LIE THEORY OF FINITE SIMPLE GROUPS AND THE
GENERALISED ROTH CONJECTURE
J L OPEZ PE~NA, S MAJID, AND K RIETSCH
Abstract. We apply a recent approach to noncommutative dierential geom-
etry on nite groups to the case of nite nonabelian simple groups and similar
groups with trivial centre. The `Lie algebra' or bicovariant dierential calculus
here is provided by an ad-stable generating set and by analogy with Lie the-
ory we consider when the associated Killing form is nondegenerate. We prove
nondegeneracy for the case of the universal calculus whenever Roth's property
holds, including for all symmetric groups Sn, all sporadic and most other nite
simple nonabelian groups. We conjecture that nondegeneracy holds more gen-
erally and prove it for the 2-cycles calculus on any Sn, and by computer for all
real conjugacy classes on nite simple nonabelian groups up to order 75,000.
In all cases we nd that the Killing form is in fact either positive denite, if
the conjugacy class consists of involutions, or otherwise has zero (evenly split)
signature. As an application of the Killing form we nd that its eigenspaces
typically decompose the conjugacy class representation into irreducibles and
we explore the possibility of bijectively assigning an irreducible representation
to a conjugacy class containing it. We prove that the conjugacy classes of Sn
containing the sign representation are those corresponding to partitions of n
into distinct odd parts and relate this observation to a classical identity of
Euler.
1. Introduction
Roth's conjecture [15] in the theory of nite groups asserts that the ad-representation
of the groupG on itself contains every complex irreducible representation ofG=Z(G)
at least once (where Z(G) is the centre). This conjecture is false in general, but
is known to be true for symmetric groups [3] and alternating groups [17], and, re-
cently, for the sporadic simple groups [7] using methods from [13]. We shall refer
to this property, when it holds, as the Roth property. Indeed, for simple groups the
exceptions according to [7] appear to be rather few and to amount to some instances
of one classical family of Lie type over nite elds of particular order. In this paper
we provide a noncommutative-geometrical point of view on this property, relating
it to the universal dierential calculus on the nite group and nondegeneracy of
the corresponding Killing form (these notions will be dened in the preliminaries
Date: March 30, 2010.
2000 Mathematics Subject Classication. Primary 81R50, 16W50, 16S36.
Key words and phrases. Finite group, Lie algebra, Riemannian geometry, conjugacy class,
noncommutative geometry, quantum group.
The 1st author is funded by an EU Marie-Curie fellowship PIEF-GA-2008-221519, the 2nd
author by a Leverhulme research fellowship and the 3rd author by an EPSRC research fellowship
EP/D071305/1.
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2 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
section of the paper). Nondegeneracy of the Killing form here appears to be weaker
than the Roth property and we conjecture that it holds at least for all simple G.
Beyond this, we expect more strongly at least when G is simple or Sn that the
Killing form is nondegenerate quite generally, not only for the universal calculus.
A general calculus on a nite group corresponds to an ad-stable subset C  G n feg
with equality in the case of the universal calculus. Here e is the group identity.
The Killing form is dened on L = CC and in the case of nite groups reduces to
the trace in the representation
K(xa; xb) := L(ab)
where fxa j a 2 Cg is a basis of L and L is the character or trace in the represen-
tation L. In our case the representation L is given by the adjoint action hence
K(xa; xb) = jZ(ab) \ Cj
where Z(g) is the centraliser of g 2 G. Complementing the case of the universal
calculus in Section 3, we focus next on the other extreme where C is a single
conjugacy class stable under group inversion (such classes are called `real' as the
character is real-valued). Section 5 proves nondegeneracy for the natural 2-cycles
calculus on symmetric groups Sn while Section 6 provides computer verication of
this conjecture for real conjugacy classes up to jGj some 75,000. This includes the
smallest sporadic group M11 and the Suzuki group Sz8. The geometric intuition is
that when G is simple any C generates G and hence could be viewed as some kind
of `Lie algebra' which we might hope to re
ect the simplicity of G.
Once we have a Killing form, nondegenerate or not, we can envisage many appli-
cations. In usual Lie theory complex simple Lie algebras have up to equivalence
a unique compact real form where the Killing form is positive denite, leading in
the geometry to a Riemannian metric with constant positive curvature. The sim-
plest nite group examples along these lines have been studied `by hand' and it
was found, for example, that S3 with its 2-cycles conjugacy class is an Einstein
space with Ricci curvature essentially proportional to the metric, while A4 with an
order 4 conjugacy class is Ricci
at [11, 12]. With such eventual geometry in mind,
we are interested in when the matrix of K in our basis C is positive denite. The
answer among real conjugacy classes appears to be precisely those which are classes
of involutions. One of these could then be a natural `compact' braided Lie algebra
structure of a nite nonabelian simple group. We nd moreover that the other real
conjugacy have the interesting feature of equal numbers of positive and negative
eigenvalues when counted with multiplicity. We will also provide rst results and
conjectures on the integrality, irreducibility and maximal eigenvalue of the Killing
form matrix, collected in Section 4.
We also explore some ideas relating to the open problem of associating an irrep to
a conjugacy class. The underlying idea is that for a simple Lie algebra the adjoint
representation is irreducible, so if a conjugacy class is like a `Lie algebra' we might
expect it to principally consist of an irrep when the group is simple or close to it
(such as the symmetric group). We are thus motivated to speculate that for Sn and
all simple groups where the Roth property holds one can bijectively associate irreps
to conjugacy classes containing them. We illustrate this in Section 4 for A5 and the
Mathieu group M11. In Section 5 we look in detail at Sn, where we rapidly establish
that such a bijection, if it exists, is not any obvious one such as taking the Young

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 3
diagram associated to a conjugacy class and constructing a Specht module from the
transposed diagram. However in Section 5.1 we nd that the related question of
which conjugacy classes contain the sign representation has a very natural answer.
We note that a rst attempt at a geometrical association between conjugacy classes
and irreps via the dierential calculus was made in [10] and was equally tentative.
An alternative strategy suggested by our positive-deniteness conjecture is to x
the noncommutative dierential geometry as dened by such conjugacy class and
to then view dierent conjugacy classes as leading to irreps in a less direct way as
some kind of `orbit method'.
Finally, Section 6 collects the computer data testing the ideas and conjectures in
earlier sections on nonabelian nite simple groups up to order 75,000.
2. Preliminaries on noncommutative differential calculi
In this section we introduce the basic elements of noncommutative dierential ge-
ometry of nite groups G in the Hopf algebraic approach. It is quite important
that these constructions are not invented in an ad-hoc manner just for groups but
that they are part of a general quantum groups approach to noncommutative dif-
ferential geometry. Suce it to say on that front that dierential structures over
any unital algebra A are expressed as a specication of the space
1 of 1-forms as
an A A-bimodule equipped with a dierential d : A !
1 obeying the Leibniz
rule. We also require that the map A
A!
1 sending a
b 7! adb is surjective
and, optionally (one says that the calculus is connected) that ker(d) is spanned by
1. When A is a Hopf algebra or `quantum group' we can require the calculus to be
covariant under left or right translation, or both. In the latter case one says that
the calculus is bicovariant. One knows that a (say) left-covariant calculus is a free
module over its space 1 of left-invariant dierentials. In the bicovariant case there
is a canonical extension to a full exterior algebra (
;d) or `noncommutative de
Rahm complex'. Every unital algebra has a universal dierential calculus dened
as
1 = ker(
: A
A ! A) and da = 1
a a
1. In the Hopf algebra case it
is automatically bicovariant. Any other calculus is isomorphic to a quotient of the
universal one.
Specialising these ideas, dierential calculi on nite sets are in correspondence with
directed graphs [1]. In the case of a nite group G the left-covariant calculi are in
correspondence with Cayley graphs dened by non-empty C  G with e =2 C (e the
group identity). Here the edges are of the form x! xa for x 2 G and a 2 C. Such
a calculus is also right covariant (hence `bicovariant') i C is ad-stable. The basis
of invariant 1-forms is fea j a 2 Cg and

