζ6 , ζ6 , ζ6 }, {ζ6, ζ6 , ζ6 , ζ6 }, {ζ6 , ζ6 , ζ6 , ζ6 }.
So the last four components are not central.
Therefore, the torsion unit with the given nontrivial partial augmentation does not satisfy the hypothesis of Lemma 4.4.1, so we can not treat it using the partial central unit criteria, as it does not have the partially central pattern.
Remark 4.4.3. We have used our computer method to verify ZC for almost all
67 groups of order less than 160, we have shown that any torsion unit of QG with non- trivial integral partial augmentations is conjugate to a partially central unit of QG that is not conjugate to an element of ZG. The exceptions are SmallGroup(72,40) and its two covering groups of order 144. The previously-known lower bound for the order of groups known to satisfy ZC was 71 (see [24]). If |G| ≤ 159, and G is not nilpotent, cyclic by abelian, or an abelian extension of a p-group, or one of the the three exceptions above, then any torsion unit of QG that produces nontrivial integral partial augmentations in the Luthar-Passi method is conjugate to a partial- ly central unit of QG that is not conjugate to an element of ZG. For some groups of order 160 and 192 not covered by an existing method, it is the Luthar-Passi method from Section 4.1 that has proved too difficult for our computers to handle.
68 Chapter 5
Torsion units of C-algebras
In this chapter we establish some basic results about torsion units of C-algebras analogous to what happens for torsion units of group rings. We will consider torsion units of R-integral C-algebras in Section 5.1, where R is a subring of the ring of algebraic integers. We obtain results on torsion units for noncommutative
C-algebras having a standard character in Section 5.2, and prove a Lagrange- type theorem concerning the orders of finite subgroups of torsion units of these in
Section 5.3. These results are similar in spirit to what occurs for integral group rings, but are also new for integral scheme rings, so we state the results in this setting as corollaries.
69 5.1 Torsion units of RB
Let (A, B, δ) be an R-integral C-algebra, where R is a subring of C containing the structure constants generated by the basis B. The R-span of B is a subring of A, which we will denote by RB. Let Q¯ be the algebraic closure of Q in C, and let
Z¯ be its subring consisting of algebraic integers. For any σ ∈ Gal(Q¯ /Q), note P P P that the map σ : A −→ A defined by σ( i αibi) = i σ(αi)bi, for all i αibi ∈ A is an algebra isomorphism as long as σ fixes the structure constants in the basis
B. In particular, since the structure constants of a C-algebra are real, complex P P conjugation induces an algebra isomorphism on A that takes i uibi 7→ i uibi.
It follows from a recent result of Bangteng Xu that if (A, B, δ) is a commuta- tive integral C-algebra then all torsion units of Z¯ B are trivial (see [37, Theorem
3.1]). We will focus on the noncommutative case, and investigate the question of whether the orders of finite subgroups of normalized units of Z¯ B divide the order of L(B). Related to this is a possible generalization of the ZC on torsion units to integral scheme rings, which would be that any normalized torsion unit of ZB should be conjugate in QB to some b ∈ L(B).
The algebraic number α is called totally real if all its conjugates are real. We begin by showing that part of the proof of Xu’s result holds in the noncommutative situation.
70 Proposition 5.1.1. Let (A, B, δ) be an integral C-algebra. Let T be the subring of
∗ Z¯ consisting of totally real algebraic integers. Let u be a unit of TB with uu = b0, where b0 is the identity element. Then u is a trivial unit.
∗ ∗ P Proof. Let u ∈ A. We have uu = b0 =⇒ 1 = (uu )0 = uiuiλii∗0. Since the
2 C-algebra is integral we have that λii∗0 ≥ 1 for all bi ∈ B, and thus |ui| ≤ 1 for all i ∈ {0, 1,..., d}.
P ¯ If we also have u = i uibi ∈ TB, then for all σ ∈ Gal(Q/Q), σ(ui) = σ(ui) for all i ∈ {0, 1,..., d}. Then 1 = σ(uu∗) = σ(u)σ(u)∗, so the same reasoning as above tells us |σ(ui)| ≤ 1 for all i and all σ ∈ Gal(Q¯ /Q). By Theorem 2.4.20, the ui are either 0 or a root of unity.
P 2 Since 1 = i |ui| δ(bi), it must be that exactly one ui is a root of unity, and the corresponding bi must be a linear element of B. (Since ui is also totally real, we in fact have shown u = ±bi.)
Corollary 5.1.2. Let (X, S ) be a finite association scheme. If u is a unit of the
∗ integral scheme ring ZS with uu = σ0 then u is a trivial unit.
A result of Blau which tells us that a primitive idempotent e of a commutative
C-algebra will satisfy e∗ = e; cf. Corollary 3.3.7. If we write a torsion unit u in the basis {es : s = 0, 1,..., d} of primitive idempotents of the commutative
71 P C-algebra A, we must have u = ζses, where ζs is a root of unity for all s. For an
¯ ∗ P ¯ ∗ P ¯ −1 element u ∈ ZB presented in this way, we have u = ζses = ζses = u . Then
Proposition 5.1.1 can be applied to show that u is a trivial unit.
Xu’s following theorem makes use of the idea of the above proposition.
Theorem 5.1.3 (Theorem 3.1 [37]). Let (A, B, δ) be a commutative C-algebra, with B = {b0 = 1A, b1,..., bd}. Assume that the structure constants of (A, B, δ) are
Pd rational numbers, and λii∗0 ≥ 1, 0 ≤ i ≤ d. If u = i=0 uibi is a unit of finite order, where ui ∈ Z¯ , 0 ≤ i ≤ d, then u = uibi, where ui is a root of unity, and bi is a unit.