= C(G):;  = n; 0 = C; 1 = spanfeag;
df =
X
a
(@af)ea; eaf = Ra(f)ea
where Ra(f) = f(( )a) and @a = Ra id are right translation and `left-invariant de-
rivative' respectively. Explicitly, ea =
P
x xdxa in terms of Kronecker -functions
at x 2 G. The higher  are determined in the simplest `Woronowicz' formulation
by skew-symmetrization with respect to a braiding operator
(ea
eb) = eaba1
ea:

4 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
The calculus is connected i.e. has H0 = C:1 i the graph is connected, which is i
C is a generating set. We refer to [8] for an introduction to the theory.
A nontrivial application of this theory to nite groups appeared in [10] where the
exterior algebra was linked to the Fomin-Kirillov algebra arising in the study of
the cohomology of
ag varieties. The theme there is a kind of extension of Schur-
Weyl duality to a duality between the geometry of the
ag variety and that of the
associated Weyl group.
Proposition 2.1. A nite group G is simple i all its bicovariant calculi are con-
nected.
Proof. Suppose that G is simple and C an ad-stable subset (dening a bicovariant
calculus). Let N = hCi the subgroup generated by C. This is clearly normal
and contains more than e (as C is nonempty), hence N = G and the calculus is
connected. Conversely, suppose that all nonempty ad-stable subsets C generate G.
Let N  G be normal and C = N n feg. This is an ad-stable subset and hCi = N
as N 6= feg is a normal subgroup, hence N = G. 
Next, in the case of nite sets, a morphism  :
1 !
10 between two dierential
calculi (a bimodule map forming a commutative triangle with the corresponding
d;d0) is fully determined if it exists. This is because (xdy) = xd0y and hence
sends an edge of the graph of
1 to the same edge in the graph 0 of
10 if this
edge exists, or else to zero. Hence such a map is a surjection i 0  . Because
of this, we see that C being a conjugacy class corresponds to
1 having no proper
quotients.
In order to develop a Lie theory, however, we need something slightly dierent.
Namely, we impose connectedness from the start and in this case the `minimal'
dierential calculus with no connected proper quotients is not necessarily given by
a conjugacy class, for example, if the group is abelian. Dually, C could be required
to be minimal but still generating.
Denition 2.2. A minimal Lie structure on a nite group G means a minimal
ad-stable generating subset C  G.
A minimal Lie structure always exists, namely one can start with C = Gnfeg (which
denes the universal calculus), this is always ad-stable and clearly generates G. We
then remove elements of C if possible until further removal is no longer possible. It
does not contain e as this could be removed and hence C would not be minimal.
Proposition 2.3. A nite group is simple i every nontrivial conjugacy class is a
minimal Lie structure. In this case the two notions are equivalent.
Proof. Suppose that G is simple and C a nontrivial conjugacy class. As hCi is
normal and nontrivial we see, as above, that C generates G. Suppose that C is
not minimal, so there exists a proper ad-stable generating subset C0  C which,
in particular, contradicts C a conjugacy class. Conversely, suppose that G is a
nite group such that every nontrivial conjugacy class is a minimal Lie structure.
Let N  G be a nontrivial normal subgroup and C = N n feg, which is therefore
ad-stable. Decompose this into nontrivial conjugacy classes and let C0 be one of

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 5
these. Then by assumption C0 is a minimal Lie structure on G, hence G = hC0i.
But hC0i  N as N was a subgroup, hence G  N and hence N = G. Hence G is
simple. Finally, suppose that G is simple and C a minimal Lie structure. If C0  C is
a proper ad-stable subset then (as above) N = hC0i is a nontrivial normal subgroup
and hence C0 generates G, which contradicts minimality. Hence C is a conjugacy
class, nontrivial as does not contain e. 
2.1. Braided Lie algebras and Killing form. In geometry the dual of the space
1 of left-invariant dierential 1-forms on a Lie group has the structure of a Lie
algebra. The corresponding notion in the quantum groups approach to noncom-
mutative dierential geometry is that of a `braided Lie algebra' [9]. This is dened
abstractly as an object L in a braided monoidal category equipped with morphisms
[ ; ] : L
L ! L (the `Lie bracket') and
: L ! L
L,  : L ! 1 forming a
coalgebra in the category (where 1 is the trivial object), subject to several axioms.
When the category is abelian, there is a certain quadratic `enveloping algebra' U(L)
associated to the braided Lie algebra and forming a bialgebra in the category. Of
further relevance to us is a `braided Killing form' dened as the braided or categor-
ical trace of two application of [ ; ]. The theory was invented particularly to solve
the problem of what is the `Lie object' underlying the standard Drinfeld-Jimbo
quantum enveloping algebras Uq(g). It is shown in [9] that at least for generic q
there is a braided Lie algebra L  Uq(g) and an algebra surjection U(L)! Uq(g).
In the classical limit L  U(g) where L = C1g recovers the classical Lie algebra in
an extended form where [1; ] = id and [ ; 1] = 0. An introduction to the theory is in
[8]. This theory was connected to noncommutative dierential geometry in [11, 6],
where the general theorem is that for any `inner' bicovariant dierential calculus
1
on a coquasitriangular Hopf algebra, there is a braided Lie algebra L isomorphic as
an object in a braided category to 1. This is the case for all standard quantum
group coordinate algebras Cq[G] and completes the picture above. The term `inner'
refers to a bi-invariant 1-form  that generates the exterior derivative in the form
d = [; ], a concept that has no analogue in classical dierential geometry. One
could view the classical limit L = C1 g as corresponding to a nonstandard higher
order dierential structure possible within noncommutative geometry.
Now let
1(G) be a bicovariant dierential calculus dened by an ad-stable subset
C and with space of left-invariant 1-forms 1. Let fxag be dual basis to the classes
of feag providing the basis of 1. The bi-invariant element  =
P
a2C ea makes
the calculus inner and the associated `braided Lie bracket' on L = 1 = CC is
[xa; xb] = xaba1 . The coproduct and counit take the form
xa = xa
xa, (xa) = 1
and the category is that of G-modules with trivial braiding, the usual `
ip'. The
associated `enveloping algebra' U(L) is a quadratic algebra with generators fxag
and relations xaxb = xaba1xa. This comes equipped with a canonical homomor-
phism U(L) ! CG to the group algebra, dened by xa 7! a. In our case because
the underlying braiding is trivial the braided trace becomes the usual trace. Then
the Killing form is
(2.1) K(xa; xb) = TraceL([xa; [xb; ]]) = jZ(ab) \ Cj = L(ab)
where Z(g) is the centraliser of g 2 G and L is the character of the conjugation
representation on L. Clearly K is ad-invariant since C is and is -symmetric, hence

6 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
^(K) = 0. It is also actually symmetric since L(ba) = L(a(ba)a1) = L(ab)
since L is a class function. We are interested in when K is nondegenerate.
Lemma 2.4. If C \ (C:c) 6= ; for some nontrivial c 2 Z(G) then K is degenerate.
In particular, if jZ(G) \ Cj > 1 then K is degenerate.
Proof. Looking at K as a matrix with rows and columns labelled by C. If b = b0c
where b; b0 2 C and c 2 Z(G) n feg then K(xa; xb) = jZ(ab) \ Cj = jZ(abc) \ Cj =
K(xa; xb0) for all a 2 C, hence K has a repeated column. If b; b0 2 Z(G) \ C are
distinct then c = b01b ts the rst part. 
Corollary 2.5. If jGj > 2 and jZ(G)j > 1 then the Killing form for the universal
calculus is degenerate.
Proof. For the universal calculus C = G n feg and in the preceding lemma we can
take any nontrivial c 2 Z(G), any b0 6= c1; e and b = b0c. Then b 2 C \ (Cc). 
If G has order 2 then K is a 1 1 matrix and is nondegenerate. We therefore only
need to investigate the universal calculus in the case where jGj > 2 and Z(G) = feg,
for example when G is simple and nonabelian.
3. Nondegeneracy for the universal calculus when Roth's property
holds
Let G be a nite group. We are going to work from the expression (2.1) derived
above, and we note rst that this formulation actually makes sense for a `Killing
form' similarly dened for any representation W and on all of CG,
KW (xa; xb) := W (ab)
where fxa j a 2 Gg is a basis of CG. It is well-known that such a symmetric
bilinear form is nondegenerate if and only if W contains every irrep of G with
positive multiplicity. This follows from semisimplicity of the group algebra CG and
general facts about semisimple algebras. (Namely, if an algebra is semisimple then
it is a direct sum of matrix blocks. If W = iniVi for some multiplicities ni then
KW has a block form with ni times the Euclidean inner product on each matrix
block. This is because an element in a matrix block corresponding to an irrep acts
as zero by left multiplication on any other block and hence in any other irrep. Hence
KW is nondegenerate i all the ni > 0). The case W = CG with the conjugation
representation is therefore nondegenerate i the conjugation representation contains
every irrep. If Z(G) is trivial then this is the Roth property. For the case of the
universal calculus we are interested in the caseW = C:(Gnfeg) where we remove the
group identity. We are also interested in restricting the Killing form to C:(G n feg)
but we defer this for the moment.
Lemma 3.1. Let G be a nite group. Suppose CG under conjugation contains every
irreducible of G with strictly positive multiplicity. Then so does W = C:(G n feg).
Proof. W spanned by set G n feg still contains a copy of the trivial representation
since this set is permuted by conjugation and hence the element  =
P
a xa, where
we sum over a 2 G n feg, is invariant. As CG = W  C:e as a G-module, if any
nontrivial representation is contained in CG then it must also be in W . 

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 7
Hence if the Roth property holds for G then it also holds for our particular W of
interest. Let n be the number of distinct irreducible representations of G, and call
these representations V1; : : : ; Vn.
Proposition 3.2. Let G be a nite group and W be a representation of G for which
every irreducible representation Vi appears in W with strictly positive multiplicity.
Then the restriction of KW to C:(G n feg) is nondegenerate.
Proof. Since the form KW is nondegenerate on CG we know that (C:(G n feg))? is
one-dimensional. We will know that KW is nondegenerate in C:(G n feg) if there
is no element in the perpendicular which also lies C:(G n feg). We prove this by
determining explicitly a vector spanning the line (C:(Gnfeg))?, and observing that
it doesn't lie in C:(G n feg).
Dene
m =
X
g2G

nX
i=1
dim(Vi)2
hVi ; W i
Vi(g)
!
xg
This is well-dened since hVi ; W i 6= 0 for all i by our assumption.
We claim that KW (xa;m) = 0 for all a 6= e. Note that the coecient of xe in m is
given by the formula
me =
nX
i=1
dim(Vi)3
hVi ; W i
;
so is always strictly positive. Therefore m does not lie in C:(G n feg) and the claim
will imply the proposition.
We use the following standard orthogonality formulas:
(3.1)
X
g2G
V (g) V 0(ga) =
(
0 if V; V 0 are distinct irreps,
jGj
dimV V (a) otherwise;
and
(3.2)
X
i
Vi(a1)Vi(a2) = 0 if a1 and a2 are not conjugate.
Note also that
W (g) =
X
i
hVi ; W iVi(g):

8 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
Now we can compute, extending linearly,
KW (m;xa) = KW
0
@
X
g2G

nX
i=1
dim(Vi)2
hVi ; W i
Vi(g)
!
xg; xa
1
A
=
X
g2G

nX
i=1
dim(Vi)2
hVi ; W i
Vi(g)
!
W (ga) =
nX
i=1
dim(Vi)2
hVi ; W i
0
@
X
g2G
Vi(g)W (ga)
1
A
=
nX
i=1
dim(Vi)
2
0
@
X
g2G
Vi(g)Vi(ga)
1
A = jGj
X
i
dim(Vi)Vi(a)
= jGj
X
i
Vi(e)Vi(a):
The last expression vanishes whenever a 6= e by (3.2). 
Corollary 3.3. Let G be a nite group for which the Roth property holds. Then
the Killing form for the universal dierential calculus on C(G) is nondegenerate.
As explained in the introduction, the Roth property holds for most simple non-
abelian groups including all the sporadic ones, see [7]. Meanwhile, the simplest
example where the Roth property holds is G = S3, the group of permutations
on three elements. This is elementary enough that we can, instructively, compute
everything in our characters approach by hand. Here Z(G) is trivial.
Example 3.4. Let G = S3. The Killing form on CG is
KW =
0
B
B
B
B
B
B
@
5 1 1 1 2 2
1 5 2 2 1 1
1 2 5 2 1 1
1 2 2 5 1 1
2 1 1 1 2 5
2 1 1 1 5 2
1
C
C
C
C
C
C
A
in a basis e; u = (12); v = (23); w = (13) = uvu; uv = (123) and vu = (132). We
obtained by writing out the group product table and evaluated W (g) = jZ(g)j 1
for each entry g 2 S3. Here
W (e) = 5; W (u) = W (v) = W (w) = 1; W (uv) = W (vu) = 2:
One can then see by computation that the lower right 55 block in KW is invertible
as required by the Proposition. To see how this arises from the character theory in
the proposition, we look at the trivial representation Vtriv, the sign (parity function)
representation Vsgn and the standard 2-dimensional representation, denoted V
.
Their character values tabulated on conjugacy classes as for W are
triv = 1; 1; 1
sgn = 1;1; 1

= 2; 0; 1
from which we see that W = 2Vtriv  Vsgn  V
. Thus the `Roth property' holds
for W as in the Lemma above, hence KW on CG is nondegenerate. We also view
this last table as the change of basis with the characteristic class functions 0 with

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 9
support feg, 1 with support fu; v; wg and 2 with support fuv; vug. Now, by
direct computation it is clear that ~m = 190 1 52 viewed as the coecients of
an element of CG lies in (C:(G n feg))? and hence spans this 1-dimensional space.
Using the table to convert back to the basis of characters, we have obtained `by
hand' an element ~m = triv + 2sgn + 8
in the perpendicular complement. This
agrees up to normalisation with the canonical element constructed in the proof of
the proposition, namely
m =
1
2
triv + sgn + 4

where the coecient of V is dim(V )2=hV ; W i. Finally, ~m has by construction a
non-zero coecient of e, so ~m does not lie in C:(G n feg).
In fact we conjecture that the conclusion of Proposition 3.2 holds even when the
Roth property fails.
Conjecture 3.5. Let G be a nite simple nonabelian group and K the Killing
form for the universal dierential calculus. Then K is nondegenerate.
Experimental evidence that this holds for the nite simple groups beyond those
implied by Proposition 3.2 is limited as even the smallest of the simple groups such
as PSU(3; 4) where the Roth property is known to fail in [7] require too much
computer memory to check directly. However, we will have evidence for a more
general conjecture where the Roth property for the relevant representation and
hence Proposition 3.2 do not apply but where the Killing form is still nondegenerate.
In particular, we will see in Section 6 that nondegeneracy holds for all conjugacy
class dierential calculi on PSU(3; 4). Also, just as the Roth property holds more
generally than for certain simple groups, eg for S3 above, we similarly further
conjecture that the nondegeneracy may hold more generally for nite groups with
trivial centre.
Finally, we also conjecture nondegeneracy at the other extreme of `minimal' dier-
ential calculi. Note that for a simple group any conjugacy class will be generating
and dene a connected dierential calculus.
Conjecture 3.6. Let G be a nite simple nonabelian group and C  G n feg a
conjugacy class closed under inversion. Then the Killing form K on CC is nonde-
generate.
This is veried by computer in Section 5 for groups up to order 75,000. This in-
cludes the rst sporadic group M11 and the exceptional group of Lie type Sz8. The
requirement on having inverses is included since the experimental evidence shows
examples of conjugacy classes without this assumption and where K is degener-
ate, although those examples are scarce and generally one has nondegeneracy even
without requiring closure under inversion.
Conjecture 3.6 and our previous Conjecture 3.5 suggest further that nondegeneracy
holds for simple G and any ad-stable subset C  G n feg closed under inverses, but
we have no direct evidence for this. We also further conjecture that nondegeneracy
can hold more generally for groups which may have centre but where Z(G) \ C is
empty and C generates the group. For example, Sn with its 2-cycles conjugacy class