P Proof. Let u be a unit of finite order. Then u = ζses, where es is a primitive
∗ P ¯ ∗ P ¯ idempotent. Since A is a commutative algebra, we have u = ζses = ζses =
−1 ∗ u , implies uu = b0. Since the structure constants are rational numbers, by
Proposition 5.1.1, u is a trivial unit.
Corollary 5.1.4. If (A, B, δ) is a commutative integral C-algebra, then every tor- sion unit of ZB is a trivial unit.
Now suppose (A, B, δ) is a noncommutative R-integral C-algebra. We are in- terested in characterizing the torsion units of RB up to conjugacy.
P Proposition 5.1.5. Let (A, B, δ) be a C-algebra. Suppose u = i uibi ∈ A is a torsion unit of multiplicative order k. Then u is conjugate in A to a unit v whose
72 b0-coefficient satisfies |v0| ≤ 1, and equality occurs iff u = ζkb0, for some k-th root of unity ζk ∈ C.
χ Proof. Let { j j : χ ∈ Irr(A), 1 ≤ j ≤ χ(b0)} be the full set of primitive orthogonal idempotents described above in terms of Corollary 3.3.5. Suppose u is a torsion unit of A with multiplicative order k. Then its image in the regular representation of A is diagonalizable, and indeed will be conjugate to a diagonal matrix whose main diagonal entries are k-th roots of unity. This means that u is conjugate to a unit v of the form
X X χ v = ζχ, j j j χ j for some k-th roots of unity ζχ, j in C.
When we isolate the coefficient of b0 in this presentation of v, we have
X X mχ 1 X X v = ζ aχ (b ) = ζ m . 0 χ, j δ(B+) j j 0 δ(B+) χ, j χ χ j χ j
P P χ The fact that χ j j j = 1 implies that
X X X + mχ = mχχ(b0) = δ(B ). χ j χ
Since the mχ are all positive real, it then follows that |v0| ≤ 1, and equality will occur if and only if all of the ζχ, j are equal, in which case v = v0b0. (Since this is central in A, we can conclude that v = u.)
73 We have the following immediate consequence of Proposition 5.1.5 for central torsion units (see [32, Cor. 1.7]).
P Corollary 5.1.6. Let (A, B, δ) be a C-algebra. Let u = i uibi ∈ A be a central torsion unit of multiplicative order k. Then the following hold.
(i) |u0| ≤ 1, and equality occurs iff u = ζkb0, for some k-th root of unity ζk ∈ C.
(ii) If the structure constants generated by the basis B are rational and u ∈ Z¯ B,
then either u0 = 0 or u is a trivial unit.
The second part of the above corollary generalizes the Berman-Higman lem- ma; cf. Lemma 3.1.1.
∗ A C-algebra is symmetric if its involution is trivial on B; i.e. bi = bi for all bi ∈ B. Symmetric C-algebras are automatically commutative, because λi jk =
λ j∗i∗k∗ for all i, j, k ∈ {0, 1,..., d}. Since the eigenvalues of a Hermitian matrix have to be real, the order of a real symmetric or complex Hermitian torsion matrix can be at most 2. In particular this applies to symmetric association schemes. We now extend the idea to symmetric C-algebras.
Theorem 5.1.7. Let (A, B, δ) be a symmetric integral C-algebra. Then every tor- sion unit of ZB is a trivial unit with order at most 2.
74 P Proof. Suppose u = i uibi ∈ V(ZB) is a torsion unit. Reasoning as in the proof
∗ of Xu’s result, [37, Theorem 3.1], we have uu = b0. Since (A, B, δ) is a symmetric
∗ 2 2 P 2 integral C-algebra, u = u. Therefore u = b0 and u = ( i ui λii∗0)b0. Therefore
2 P 2 (u )0 = i ui λii∗0 = 1. But for all bi ∈ B, 0 < λii∗0 ∈ Z, therefore, ui can be nonzero for only one bi, and λii∗0 = 1 for this bi.
Remark 5.1.8. Since C-algebras are semisimple, the ones of rank 2, 3 or 4 are automatically commutative. So we know by Xu’s result [37, Theorem 3.1] that torsion units of Z¯ B will be trivial for (A, B, δ) an integral C-algebra of rank ≤ 4.
We do not yet know if there is a noncommutative C-algebra of rank 5.
5.2 Torsion units for integral C-algebras with a
standard character
Let (A, B, δ) be a C-algebra with the distinguished basis B = {b0, b1, b2,..., bd}.
Let λi jk be the structure constants relative to the basis B. In this section we shall consider C-algebras with standard character; cf. Section 3.3. We know that the C- algebra (A,B, δ) has a standard character if there is an ∗-algebra homomorphism
+ Γ : A → Mn(C) of degree n = δ(B ) whose character is equal to the standard feasible trace ρ. This is equivalent to the condition that all multiplicities mχ for
75 χ ∈ Irr(A) are nonnegative integers. The next lemma proves that a standard representation of a C-algebra is always a faithful representation.
Lemma 5.2.1. Let (A, B, δ) be a C-algebra that has a standard character. Suppose
Γ is a standard representation of (A, B, δ). Then {Γ(bi): bi ∈ B} is a linearly independent set.
Proof. Let B = {b0 = 1, b1, ..., bd} and let {λi jk}i, j,k be the structure constants rel-
Pd ative to the basis B. Suppose 0 µiΓ(bi) = 0 where µi ∈ C. If µ j , 0, then multiplying both sides of the latter equation by Γ(b j∗ ) will yield
µ0Γ(b j∗ ) + µ1Γ(b1b j∗ ) + ... + µ jΓ(b jb j∗ ) + ... + µdΓ(bdb j∗ ) = 0.
+ Applying the trace function to both sides, we get µ jλ j j∗0δ(B ) = 0. As the values of the degree map of a C-algebra are positive on elements of B, we must have µ j = 0, which is a contradiction. Therefore, {Γ(bi): bi ∈ B} is a linear independent set.