10 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
(which generates the group) has nondegenerate Killing form yet the analogue of the
`Roth property' clearly does not hold for W = CC as most irreps are not contained
in it. We will prove these assertions in Section 5, but for now we directly look at
the simplest case of S3 to illustrate why we require C to generate the group.
Example 3.7. For G = S3 and C the order 3 conjugacy class of 2-cycles (which
generates the group), the Killing form is nondegenerate. Here G acting by conju-
gation permutes the elements of C = fu; v; wg and hence W = Vtriv  V
. This
does not have the Roth-type property used in Proposition 3.2 and indeed on CG
we have W = 3; 1; 2 on the three classes and hence
KW =
0
B
B
B
B
B
B
@
3 1 1 1 0 0
1 3 0 0 1 1
1 0 3 0 1 1
1 0 0 3 1 1
0 1 1 1 0 3
0 1 1 1 3 0
1
C
C
C
C
C
C
A
in basis order e; u; v; w; uv; vu as before, which is degenerate. However, the 3  3
submatrix arising from its restriction to C is a multiple of the Euclidean inner
product [10] and is invertible. Incidentally, its restriction to the other class and to
the union of the two nontrivial classes is also nondegenerate.
Example 3.8. For G = S3 and C the order 2 conjugacy class of 3-cycles (which
does not generate the group), the Killing form is degenerate. Here each element
u; v; w acts to
ip the elements of C while uv; vu; e of course act trivially. Hence
W = Vtriv  Vsgn again does not have the Roth-type property. In this case W =
2; 0; 2 on the three classes. Hence
KW =
0
B
B
B
B
B
B
@
2 0 0 0 2 2
0 2 2 2 0 0
0 2 2 2 0 0
0 2 2 2 0 0
2 0 0 0 2 2
2 0 0 0 2 2
1
C
C
C
C
C
C
A
which is degenerate. Its 2  2 submatrix corresponding to the restriction to C is
also degenerate (as is its restriction to the other nontrivial conjugacy class and to
the sum of the two classes).
4. Eigenvalues of the Killing form operator associated to a
conjugacy class
We now look at some general results about the Killing form of particular relevance
to the case of a conjugacy class dierential calculus. Let G be a a nite group and C
an ad-stable subset of Gnfeg. We note that the `Euclidean metric' (xa; xb) = a;b
given by the Kronecker delta-function in basis C is also ad-invariant. We regard
both ;K as maps CC ! (CC) and hence the composite 1 K, which we also
denote K, is a map CC ! CC. Here explicitly
K(xa) =
X
b2C
K(xa; xb)xb:

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 11
By construction this is also ad-invariant. Hence its eigenspaces provide a natural
decomposition of CC into subrepresentations. Note that as K is real and symmetric
in our basis it has a full diagonalisation with real eigenvalues over R. However, it
can also be viewed as a hermitian matrix or self-adjoint operator over C. Also the
entries of K are non-negative integers.
Proposition 4.1. Suppose V is an irreducible representation of CG which is de-
ned over Q. So V = VQ
QC for an irreducible representation VQ of QG. Further-
more, suppose C is a conjugacy class in G such that V occurs in the conjugation
representation CC.
If the isotypical component of V in CC is contained in a single eigenspace of the
Killing matrix K of C, then the corresponding eigenvalue lies in Z.
Proof. Choose an element x 2 C and consider the map
 : CG! CC : g 7! gxg1
which is a G-equivariant surjection from the left-regular representation to the con-
jugation representation of G. Let AV  CG denote the block of the irreducible
representation V . By block decomposition of CG and Schur's lemma it follows
that  restricts to a surjection from AV to the isotypical component of V in CC.
However, since V was dened over Q it follows that AV has a basis that lies inside
QG. Moreover by surjectivity of jAV onto the isotypical component there exists
such a basis element b whose image (b) is nonzero. By the assumptions, b is an
eigenvector of the Killing matrix. Moreover b has rational coecients as a vector
in CC. Since the entries of K are integral and b is rational it follows that the
eigenvalue of b lies in Q. On the other hand the integrality of K implies that the
eigenvalues are all algebraic integers. So the eigenvalue of b is a rational number
and an algebraic integer. Therefore it must lie in Z. 
Note that this proposition implies in particular, that if V is a complex representa-
tion of G dened over Q which occurs in CC with multiplicity 1, then it lies in an
eigenspace of K with eigenvalue in Z. Since all representations of the symmetric
group are dened over Q (over Z even), we have the following corollary.
Corollary 4.2. Let C be a nontrivial conjugacy class of Sn. If an irreducible
representation of Sn occurs in the conjugation representation CC with multiplicity
one, then it embeds into an eigenspace for the corresponding Killing form with
eigenvalue in Z. 
Note that an irreducible representation is rational if all its character values are
rational. This is because the matrix entries can be obtained by projection via
central idempotents in the group algebra with coecients dened by the characters.
Similarly the character determines if an irreducible representation is complex in the
sense of not real.
Proposition 4.3. Let C  G n feg be an ad-stable subset.
(1) If a complex irreducible representation V occurs in CC in an eigenspace
of the associated Killing form matrix, then so does its dual representation
(with complex conjugate character).

12 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
(2) If we consider the inverse conjugacy class C1 then the eigenvalues of the
Killing form matrix for C1 are the same as the ones obtained for C, and
the decompositions of the respective eigenspaces into irreps are isomorphic.
Proof. Let C be an ad-stable subset. The conjugation representation is clearly
dened over R by working in the basis C (indeed, over Z). Since K is real and
symmetric in the basis C its eigenspaces are also dened over R, and hence real as
subrepresentations of the conjugation representations. This implies the rst part.
For the second part we consider inversion as a bijection between the two ad-stable
subsets. Let a; b; c 2 C. Clearly c commutes with ab i c1 commutes with b1a1.
But as the Killing forms are symmetric, we see that the Killing forms have the
same matrices in their respective bases. If v 2 CC is expanded in the basis C we
dene ~v to be the corresponding vector in CC1 with the same coecients in the
corresponding basis, i.e. v; ~v are represented by the same column vector in their
respective bases. One may readily see that the matrices for the action of an element
of g in the two cases are also identical. This implies the second part. 
One can extend the rst part to a general Galois theory argument that if an ir-
rational representation occurs then so do its various `conjugates', with the corre-
sponding change of Galois root in the eigenvalue (we will see this for A5 below).
Next, we look at the trivial representation in CC canonically present as spanned by
the sum of the elements of C. By a slight abuse of notation we denote this element
by  (its analogue as a left-invariant 1-form makes the calculus inner). We recall
that a matrix with non-negative entries is called irreducible if for all indices i; j
there exists m 2 N such that the matrix entry (Km)ij 6= 0. This is equivalent to
connectedness of the graph on the set of indices dened by an edge whenever the
entry Kij 6= 0.
Proposition 4.4. Let G be a nite group and C  Gnfeg a conjugacy class. Then
K has a (positive) integral maximal eigenvalue c, given by the sum of any row of
K. Moreover, K splits onto r irreducible components i the eigenspace associated
to c has dimension r and in this case all other eigenspace dimensions are divisible
by r. In particular, if K is irreducible then the eigenspace associated to c is 1-
dimensional, generated by the eigenvector  =
P
a xa.
Proof. K() =
P
a;bK(xa; xb)xb =
P
b cbxb where cb is the sum of the b'th column
of the matrix of K. However, cgbg1 =
P
aK(xa; xgbg1) =
P
aK(xg1ag; xb) = cb
after a change of variables. Hence cb is independent of b 2 C in the case of a
conjugacy class. Hence  is an eigenvector ofK with eigenvalue as stated. Moreover,
ifK is irreducible then by Perron-Frobenius theory there is a 1-dimensional maximal
eigenspace with eigenvalue the column sum of K, i.e. with eigenvector . If K is not
irreducible then after a reordering of the basis it can be presented as a direct sum.
Iterating this, we reduce K to a direct sum of some number r > 1 of irreducible
blocks. In fact each block will be, after reordering, a copy of the same irreducible
matrix. This follows from ad-invariance of K as follows. Consider an element in G
that conjugates a corner of the rst block to the corresponding corner of another.
All the indices relating to the rst block belong to the same connected component
of the graph and, by assumption, they are not connected to any of the indices

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 13
for the other blocks, and this notion is ad-invariant, as K is. Hence the indices
relating to the conjugated rst block must be connected to themselves and not to
the rst block. Hence the rst block maps over to the conjugated block, and all its
entries are the same when suitably ordered, again by ad-invariance of K. Once K
has been presented as r blocks Ki, its eigenvectors will consist of r parts forming
eigenvectors for each block with the same eigenvalue. However, since these blocks
are all irreducible and have the same row sum as K, they will each have the same
maximal eigenvalue as K, and any other eigenvalues will be strictly lower. This
implies the facts stated. 
Note that the diagonal of K is always non-zero as a commutes with a2 for all a 2 C.
Hence Km+1 can only have the same or more positive entries as Km, so in our case
irreducible is equivalent to the existence of m 2 N such that all entries of Km are
positive, i.e. the primitivity of the matrix K.
Conjecture 4.5. Let G be a nite nonabelian simple group and C  G n feg a
conjugacy class not consisting of involutions. Then the associated K is irreducible.
This is surmised by looking at nite simple groups up to order 75,000. The only
observed reducible cases are the classes of involutions for G = PSL(2; 2k), G =
PSU(3; 2k) or G = Suz(22k1) for k  2 up to the order that we could check.
These are all groups of Lie type over nite elds of characteristic 2 and we further
conjecture that this is always the case. Results for Sn will be discussed in Section 5.
Next, we consider the question: for which conjugacy classes is K not only nonde-
generate but positive denite? We have explained in the introduction that if CC is
some kind of braided Lie algebra we might expect a simple group to have a unique
`compact real form', i.e. a choice of CC with K positive denite. Whilst this is not
generally true, we can conjecture a close enough result:
Conjecture 4.6. Let G be a nite nonabelian simple group and C  G n feg a
conjugacy class closed under inverses. If C consists of involutions, then the Killing
form is positive denite. Otherwise the Killing form has zero signature (i.e. the
same number of positive and negative eigenvalues counted with multiplicities).
This is supported by computer verication to order 75,000. Note that by the Feit-
Thompson theorem every (nonabelian) nite simple group must contain involutions,
so a conjugacy class in the terms of the conjecture will always exist. Uniqueness
is not true in general, since various simple groups (eg. the altenating groups An
for n  8) contain several dierent conjugacy classes of involutions. There are also
some examples, see the conjugacy classes 3A and 3B in PSU(4; 2) in Section 6,
that yield positive denite Killing forms without being closed under inverses. We
also expect a behaviour similar to the conjectured one to hold for groups such as
Sn with C generating, see the next section.
The second part of Conjecture 4.6 is not supported by the analogy but it perhaps
reminiscent of the split real form in Lie theory. Finally, bearing in mind that a
braided-Lie algebra in the case of a classical Lie group has the form L = C1 g 
U(g) where g is in the adjoint representation and is irreducible when the Lie group
is simple, we might hope that CC will decompose in a similar way and that this
might produce an irreducible representation analogous to g after removal of the

14 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
C  A4 jCj Decomposition (eigenvalue) Suggested irrep
(12)(34) 3 1(9) 1(0) 1(0) 1; 1
(123) 4 1(4) 3(0) 3
(124) 4 1(4) 3(0) 3
C  A5 jCj Decomposition (eigenvalue) Suggested irrep
(12)(34) 15 1(21) 4(21) 5(12) 5(12) 4
(123) 20 1(34) 4(24) 5(18) 4(12) 3(22) 3(22) 5
(12345) 12 1(24) 5(12) 3(10 + 2
p
5) 3(10 2
p
5) 3; 3
(12354) 12 1(24) 5(12) 3(10 + 2
p
5) 3(10 2
p
5) 3; 3
Table 1. Decomposition of span of conjugacy classes into irreps
(eigenvalue of Killing form in brackets), for A4; A5. Here irreps are
labelled by their dimensions and other symbols, with a bar for the
conjugate representation.
trivial representation. This is not exactly what happens but we propose to use
the Killing form to pick out an irreducible component in CC, and we shall explore
this for Sn with its 2-cycles class in the next section. By thinking of any conjugacy
class as a braided-Lie algebra and using the Killing form we might hope to associate
irreps to conjugacy classes in a natural way. This is not so much a matter of precise
conjectures at the moment but a programme in the form of a couple of `principles'
which we propose.
Principle 4.7. (Nondegeneracy principle.) For large nite nonabelian simple
groups G the decomposition into eigenspaces of K is typically into irreps or into
complex conjugate pairs of irreps.
This is certainly not always true but we shall see how it works for the M11 group.
Note that the strong form stated implies that where an irrep occurs with multi-
plicity then these copies are generally distinct as well, i.e. K typically resolves or
`separates' the isotypical components in the process.
Principle 4.8. (Correspondence principle.) Let G be a nite nonabelian simple
group for which Roth's property holds. There is often a `reasonable' way up to
choice of Galois and complex conjugates to bijectively assign nontrivial irreps to
non-trivial conjugacy classes containing them in the conjugation representation, in
such a way that if a complex irrep is assigned to a conjugacy class not stable under
inversion then its complex conjugate is assigned to the inverse class, and where
possible the assigned irrep occurs without multiplicity.
We certainly nd examples where avoiding multiplicities is not possible, and in this
case a weaker requirement is to have a distinct eigenvalue of K for the chosen irrep
or complex conjugate pair of irreps.
Example 4.9. Table 1 provides a rst impression by looking at A5. We used
Sage and some deductive reasoning to determine the decomposition of conjugacy
classes into irreps and their eigenvalues. There are some accidental degenerations,
one of them because K is reducible, but otherwise the eigenspaces of K are irreps
or sums of `conjugate' irreps (though not in this case complex conjugate). And

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 15
in the last column we have also selected an irrep to associate to each conjugacy
class in such a way that there is a 1-1 correspondence. This is a more or less
unique assignment if we want the 5 and the 4 to enter in their respective conjugacy
classes without multiplicity and if we intuitively require the `conjugate' pair of 5-
cycle classes to map to the `Galois conjugate' pair of irreps (which of the pair of
classes is assigned to which of 3; 3 is a residual ambiguity). The eigenvalues have
been deduced from the knowledge (see above) that the trivial representation has
maximal eigenvalue followed by comparison of dimensions between eigenspaces and
irreps. This method does not fully identify the eigenvalue when distinct irreps with
the same dimension occur, but in the case of 3; 3 this is a matter of denition as to
which is called which. However, having dened one as 3 one may verify by explicit
calculation that it appears in the last line of the table with the other eigenvalue. The
eigenvalues in these classes are not integer and indeed the eigenspaces are examples
of real irrational representations (however, when they both appear with the same
eigenvalue then this is an integer). Also note that there is a unique conjugacy class
where K is positive denite and the equal dimensions of the combined positive and
negative eigenspaces in each of the other cases. We contrast both features with A4,
also in the table, where aside from the trivial representations all eigenvalues are
zero. Note that A4 is not simple and indeed there is no way to associate nontrivial
irreps and nontrivial conjugacy classes containing them.
Example 4.10. Table 2 illustrates that the same ideas hold in the simplest sporadic
group, M11, again using Sage and deductive reasoning to construct the table. Here
again we nd for the most part that the eigenspace decomposition is generally a
decomposition into irreps or conjugate pairs of irreps (the sole exception is the
10,11 irreps in the class 2A). We use ATLAS labelling of conjugacy classes as in
Section 6. We also nd a more or less unique 1-1 correspondence between conjugacy
classes and irreps as shown, as follows. We see as explained above that inverse pairs
of conjugacy classes have the same spectrum and for any complex conjugate pair
of irreps if one occurs in a conjugacy then so must the other and with the same
eigenvalue. In view of this, inverse pairs of classes are naturally assigned to a pair
of complex conjugate irreps as shown (which of each pair goes to which is then
largely a matter of denition). This forces us to assign 10; 10 as shown as it
does not occur in the other inverse pair, and hence 16; 16 to that. Both then occur
without multiplicity. Of the remaining nontrivial irreps the only one which occurs
without multiplicity in the 2nd row of the table is 11, so we are led to assign that.
Similarly in the class 3A, only the 10 occurs without multiplicity so we assign that.
Of the remaining classes only the 55 appears in the class 2A without multiplicity,
so we assign that. However, the assignment of the 44 and 45 is not determined by
these arguments and this represents the residual ambiguity. They both occur with
multiplicity in the remaining two conjugacy classes. One of these, the 5A, has no
nontrivial irreps occurring with integer eigenvalues at all, so this provides no guide
either. Note also the unique conjugacy class 2A that has positive denite K and
the equal dimensions of the combined positive and negative eisgenspaces in each
of the other classes, other than the mutually inverse pair 8A; 8B and the mutually
inverse pair 11A; 11B.
Table 2 also shows the need for a more compact account of the Killing form eigen-
values and we do this in Figure 1 in the form of a spectrogram. This shows all the