Our next lemma is an analogue of Berman-Higman’s proposition on torsion units of group rings; cf. [32, Proposition 1.4].
Lemma 5.2.2. Let (A, B, δ) be an integral C-algebra of order n that has a standard
¯ P character. Suppose u is a normalized torsion unit of ZB, and write u = i uibi for some ui ∈ Z¯ . Then u0 , 0 implies that u = b0.
76 Proof. Let ρ be the standard character of A, so n if bi = b0, ρ(bi) = 0 otherwise .
Suppose u has finite order k. Then Γ(u) is a diagonalizable matrix all of whose
fi n eigenvalues are k-th roots of unity. Let spec(Γ(u)) = {ζk }i=1, where ζk is a fixed n P fi primitive k-th root of unity in C. Now ρ(u) = ζk = u0 · n. Then i=1
Xn fi |u0|n = | ζk | ≤ n, i=1
fi f1 and |u0|n = n ⇐⇒ all ζk ’s are equal to ζk = u0.
If σ ∈ Gal(Q¯ /Q), then σ defines an algebra automorphism of A given by P σ(u) = i σ(ui)bi because the structure constants relative to B are integral. So
σ(u) will also be a normalized torsion unit of order k for every σ ∈ Gal(Q¯ /Q), and σ(u)0 = σ(u0). By the above reasoning, we see that |σ(u)0| ≤ 1 for all
σ ∈ Gal(Q¯ /Q), and so by Theorem 2.4.20, u0 must be either 0 or a root of unity.
When u0 is a root of unity, then all eigenvalues of Γ(u) are equal to this, and we have Γ(u) = u0 · I = u0Γ(b0).
Since Γ is a faithful representation, we have u = u0 · b0. As δ(u) = 1 we have u0 = 1 and so u = b0.
77 P Corollary 5.2.3. Let (X, S ) be a finite association scheme. Suppose u = usσs ∈ s∈S tor V(ZS ) . Then u1 , 0 implies that u = σ0.
Remark 5.2.4. We note that for a commutative C-algebra Lemma 5.2.2 is a con- sequence of Corollary 5.1.6.
The conclusion of Lemma 5.2.2 fails if we extend the coefficients beyond rings of algebraic integers, see the following example for the verification of this claim.
Example 5.2.5. Let (A, B, δ) be the integral table algebra whose distinguished
2 2 basis B = {1 := b0, b1, b2} satisfies b1 = b0, b2 = 2b0 + 2b1 and b1b2 = b2b1 = b2.
This table algebra is the adjacency algebra of the association scheme of order 4 and rank 3, so it has a standard character. Its adjacency matrices are: 1 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 1 1 b0 = , b1 = , b2 = . 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 − 2 1 1 Let u = b0 + b1 + b2. Then u = 4b0, so 2 u is a torsion unit with δ( 2 u) =
1 − 1 2 ( 1+1+2) = 1. Therefore, Z[ 2 ]B has nontrivial normalized torsion units whose support includes b0.
The next proposition shows that if the support of a normalized torsion unit of
Z¯ B includes a linear element that commutes with the unit, then the given unit must be a trivial unit.
78 Proposition 5.2.6. Let (A, B, δ) be an integral C-algebra that has a standard char-
tor acter. Let u ∈ V(Z¯ B) . If bi ∈ L(B) ∩ supp(u) and bi commutes with u, then u = bi.
tor Proof. Let u ∈ V(Z¯ B) . Now let bi ∈ L(B) ∩ supp(u) for which bi commutes
¯ 0 −1 with u. Then bi is a unit of ZB and ui , 0. Let u = bi u. Since bi commutes with
0 0 0 u, u has finite order. Since (u )0 = ui , 0, we must have u = b0 by Lemma 5.2.2.
Therefore, u = bi is a trivial unit, as claimed.
tor Corollary 5.2.7. Let u ∈ V(ZS ) . If s ∈ Oϑ(S ) ∩ supp(u) and σs commutes with u, then u = σs is a trivial unit of ZOϑ(S ).
The center of the finite association scheme (X, S ) is defined to be Z(S ) = {t ∈
S : σtσs = σsσt, for all s ∈ S }. The scheme (X, S ) is a commutative scheme if
Z(S ) = S . The next corollary is immediate from Corollary 5.2.7.
Corollary 5.2.8. 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 ).
There is one further noncommutative situation in which we know central tor- sion units are trivial.
Theorem 5.2.9. Let (A, B, δ) be an integral C-algebra that has a standard char- acter. Let p be a prime integer. Suppose that for all bi ∈ B\L(B), λii∗0 is divisible
79 by p. Then every central torsion unit of Z¯ B is a trivial unit.
Proof. Let u be a central torsion unit of Z¯ B with multiplicative order k. Suppose u is not trivial. By Proposition 5.2.6, the intersection of L(B) with supp(u) must be Pk empty. Let supp(u) = {b1,..., bk} and write u = uibi. Our assumption implies i=1 that for all bi ∈ supp(u), p divides λii∗0. But then
k ¯ 1 = (u )0 ∈ spanZ{((u j1 b j1 )(u j2 b j2 ) ... (u jk b jk ))0 : ji ∈ {1,..., k} for 1 ≤ i ≤ k} ⊆ pZ, a contradiction.
Corollary 5.2.10. 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.
The above corollary applies in particular to the important subclass of p-valenced association schemes, for which every ns > 1 is a power of the prime p.
The following lemma for C-algebras is an analog of [32, Lemma (37.5)].
Lemma 5.2.11. Let K and F be infinite fields with K ⊆ F. Let (A, B, δ) be a
+ C-algebra, with B = {b0, b1,..., bd} and δ(B ) = n. Suppose that T1 and T2 are two finite subgroups of units of KB. Then
T1 ∼ T2 in FB =⇒ T1 ∼ T2 in KB.