16 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
C M11 jCj Decomposition (eigenvalue)
Suggested
irrep
2A 165 1(489) 44(223:) 10(192) 11(192) 55(136) 44(122:) 55
3A 440
1(946) 11(609:) 10(572) 44(484:) 44(470:)
55(428) 11(404:) 44(344:) 45(413:) 10(420)
 10(420) 55(426) 55(444) 45(448:)
10
4A 990
1(2108) 10(1195:) 44(1047:) 45(1043:) 44(1013:)
11(1010) 55(996:) 44(988:) 16(986) 16(986)
45(982:) 55(959:) 55(952) 44(949:) 10(918:)
44(910:) 55(933:) 55(940:) 45(977:) 55(984:)
44(987:) 45(1004:) 16(1014) 16(1014)
55(1025:) 10(1026) 10(1026) 45(1062:)
11
5A 1584
1(3096) 11(1828:) 45(1712:) 55(1686:) 44(1685:)
10(1673:) 11(1649:) 44(1628:) 55(1588:) 45(1586:)
16(1575:) 16(1575:) 55(1574:) 44(1570:) 44(1556:)
44(1544:) 11(1537:) 55(1534:) 16(1532:) 16(1532:)
10(1526:) 55(1515:) 44(1506:) 45(1496:) 45(1493:)
55(1508:) 10(1527:) 10(1527:) 55(1534:)
16(1542:) 16(1542:) 45(1548:) 44(1549:)
55(1562:) 45(1567:) 55(1572:) 16(1573:)
 16(1573:) 45(1574:) 10(1576:) 10(1576:)
45(1591:) 55(1617:) 44(1622:) 45(1638:)
55(1775:)
44; 45
6A 1380
1(2568) 44(1532:) 11(1465:) 44(1380:) 10(1366:)
44(1359:) 55(1335:) 44(1324:) 16(1320) 16(1320)
11(1315:) 10(1313:) 55(1305:) 44(1298:) 45(1298)
55(1284) 55(1276) 55(1276:) 45(1256) 44(1251:)
11(1226:) 55(1173:) 45(1214:) 55(1259:)
10(1274:) 10(1274:) 45(1285:) 45(1299:)
55(1322:) 45(1323:) 44(1326) 16(1336)
 16(1336) 10(1357:) 10(1357:) 45(1365:)
44(1376) 55(1380:) 55(1425:)
45; 44
8A
8B
g 990
1(920) 10(106:) 44(103:) 55(57:) 16(43:) 16(43:)
45(31:) 55(28:) 44(25:) 55(24:) 44(20:) 10(14)
 10(14) 45(13:) 55(5:6) 10(3:1) 44(0:6) 55(8:)
11(10) 45(14:) 16(21:) 16(21:) 55(28:)
44(31:) 45(35:) 44(54:) 55(64:) 45(94:)
10; 10
11A
11B
g 720
1(575) 45(96:) 44(79:) 55(75:) 11(35) 55(29:) 44(21:)
45(11:) 55(0:7) 45(7:3) 45(7:6) 16(10) 16(10)
55(16:) 44(43:) 45(48:) 55(64:) 44(67:)
16; 16
Table 2. Decomposition of span of conjugacy classes into irreps
(eigenvalue of Killing form in brackets), for Matheiu group M11.
Classes are labelled by the order of elements and other symbols.
Irreps are labelled by their dimensions and other symbols, with a
bar for the conjugate representation. Non-integer eigenvalues are
irrational and shown truncated with a period.

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 17
eigenvalues of the irreps occurring in the various conjugacy classes together as an
overall picture or bar-code of the group, as well as an example of the contribution
from just one class. Complex conjugates have the same spectrum so we do not list
them and the thickness of lines indicates a preponderance of nearby eigenvalues.
The combined spectrum is that of the direct sum D of all the dierent K acting on
a Hilbert space H which is the direct sum of all the spans of conjugacy classes. The
latter if we include the span of feg carries a faithful representation of the algebra
of functions C(G) and the triple (C(G);H; D) can be viewed as a discrete `spectral
triple' cf. [1] although obeying a more general set of axioms. This is a direction to
be explored more fully in a sequel. One may compute
[f;D] =
X
g2Gnfeg
eg@
gf; eg(xa) :=
jCj
jGj
K(xa; xgag1)xgag1 ; fxa := f(a)xa
for all f 2 C(G) and a 2 C. Here
(@gf)(a) := f(gag1) f(a)
denes `vector elds' for the conjugation action. We also have feg = egf(g( )g1)
for the noncommutativity of 1-forms and functions. In this sense [f;D] then has
the form of an exterior dierential with vielbein lengths given by K. The jCj=jGj
weighting is to adjust for overcounting by the order of the isotropy group and could
be omitted.
Finally, for completeness we mention that a more usual application of the Killing
form in Lie theory is to the construction of the quadratic Casimir as a central
element in the enveloping algebra. We can indeed do something similar whenever
K is nondegenerate.
Denition 4.11. When the Killing form K is nondegenerate we let Kab be its in-
verse matrix in basis C. Then the quadratic Casimir is dened as ~C =
P
a;bK
abxaxb 2
U(L). Its image in the group algebra C =
P
a;bK
abab is also called the Casimir
element.
Unlike K itself, C denes an operator not only on CC but on every representation.
We can of course use its values to induce a labelling of irreducibles. In the case of
Sn this is hardly necessary but could be useful for other nite groups equipped with
nondegenerate Killing form, much as in Lie theory. On the other hand we can also
decompose C in terms of other ` elements' consisting of sums over other conjugacy
classes. So this coordinatisation is a change of basis from using the various  for
dierent conjugacy classes, i.e. from using the so-called `central characters' for the
pairing of an irrep with a conjugacy class. Here the value of C =
P
a2C xa in an
irrep V is given by
C jV = jCj
V (a)
V (e)
; a 2 C:
However, more geometrically, the action of the quadratic Casimir is the Laplace-
Beltrami operator and this is analogous to the ability to view a classical quadratic
dierential as a `linear' dierential with respect to a dierent (noncommutative)
dierential calculus.