80 + Proof. Let δ(B ) = n, |B| = d, and |T1| = m. Suppose w ∈ U(FB) and that for all
w u α ∈ T1, α = βα ∈ T2. We wish to find u ∈ U(KB) with α = βα. Consider the P equations αu = uβα, and write u = xsbs.
As α runs over T1 we have
P P P P αu = uβα ⇐⇒ ( αsbs)( xtbt) = ( xtbt)( βα,sbs) s t t s P P P P P P ⇐⇒ αs xt λstrbr = xtβα,s λtsrbr s t r s t r P P P P P P ⇐⇒ ( αsλstr xt)br = ( xtβα,sλtsr)br r s t r s t
⇐⇒ ([αsλstr]s,t[xt]t) = ([βα,sλtsr]s,t[xt]t), ∀ 0 ≤ r ≤ d.
Thus this reduces to a system of m homogeneous linear equations in d vari- ables over K of the form Mα,u[xt]t = 0. Let {v1,..., v`} be a basis of the solution space in Kd, with 1 ≤ `. This is also a basis for the solution space in Fd. For P P every solution yivi = [xt]t, the element x = xtbt will satisfy αx = xβα, for all i t
α ∈ T1.
d P Suppose that for every solution [xt] in K , the element x = xtbt is a ze- t ` P P ro divisor in KB. Let γ : K → KB be the map γ( yivi) = γ([xt]t) = xtbt. i t
Consider KB as a subset of Mn(K), and consider the polynomial φ(y1,..., y`) = P P det(γ( yivi)) = det( xtbt). Our assumption implies that this polynomial vanish- i t es at infinitely many points in K`, and hence it must be identically zero. However,
w the existence of the unit w of FB with T1 = T2 implies that the polynomial does
81 not vanish on F`, which is a contradiction. It follows that there is at least one point
` P P (y1,..., y`) in K for which yivi = [xt]t and xtbt is not a zero divisor in KB. i t P It then follows that u = xtbt is a unit of KB for which αu = uβα for all α ∈ T1. t
Therefore, T1 and T2 are conjugate in KB.
This lemma means that in order for two finite subgroups of ZB to be conjugate in QB, it is enough that they be conjugate in CB. This can be determined using their spectrum in every irreducible representation, as we have done in our com- puter implementation of the Luthar-Passi method for integral group rings. What is missing for this approach in the C-algebra or scheme settings is a reasonable interpretation of conjugacy classes and partial augmentations.
5.3 Lagrange’s theorem for normalized torsion
units of Z¯ B
The Lagrange theorem for finite groups states that for a finite group G, the order of any finte subgroup H of G divides the order of group G. This theorem has been generalized to the case where H is a finite subgroup of V(ZG) for a finite group G
[32, Lemma (37.3)]. In this section we prove this theorem for Z¯ B, where Z¯ is the ring of algebraic integers and B is a distinguished basis for a Z¯ -integral C-algebra.
82 We begin by extending a result for group algebras over fields of characteristic
0 to C-algebras over fields of characteristic 0.
Theorem 5.3.1. Let (A, B, δ) be a C-algebra that has a standard character. Let
K be a subfield of C containing the structure constants relative to the basis B. Pd Suppose e is an idempotent of KB, and write e = eibi with ei ∈ K. Then i=0 m + e0 = n ∈ Q, where n = δ(B ) and m is the rank of the image of e in any standard representation.
Proof. Let Γ be the standard representation and ρ be the standard character of
KB. As e is an idempotent, we know that spec(Γ(e)) = [1(m), 0(n−m)] is a multiset, where m = rank(Γ(e)). Thus
Xd ρ(e) = eiρ(bi) = e0 · n = m. i=0
m 1 n−1 0 Therefore, e0 = n ∈ {0, n ,..., n , 1}. Furthermore, e0 = 0 = n ⇐⇒ e = 0 and
n e0 = 1 = n ⇐⇒ e = b0.
Corollary 5.3.2. Let K be a field of characteristic 0 and (X, S ) be a finite associ- P ation scheme of order n. Let e = esσs , 0, 1 be a nontrivial idempotent of KS . s∈S m Then e1 = n ∈ Q, 0 < e1 < 1, where n = |X| and m is the rank of e as the matrix in the standard representation.
83 Corollary 5.3.3. Let K be an algebraic number field with ring of integers R. Sup- pose (A, B, δ) is an R-integral C-algebra that has a standard character. Then the only idempotents of RB are 0 and 1.
+ + Proof. Let e ∈ RB be an idempotent. Then e0 ∈ Q ∩ R = Z . By Theorem 5.3.1, this implies e0 = 0 or 1, and by considering the rank of Γ(e) in these respective cases we have e = 0 or 1.
P Let G be a finite group, and ZG be its integral group ring. If u = ugg ∈ ZG g∈G P with ug ∈ Z for all g ∈ G, then the augmentation of u is ug ∈ Z. Viewing G as g∈G a C-algebra basis, we see that L(G) = G, and so the augmentation map coincides with the degree map. It is a well-known fact about normalized units of group rings that any finite subgroup of normalized units of ZG is linearly independent in CG; cf. [32, Lemma (37.1)]. It is natural then to inquire about what happens in the case of integral C-algebras.
Lemma 5.3.4. Let (A, B, δ) be an integral C-algebra that has a standard charac- ter. Then any finite group of normalized units of Z¯ B is a set of C-linearly indepen- dent elements of A.
Proof. Let T = {u1 = b0, u2,..., u`} be a finite group of units contained in V(Z¯ B).
Suppose c1ui1 + ... + ckuim = 0 is an expression of minimal length, where the ui j
84 are elements of T and the coefficients c j ∈ Z¯ are not all 0. Since T is a group,
we can assume without loss of generality that ui1 = b0. Expressing the ui j for P j = 2,..., m, as ui j = s ui j,sbs, we have by Lemma 5.2.2 that ui j,0 = 0 for j = 2,..., m. It follows that
0 = (c1b0 + c2ui2 + ··· + cmuim )0 = c1, contradicting the minimal length assumption. Therefore, T is a linearly indepen- dent set.