18 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
Figure 1. Killing forms spectrogram of Mathieu group M11 and
below it of sample conjugacy class 5A.
Example 4.12. For A5 with its 2-2-cycles calculus we have K reducible into 5
identical 3 3 blocks Ki in a suitable basis order, with inverse of each block
K1i =
1
84
0
@
6 11
1 6 1
11 6
1
A :
From this one nds
C =
15
14
e
1
42
(12)(34)
involving the same conjugacy class as dening the calculus. By contrast, for S4
with its 2-cycles dierential calculus we follow the above steps starting with the
Killing form. This is also reducible and consists of 3 identical 2  2 blocks in a

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 19
suitable basis order with
K1i =
1
16

3 1
1 3

and resulting quadratic Casimir
C =
9
8
e
1
8
(12)(34):
This is a combination of e and the  for the 2-2-cycles conjugacy class and is more
manifestly `quadratic' for our 2-cycles calculus as expected for the natural Laplace-
Beltrami operator on the nite group.
5. The Killing form and conjugation representation for Sn
Although we are primarily interested in simple groups, the symmetric groups are
so suciently close and more readily computable that they are useful as a rst
indication. Also in noncommutative geometry S3 in particularly with its 2-cycles
class has the same noncommutative de Rham cohomology as Cq[SU2] and has the
avour of some kind of discrete model of that. In this section we develop our theory
for general Sn.
First we note that in the case of Sn for n > 4, with the 2-cycles conjugacy class,
one can see from the formulae for the Killing form in [10] that K itself has all
entries strictly positive. Hence the Proposition 4.4 applies in this case and there
is a unique maximal eigenvalue, with eigenspace spanned by . For S3 and S4, K
is reducible, and  is a maximal eigenvector but each eigenvalue has multiplicity 3
(as one can see from Table 3 below). Our main result in this section is, for Sn with
its 2-cycles class, an explicit decomposition of CC into irreducible representations
in a manner compatible with the eigenspace decomposition under K. We also nd
explicit formulae for the eigenvalues. As an application we obtain that the Killing
form matrix K for this conjugacy class is positive denite.
Additionally at the end of the section we give a complete analysis for which conju-
gacy classes contain the sign representation.
First recall that irreducible representations of Sn, called the Specht modules, are
indexed by partitions  ` n, and a partition is represented by its Young diagram
or shape. For example the partition (4; 2; 1) of 7 is represented by
:
The Specht module associated to such a partition, or shape, is constructed ab-
stractly for example in [4, 16].
We will use another construction [5] of the Specht module S, namely as a subrepre-
sentation of the regular representation. We recall this construction now. Note that,
since S occurs in the regular representation with multiplicity equal to dimS, the
construction for given  naturally depends on an additional choice. Namely we
choose a tableau of shape , a one-to-one labelling of the boxes by the integers
f1; : : : ; ng. As usual, the symmetric group Sn acts on the set of tableaux by per-
muting the entries.

20 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
To the chosen tableau T we associate its `row subgroup' R(T ), consisting of per-
mutations in Sn preserving the row sets, and its `column subgroup' C(T ) of per-
mutations preserving the column sets.
For example, for S7, for the tableau T of shape  = (4; 2; 1) given by
4 2 1 3
5 6
7
;
the row sets are f1; 2; 3; 4g; f5; 6g; f7g giving the product of permutation groups,
R(T ) = Perm(f1; 2; 3; 4g) Perm(f5; 6g);
and the column sets are f4; 5; 7g; f2; 6g; f1g; f3g, so that
C(T ) = Perm(f4; 5; 7g) Perm(f2; 6g):
To the tableau T we can then associate an element of the group algebra CSn called
the `Young symmetrizer' cT = bTaT where
aT =
X
2R(T )
; bT =
X
2C(T )
(1)`()
and l() is the length of a permutation. To nish the construction we consider the
left ideal in the group algebra dened by the Young symmetrizer,
ST := CSn cT :
This denes a subrepresentation of the left regular representation. Clearly the right
action of Sn on CSn provides Sn-equivariant isomorphisms between the modules
ST for varying T . The isomorphism of any ST with the abstract Specht module
S constructed in [4, 16] is straightforward to write down. Moreover, the sum of
the subspaces ST for varying tableaux T is the block of S inside CSn.
Note that there are many more tableaux than the multiplicity of S, so this is not a
direct sum. Let SY T () denote the set of standard Young tableaux, that is tableaux
whose entries are strictly increasing in rows and in columns. Then the block of S
inside CSn is precisely the subspace
M
T2SY T ()
ST :
We now use this theory to give a concrete construction of irreducible subrepresen-
tations inside a conjugation representation. For any partition  = (1; : : : ; k) of n
we have a corresponding conjugacy class C in Sn, namely the one with cycle type
. Explicitly C is the conjugacy class containing the element
a = (1; : : : ; 1)(1 + 1; : : : ; 1 + 2) : : : (n k + 1; : : : ; n):
If we let Za denote the centraliser of a and identify Sn=Za = C via Za 7!
a1, then we obtain a Sn-equivariant homomorphism from the left regular
representation to the conjugation representation,
(5.1)  : CSn ! C(Sn=Za) = CC;
coming from the linear extension of the quotient map Sn ! Sn=Za .

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 21
Lemma 5.1. Suppose  and  are partitions of n and all notations as above.
The Specht module S occurs as a subrepresentation of the conjugation represen-
tation CC if and only if there exists a standard Young tableau T of shape  for
which cT
a 6= 0 in CC.
In that case, the subrepresentation is explicitly realized as the subspace (ST ), where
 is the projection map from (5.1).
Proof. If there is a tableau T for which cT
a 6= 0, then the restriction of the map
 from (5.1) to the subrepresentation ST of CSn denes a non-zero Sn-equivariant
map ST ! C. Since ST is irreducible and isomorphic to S it follows that this
map must be an isomorphism onto its image, and therefore (ST ) is a submodule
of the conjugation representation isomorphic to S.
On the other hand, if cT
a = 0 for all T 2 SY T (), then the entire block of
S in CSn lies in the kernel of , and therefore the irreducible representation S
does not occur in the image of . Since  is surjective, this means that S is not a
subrepresentation of CC. 
We remark that this lemma also holds, of course, with CC replaced by any cyclic
CSn-module, with the same proof.
Our rst application of this construction is to the the 2-cycles class. In the example
of S3, the 2-cycles class C(2;1) has three elements and it is straightforward to see
that the conjugation representation, CC(2;1), is the (dening) three-dimensional
permutation representation of S3. In terms of Specht modules this representation
decomposes as
(5.2) CC(2;1) = S
(3)  S(2;1):
That is, the trivial representation plus the standard 2-dimensional representation.
The general case is not much dierent, as we observe below.
We use the notation (2; 1n2) for the partition (2; 1; : : : ; 1) which represents the
2-cycles class in Sn.
Proposition 5.2. Consider Sn for n > 2 with the 2-cycles class C = C(2;1n2). For
n = 3 the decomposition of CC into irreducibles is given in equation (5.2).
(1) For n > 3 the decomposition of the conjugation representation CC into
irreducible representations is given by
CC = S(n)  S(n1;1)  S(n2;2):
Here the rst two Specht modules S(n); S(n1;1) are the trivial representation
and the standard (n 1)-dimensional representation, respectively.
(2) Each irreducible submodule of CC lies in an eigenspace for the Killing
form matrix K with eigenvalues as follows. The eigenvalue of K for the
eigenspace containing S(n) (spanned by the element ) is
1
4
(n4 10n3 + 41n2 72n+ 48):
The eigenvalue of K in the eigenspace containing S(n1;1) is
n2 6n+ 12:

22 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
Suppose n > 3. Then the eigenvalue of K on the eigenspace containing
S(n2;2) is 2n.
Proof. (1) could be checked using character theory. But we will rather dene ex-
plicit embeddings of the Specht modules, by the method of Lemma 5.1, in order to
be able to compute the eigenvalues of K in the later parts of the proof.
Of course the trivial representation embeds into CC(2;1n2) as the subspace spanned
by the element  =
P
a2C a, and has multiplicity 1.
For the standard representation S(n1;1) we consider the subspace (ST1) of CC(2;1n2)
for  from (5.1) corresponding to the tableau
T1 =
1 2 3 : : : n 1
n
This is the submodule of CC obtained by applying CSn to the vector cT1
(12). Up
to an overall multiple, which we drop, this vector works out to be
vT1 = (12) + (13) +


+ (1; n 1) (2; n) (3; n)