Corollary 5.3.5. 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.
For a finite group G, the order of any finite subgroup H of V(ZG) divides the order of G (see [32, Lemma (37.3)]). We can give an analogue of this theorem for integral C-algebras that have a standard character.
Theorem 5.3.6. Let (A, B, δ) be an integral C-algebra that has a standard char- acter. Suppose that this C-algebra has order n = δ(B+) and size r = |B|. Then the order of any finite subgroup T of V(Z¯ B) divides n and is at most r.
Proof. Since Z¯ B is a free module with basis B, any linearly independent subset of Z¯ B has at most r = |B| elements. By Lemma 5.3.4, T is a linearly independent subset, so |T| ≤ r.
85 1 P + 2 Now let e = |T| t = (T )/|T|. Since T is a finite group, we have e = e . t∈T Let Γ be a standard representation of A and let ρ be the standard character. Since e2 = e , 0, spec(Γ(e)) = [1(m), 0(n−m)], where m is the rank of the matrix Γ(e).
¯ + 1 + 1 + n Therefore, ρ(e) = m ∈ Z . Also ρ(e) = |T| ρ(T ) = |T| n(T )0 = |T| , since the
+ n argument of Lemma 5.3.4 implies (T )0 = 1. Therefore m = |T| , hence |T| divides n.
Corollary 5.3.7. 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 |.
If H is a subgroup of V(ZG) for a finite group G with |H| = |G|, then ZG = ZH; cf. [32, Lemma (37.4)]. It is not necessarily true in this case that G be isomorphic to H as groups. The following lemma proves an analogous result for schemes.
Lemma 5.3.8. 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 Lemma 5.3.4, T is linearly independent and thus QS = QT. It follows that ZS ⊇ ZT and mZS ⊂ ZT for some positive integer m.
Let T = {t1 = σ0, t2,..., tr} and let s ∈ S . Then
Xr mσs = citi, for some ci ∈ Z. i=1
86 We wish to show that each ci is a multiple of m. For each j ∈ {1,..., r}, we have
−1 X −1 mσst j = c jσ0 + ci(tit j ). i, j
Since by Lemma 5.2.2, (t t−1) = 0 for i j, the coefficient of σ on the right i j 1 , 0 hand side is c j whereas on the left hand side it is a multiple of m. It follows that m | c j for j = 1,..., r. Therefore, σs ∈ ZT for all s ∈ S , and hence ZS = ZT.
While thin association schemes give immediate examples where the conclu- sion of the preceding theorem holds, we are uncertain as to whether ZS can pos- sess 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 5.3.9. Let (X, S ) be the association scheme as27-5 in Hanaki and
Miyamoto’s classification of small association schemes (see [14]). This is a com-
∗ mutative non-symmetric scheme of order 27 and rank 3. We have S = {1X, s, s },
2 where ns = ns∗ = 13, and the structure constants of S are determined by σs =
2 6σs + 7σs∗ , σsσs∗ = 13σ0 + 6σs + 6σs∗ , σs∗ = 7σs + 6σs∗ . The character table for
87 the scheme is: ∗ 1X s s mχ
χ1 1 13 13 1
2 2 χ2 1 −ζ3 + 2ζ3 2ζ3 − ζ3 13
2 2 χ3 1 2ζ3 − ζ3 −ζ3 + 2ζ3 13
Analysis of the character table of S shows that QS QC3, where C3 is a cyclic group of order 3. Let δ be the irreducible character of CS corresponding to the valency map, and let ψ, ψ be the other two irreducible characters of CS .
Let {eδ, eψ, eψ} be the centrally primitive idempotents of CS , the character formula for which can be found in Theorem 3.3.4. An element v of CS with order 3 and valency 1 is given by
2 v = eδ + ζ3eψ + ζ3 eψ, and since v is fixed by complex conjugation, v ∈ QS . Using the character formula for centrally primitive idempotents of CS , we find that
1 v = (−4σ − σ + 2σ ∗ ), 9 0 s s
2 ∗ 2 and v = v . So if T = {σ0, v, v }, then T is a finite subgroup of normalized units
⊆ 1 of QS for which QS = QT. In this case ZS (ZT Z[ 9 ]S .
Proposition 5.3.10. Let (A, B, δ) be a C-algebra with standard character and dis-
88 tinguished basis B = {b0, b1} of order n and rank 2. Then 1 if n ≥ 3, and |V(ZB)tor| = 2 if n = 2.
Proof. Let u be a normalized torsion unit of ZB with multiplicative order k. Our
Lagrange theorem for C-algebras with standard character implies k divides n and k ≤ 2. So we are done if n is odd. Suppose k = 2. Since B is symmetric and
∗ 2 ∗ u ∈ ZB, u = u . Therefore, u = b0 implies that uu = b0, and so by Proposition
5.1.1, u = bs for some s ∈ B with δ(bs) = 1. Such an element of the C-algebra of rank 2 with s , 0 only exists when n = 2.
For symmetric C-algebras of rank 3, we have already seen that normalized torsion units must be trivial with order at most 2. Non-symmetric association schemes of rank 3, such as the one seen in the example above, arise naturally from strongly regular directed graphs.
Proposition 5.3.11. Let (A, B, δ) be a C-algebra with standard character of order
∗ n > 2 and rank 3 with distinguished basis B = {b0, b1, b1}. If δ(b1) > 1, then
|V(ZB)tor| = 1.
Proof. Suppose u ∈ V(ZB) is a normalized torsion unit with u , b0. By Lemma
∗ ∗ 5.2.2, supp(u) = {b1, b1}, so u = αb1 + βb1 for some α, β ∈ Z. Since δ(u) = 1, we
89 ∗ have 1 = αδ(b1) + βδ(b1) = (α + β)δ(b1), which is not possible as α, β ∈ Z.
5.4 Applications to Schur rings and Hecke algebras
Schur rings and integral Hecke algebras (i.e. double coset algebras) are interesting classical examples of scheme rings. In this section we see how the general results of Section 5.3 apply in these settings.
5.4.1 Schur rings
Let G 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
(i) {1G} ∈ F ;
∗ −1 −1 (ii) for all U = {g1,..., gk} ∈ F , U = {g1 ,..., gk } ∈ F ; and
(iii) for all U, V, W ∈ F , there exists nonnegative integers λUVW such that
X UbbV = λUVW Wb, W∈F P where Ub = g∈U g denotes the sum of the elements of U in the group ring
ZG.
90 The Schur ring ZF is defined to be the Z-span of {Ub : 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 a Schur 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 ring ZF is isomorphic to an integral scheme ring.
Given the group G and Schur ring partition F , for U ∈ F we set
Uτ = {(x, y) ∈ G × G : xg = y for some g ∈ U}.
Let F τ = {Uτ : U ∈ F }. Using the properties of the Schur ring partition F , it is straightforward to show that (G, F τ) is an association scheme of order n = |G| and rank r = |F |. Furthermore, ZF' Z[F τ] as rings, where the isomorphism is produced by the restriction of the regular representation of G to ZF . The restric- tion of the augmentation map on the group ring to ZF corresponds to the valency map of Z[F τ] under this isomorphism. The following corollary is the application of our Lagrange theorem for scheme rings to this special setting.
Corollary 5.4.1. Let F 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 |.
91 5.4.2 Integral Hecke algebras
Let H 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 for g ∈ G. Corresponding to each double coset HgH for g ∈ G, let
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 sheme is known as a Schurian scheme, and its rational adjacency algebra Q[G//H] is ring isomorphic to the ordinary Hecke
1 P algebra eHQGeH, where eH = |H| h∈H h. (For details see [15], 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 5.4.2. Let H 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.
92 Chapter 6
Future Work
Among other results on normalized torsion units of C-algebras we established a
Lagrange-type theorem on normalized torsion units of C-algebras. As this disser- tation is one of the first treatments of torsion units for integral C-algebras, there is a great deal of potential for further development along the lines of what has been done for integral group rings. In the future we shall try to establish a Lagrange- type theorem for integral C-algebras with no standard character. In our compu- tational approach, we shall try to raise the bound for small groups satisfying ZC to 215, because 216 = 2333 is expected to be the next difficult case. Prelimi- nary results in our computer search show difficulties occur at orders 160, 192 and
200. We would like to know if there is an analog of the Luthar-Passi method for
93 attacking the ZC in the scheme or C-algebra settings.
In the remainder of the chapter we gather preliminary results for these future investigations.
6.1 Categorical aspects
We define the quasi-direct product of bases of C-algebras analogous to the defini- tion of quasi-direct product of schemes; cf. [38, Chapter 7].
Lemma 6.1.1. Let (Ai, Bi, δi) be C-algebras with the distinguished bases B1 =
{b0, b1,..., bd1 } and B2 = {c0, c1,..., cd2 }. Let B = B1 × B2 = {bi ⊗ cu : bi ∈
∗ B1, cu ∈ B2}. If (bi ⊗ cu) = bi∗ ⊗ cu∗ and other operations are componentwise then
B is a basis for a C-algebra (A, B, δ), where δ(bi ⊗ cu) = δ1(bi) ⊗ δ2(cu).
Proof. Since operations are componentwise, b0 ⊗ c0 is the identity element of A.
P P 0 For any bi ⊗ cu, b j ⊗ cv ∈ B, let bib j = λi jkbk and cucv = λuvwcw, where λ s k w P P are real. Then (bib j ⊗ cucv) = λi jkbk ⊗ λuvwcw implies (bi ⊗ cu)(b j ⊗ cv) = k w P P λi jkλuvw(bk ⊗ cw). If λ(i,u),( j,v),(k,w) = λi jkλuvw, then for λ(i,u),( j,v),(0,0) , 0, we k w ∗ ∗ have λi j0λuv0 , 0, implies λi j0 , 0, λuv0 , 0. It follows that j = i , v = u and
∗ ∗ ( j, v) = (i , u ). As λ(i,u),(i∗,u∗),(0,0) = λii∗0λuu∗0 and λii∗0 = λi∗i0 > 0, λuu∗0 = λu∗u0 > 0, so λ(i,u),(i∗,u∗),(0,0) = λ(i∗,u∗),(i,u),(0,0) > 0. Hence (A, B, δ) is a C-algebra.
94 Example 6.1.2. Let (A, B, δ) be a C-algebra with distinguished basis B = {b0, b1,
..., bd}, and let C2 be a cyclic group of order 2, viewed as a distinguished basis of
CC2, a C-algebra with trivial degrees. Then B × C2 is a distinguished basis for a
C-algebra.
The algebraic significance of the quasi-direct product of bases B1 and B2 is that it is a C-algebra basis of the tensor product algebra of A1 and A2. For the association scheme and C-algebra settings, this is analogous to the direct product of groups being a basis for the tensor product of their group algebras: i.e. C[G1 ×
G2] C[G1] ⊗ C[G2].
Definition 6.1.3. Let (A1, B1, δ1) be a C-algebra with distinguished basis B1 =
{b10, b11,..., b1d}, and let (A2, B2, δ2) be another C-algebra with distinguished ba- sis B2 = {b20, b21,..., b2n}. A C-algebra homomorphism φ : A1 −→ A2 is an algebra homomorphism if it satisfies
(i) φ(1) = 1,
∗ ∗ (ii) φ(b1i) = φ(b1i) for all i = 0, 1,..., d,
(iii) for all i = 0, 1,..., d, there are nonnegative real numbers µb1i,b2 j such that Pn φ(b1i) = µb1i,b2 j b2 j, and j=0
95 (iv) for all b1i, b1 j ∈ B1, supp(φ(b1i))∩ supp(φ(b1 j)) , ∅ implies that there exists
ρb1i,b1 j > 0 such that φ(b1i) = ρb1i,b1 j φ(b1 j).
Definition 6.1.4. Let (X, S ) and (Y, T) be schemes. A scheme homomorphism from
(X, S ) to (Y, T) is a pair φ = (φX, φS ) of functions φX : X −→ Y and φS : S −→ T satisfying:
(i) if (x1, x2) ∈ s then (φX(x1), φX(x2)) ∈ φS (s), and
(ii) for all w, z ∈ X and s ∈ S, if (φX(w), φX(z)) ∈ φS (s) then there exists
(x1, x2) ∈ s such that (φX(x1), φX(x2)) = (φX(w), φX(z)).
Definition 6.1.5. A bijective scheme homomorphism is called an isomorphism.
An scheme isomorphism that preserves structure constants is called an algebraic isomorphism.A normalized algebraic scheme isomorphism preserves valencies as well.
6.2 When all units of ZB are trivial
We want to know for which integral C-algebra bases B is it true that V(ZB) =
L(B). This is the “trivial units problem” for integral C-algebras. The next result is a generalization of [32, Proposition (2.1)] which states that if G is a group and
∗ ∗ ∗ G = G × C2 then U(ZG) = ±G =⇒ U(ZG ) = ±G .
96 Proposition 6.2.1. Let (A1, B, δ) be a C-algebra with the distinguished basis B =
{b0, b1,..., bd} Let A2 = CC2, where C2 = hxi is a group of order 2. Then U(ZB) =
±B =⇒ U(Z[B × C2]) = ±(B × C2).
Proof. Let α, β, γ, δ ∈ ZB be such that (α + βx)(γ + δx) = 1, then
(αγ + βδ) + (αδ + βγ)x = 1.
Upon comparing coefficients, we have
αγ + βδ = 1, αδ + βγ = 0.
Therefore
(α + β)(γ + δ) = 1, (α − β)(γ − δ) = 1.
Since U(ZS ) = ±S , therefore
α + β = ±u, α − β = ±v, u, v ∈ B.
Thus 2α = ±u ± v. Since the coefficient in left hand side is 2, and the only way for ±u ± v to have even coefficient is u = ±v. It follows that α = 0 or β = 0. In any case, α + βx is trivial.
97 6.3 Normalized automorphisms of QB
Definition 6.3.1. Let (A, B, δ) be a C algebra. An automorphism of QB is a nor- malized C-algebra homomorphism if it preserves the degree.
The fact that normalized automorphisms of integral group rings preserve class sums is motivating this line of investigation (see [32, Theorem (36.5)]). The fol- lowing theorem shows that normalized algebraic isomorphism preserves not only valencies but also thin radicals under some conditions.
Theorem 6.3.2. Let B1 and B2 be commutative C-algebras of the same order. Let
φ : CB1 −→ CB2 be a normalized C-algebra isomorphism. Then φ(Z[L(B1)]) =
Z[L(B2)] implies that φ(L(B1)) = L(B2).
tor Proof. Since torsion units of degree 1 are trivial in this case, φ(V(ZB1) ) =
tor V(ZB2) implies that φ(L(B1)) = L(B2).
As an application to Corollary 5.2.10, the following theorem is a partial ana- logue to [32, Theorem (36.3)] for schemes. If S 0 is a closed subset of an associa- tion scheme then Z(S 0) denotes the center of S 0.
Theorem 6.3.3. Let S and T be p-valanced association schemes for some prime p with nS and nT finite. Let φ : S −→ T be a normalized algebraic isomorphism,
98 and let its linear extension to ZS also be denoted by φ. If φ : ZS −→ ZT then
Z(Oϑ(S )) ' Z(Oϑ(T)).
Proof. Let φ be the normalized algebraic isomorphism between ZS and ZT. As
φ is normalized, for any σs ∈ Z(Oϑ(S )) with order k, φ(σs) has valency 1, is a unit of order k, and φ(σs) ∈ Z(ZT). Since φ(σs) has valency 1, by Corollary
5.2.10, φ(σs) ∈ T, hence lies in Z(Oϑ(S )). Therefore φ(Z(Oϑ(S ))) ⊆ Z(Oϑ(T)).
Conversely, let σt ∈ Z(Oϑ(T)) with order k. Since φ is a normalized isomorphism,
−1 −1 −1 φ (σt) ∈ Z(ZS ) and φ (σt) has valency 1, so by Corollary 5.2.10, φ (σt) ∈ S .
Therefore, φ(Z(Oϑ(S ))) ⊇ Z(Oϑ(T)). Hence φ(Z(Oϑ(S ))) = Z(Oϑ(T)).
99 Bibliography
[1] Z. Arad, E. Fisman, M. Muzychuk, Generalized table algebras, Israel J.
Math. 114 (1999) 29-60.
[2] Javad Bagherian and Amir Rahnamai Barghi, Standard character condi-
tion for C-algebras, arXiv:0803.2423 [math.RT].
[3] Harvey I. Blau, Association schemes, fusion rings, C-algebras, and
reality-based algebras where all nontrivial multiplicities are equal, J. Al-
gebraic Comb., 31 (2010), no. 4, 491-499.
[4] Harvey I. Blau, Table algebras, European J. Combin., 30 (2009), no. 6,
1426-1455.
[5] Harvey I. Blau, Quotient structures in C-algebras, J. Algebra, 175 (1995),
no. 1, 24-64; Erratum: 177 (1995), no. 1, 297-337.
100 [6] Harvey I. Blau and Bangteng Xu, On homogeneous integral C-algebras,
J. Algebra 199, (1998), 61-66.
[7] Mauricio Caicedo, Leo Margolis, and Angel´ Del Rio,´ Zassenhaus con-
jecture for cyclic-by-abelian groups, J. London Math. Soc., (2)88 (2013),
65-78.
[8] J. A. Cohn and D. Livingstone, On the structure of group algebras, I,
Canad. J. Math., 17 (1965), 583-593.
[9] Charles W. Curtis and Irving Reiner, Representation Theory of Finite
Groups and Associative Algebras, John Wiley and Sons Inc., New York,
1962.
[10] Charles W. Curtis and Irving Reiner, Methods of representation theory.
Vol. I, with applications to finite groups and orders, John Wiley and Sons
Inc., New York, 1981.
[11] David S. Dummit and Richard M. Foote, Abstract Algebra, John Wiley
and Sons Inc., New York, 2004, A Wiley-Interscience Publication.
[12] The GAP Group, GAP - Groups, Algorithms, and Programming, Version
4.6.2, Aachen, St. Andrews, 1999, (http://www.gap-system.org).
101 [13] D. G. Higman, Coherent algebras, Linear Algebra Appl., 93 (1987), 209-
239.
[14] A. Hanaki and I. Miyamoto, Classification of small association Schemes,
(07/01/2014), (http://www.shinshu.jp/˜hanaki/as).
[15] Akihide Hanaki and Mitsugu Hirasaka, Theory of Hecke algebras to asso-
ciation schemes, SUT Journal of Mathematics, 38(1) (2002), 61-66.
[16] Allen Herman, Seminar Notes: Algebraic Aspects of Association
Schemes and Scheme Rings, 2011, (http://uregina.ca/˜hermana/ASSR-
Lecture9.pdf).
[17] Allen Herman and Gurmail Singh, Revisiting the Zassenhaus Conjecture
on torsion units for the integral group rings of small groups, Proceedings.
Mathematical Sciences. Indian Academy of Sciences, 125 (2), 2015, 167-
172.
[18] Allen Herman and Gurmail Singh, Torsion units of integral C-algebras,
JP Journal of Algebra, Number Theory, and Applications, 36 (2), 2015,
141-155.
102 [19] Allen Herman and Gurmail Singh, On the Torsion Units of Integral Adja-
cency Algebras of Finite Association Schemes, Algebra, Vol. 2014, 2014,
Article ID 842378, 5 pages. (Open Access)
[20] Martin Hertweck, On the torsion units of some integral group rings, Alge-
bra Colloquium, 13 (2006), no. 2, 329-348.
[21] Martin Hertweck, Torsion units of integral group rings of certain
metabelian groups, Proc. Edinburgh Math. Soc., 51 (2008), 363-385.
[22] Martin Hertweck, The orders of torsion units in integral group rings of
finite solvable groups, Comm. Algebra, 36 (2008), no. 10, 3585-3588.
[23] Martin Hertweck, On torsion units in integral group rings of Frobenius
groups, (07/22/2012), arXiv:1207.5256 [math.RT].
[24]C.H ofert¨ and W. Kimmerle, On torsion units of integral group rings
of groups of small order, in Groups, Rings and Group Rings, pp. 243
- 252, Lecture Notes in Pure and Applied Mathematics, Vol. 248, CRC
Press/Chapman and Hall, 2006.
[25] G. Higman, Units in Group Rings, D. Phil. Thesis Univ. Oxford (1940).
103 [26] L. Klingler, Construction of a counterexample to a conjecture of Zassen-
haus, Commun. Algebra, 19 (1991) pp. 2303-2330.
[27] I.S. Luthar and I.B.S. Passi, Zassenhaus conjecture for A5, Proc. Indian
Acad. Sci. Math. Sci., 99 (1989), no. 1, 1-5.
[28] Z. Marciniak, J. Ritter and S.K. Sehgal and A. Weiss, Torsion units in
integral group rings of some metabelian groups, II Journal Of Number
Theory, 19 (1987) pp. 340-352.
[29] Amir Rahnamai Barghi and Javad Bagherian, Standard character condi-
tion for table algebras, Electronic J. Combinatorics 17, no. R113 (2010).
[30] K. Roggenkamp, L. Scott, Isomorphisms for p-adic group rings, Ann.
Math., 126 (1987) pp. 593-647.
[31] Roggenkamp, Klaus W. Observations on a conjecture of Hans Zassenhaus.
Groups-St. Andrews 1989, Vol. 2, 427-444, London Math. Soc. Lecture
Note Ser., 160, Cambridge Univ. Press, Cambridge, 1991.
[32] S.K. Sehgal, Units in integral group rings, Pitman Monograph and Sur-
veys in Pure and Applied Mathematics, Vol. 69, Longman Scientific and
Techical, Harlow, 1993.
104 [33] Sehgal S K, Topics in group rings, New York: Dekker, 1978 .
[34] A. Weiss, Rigidity of p-adic p-torsion, Ann. of Math., 127 (1988) pp. 317-
332.
[35] A. Weiss, Torsion units in integral group rings, J. reine angew. Math. 415
(1991), 175-187.
[36] Prasolov, Victor V. Polynomials. Translated from the 2001 Russian second
edition by Dimitry Leites. Algorithms and Computation in Mathematics,
11. Springer-Verlag, Berlin, 2004.
[37] Bangteng Xu, On isomorphisms between integral table algebras and ap-
plications to finite groups and association schemes, Comm. Algebra, 42
(2014), no. 12, 5249-5263.
[38] Paul-Hermann Zieschang, Theory of Association Schemes, Springer-
Verlag, Berlin, 2005, Springer Monographs in Mathematics.
105