(n 1; n):
Since vT1 6= 0 we have found a copy of S
(n1;1) in CC.
For the next representation S(n2;2) we consider the subspace (ST2) of CC for 
from (5.1) and the tableau
T2 =
1 2 3 : : : n 2
. . . n
This is the submodule of CC obtained by applying CSn to the vector cT2
(12). Up
to an overall multiple this vector works out to be
vT2 = (12) (2; n 1) (1; n) + (n 1; n);
and since vT2 6= 0 we have found a copy of S
(n2;2) in CC.
That we have thereby completely decomposed CC follows by dimension count:
dimS(n) + dimS(n1;1) + dimS(n2;2) = 1 + (n 1) +
n(n 3)
2
=

n
2

= dimCC;
where dim(S(n2;2)) is computed for example by the hook formula. (A prettier
proof of the above sum decomposition for
n
2


into dimensions of Specht modules
goes via symmetric functions). This concludes the proof of (1).
That the irreducible subrepresentations lie in eigenspaces of K follows immediately
from the fact that in the decomposition of CC each irreducible representation occurs
with multiplicity at most one. We can now compute the eigenvalues.
For the trivial representation we compute the column sum
P
aK((12); a) over all
2-cycles. In the basis of the `triangular' listing
(12)
(13); (23)
(14); (24); (34)
(15); (25); (35); (45)

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 23
...
...
(1n); (2n); (3n); (4n);


; (n 1; n)
we have for a the choice (12), or a lies in the size 2(n 3) region on the left where
a has one entry in common with (12), or a lies in the triangle to the right of size
(n2)(n3)=2 where a is disjoint from (12). Using the values of K for these three
cases in [10], we nd

n
2

+ 2(n 2)

n 3
2

+
(n 2)(n 3)
2

n 4
2

+ 2

which computes as stated.
For the standard representation we use the vector we constructed in the proof of
(1),
vT1 = (12) + (13) +


+ (1; n 1) (2; n) (3; n)


(n 1; n);
which involves the left and bottom slopes of the triangle leaving out the common
vertex. Then the eigenvalue computed as the coecient of (12) in K(vT1) is
K((12); (12)) + (n 3)K((12); (13))K((12); (2; n)) (n 3)K((12); (3; n))
=

n
2

+ (n 4)

n 3
2

+ (n 3)

n 4
2

+ 2

which comes out as stated. Both formulae, although computed for n > 4 in the
above counting, also give the right answer for n = 2; 3; 4, as computed by hand.
For the representation S(n2;2) we use the vector
vT2 = (12) (2; n 1) (1; n) + (n 1; n)
from the proof of (1) and compute the eigenvalue as the (12) coecient of K(vT2),
i.e. as
K((12); (12))K((12); (2; n 1))K((12); (1; n)) +K((12); (n 1; n))
=

n
2

2

n 3
2

+

n 4
2

+ 2 = 2n:

Corollary 5.3. The Killing form for Sn, n > 2 with the 2-cycles conjugacy class C
is non-degenerate and in fact positive denite. Moreover the decomposition of CC
into irreps consisting of the trivial and the standard representation, and the repre-
sentation S(n2;2), coincides for n > 6 with the decomposition of K into eigenspaces
of respectively the maximal, next to maximal and smallest eigenvalues.
Proof. Looking at the three expressions for the eigenvalues in the lemmas above it
is evident that they have dierent leading powers of n and hence are distinct for
all n bigger than some value. By inspection, the only degeneracies are n = 3 when
the trivial and the standard representation have the same eigenvalue of K, n = 4
when the eigenvalues of the trivial and the S(n2;2) coincide, being smaller than the
eigenvalue of the standard representation, and n = 6 when the eigenvalues of the
standard representation and of S(n2;2) coincide. After that, the eigenvalue of the
trivial exceeds that of the standard representation which exceeds that of S(n2;2)
as stated. As all the eigenvalues are positive we conclude that K is non-degenerate
(and positive denite when extended as a hermitian inner product). 

24 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
One useful application of the K eigenvalue decomposition is that when it succeeds it
directly provides subspaces of the conjugacy class representation which decompose
it into irreps, just by linear algebra and entirely bypassing combinatorics. We
illustrate this for S7 and this also serves as a direct check of the above.
Example 5.4. For S7 with its 2-cycles conjugacy class, we use the explicit formula
for the Killing form in [10] and MATHEMATICA to compute the eigenvalues and
eigenspaces. The eigenvalues are
f131; 19; 19; 19; 19; 19; 19; 14; 14; 14; 14; 14; 14; 14; 14; 14; 14; 14; 14; 14; 14g:
The unique `Perron-Frobenius' eigenvector for the 131 eigenvalue is  as it must be.
The eigenvectors for the 19 eigenvalue are, with x = 2=5; y = 3=5,
e1 = (y; y; x; y; x; x; y; x; x; x; y; x; x; x; x; 1; 0; 0; 0; 0; 0)
e2 = (y; x; y; x; y; x; x; y; x; x; x; y; x; x; x; 0; 1; 0; 0; 0; 0)
e3 = (x; y; y; x; x; y; x; x; y; x; x; x; y; x; x; 0; 0; 1; 0; 0; 0)
e4 = (x; x; x; y; y; y; x; x; x; y; x; x; x; y; x; 0; 0; 0; 1; 0; 0)
e5 = (x; x; x; x; x; x; y; y; y; y; x; x; x; x; y; 0; 0; 0; 0; 1; 0)
e6 = (x; x; x; x; x; x; x; x; x; x; y; y; y; y; y; 0; 0; 0; 0; 0; 1)
in a basis ordered as above. On inspection we can see that conjugation by (12); (23); (34); (45); (56)
interchanges the ei as in the standard representation namely (12)e1 = e2; (12)e2 =
e1 and xing the others, etc. The action of (67) is more complicated and comes out
as (67)ei = ei + v for i = 1;


; 5 and (67)e6 =
P6
i=1 ei + v where
v = (0; 0; 0; 0; 0; 0; 0; 0; 0; 0;x;x;x;x;x; x; x; x; x; x; 0):
This obeys (67)v = v. We now change to a new basis ~ei = ei + v2 which does not
change the form of the action of (12);


; (56) by transposition but now (67)~ei = ~ei
for i = 1;


; 5. We dene ~e7 = (67)~e6 to identify the representation as the standard
one embedded in C7. The eigenvectors for the 14 eigenvalue can similarly be seen
to form an irrep, the S(n2;2) one.
Apart from accidental degeneracies and reversals for small n, we see that the next-
to-maximal eigenvalue associates to the 2-cycles conjugacy class the standard repre-
sentation, which is the Specht module associated to the transposition of the Young
diagram describing the conjugacy class. Such an association also assigns the trivial
representation to the trivial conjugacy class as expected. The methods we have
developed are general and can be applied in principle to other conjugacy classes
allowing us to study which if any irreps are picked out in a stable range with a view
towards extending such a 1-1 correspondence. Such an extended correspondence,
however, cannot be the obvious one because we nd that not every Specht module is
actually embeddable in the adjoint representation of the `expected' conjugacy class
under the association given by transposition of the Young diagram. Specically,
we show in Section 5.1 that the sign representation does not occur in the n-cycle
class for all even n. At least for small n we can see, however, that an extended
correspondence is possible.

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 25
Example 5.5. A complete analysis for S3; S4 using methods as above is sum-
marised in Table 1. One could obtain these results from Sage but here we obtain
them as an illustration of the general theory above to conjugacy classes of dier-
ent shape. We focus on S4 and we rst realise the relevant Specht modules S as
submodules of the left regular representation CS4 explicitly as
S(4) = = CS4 c 1 2 3 4 ; S
(3;1) = = CS4 c 1 2 3
4
; S(2;2) = = CS4 c 1 2
3 4
S(2;1;1) = = CS4 c 1 4
2
3
; S(1;1;1;1) = = CS4 c 1
2
3
4
These give rise to explicit embeddings of these Specht modules into the conjugation
representation as follows.
(1) For the (2; 2)-cycle conjugacy class in S4 we have an explicit embedding of
the Specht module S(2;2) given by applying CSn to the vector
c 1 2
3 4

(12)(34) = 8 ((12)(34) (23)(14)) :
As this vector is non-zero, this 2-dimensional representation is contained
and by adding up the dimensions
CC(2;2) =  :
Next, the Killing form matrix has all entries 3 because the product of any
2-2-cycles here is another 2-2-cycle and all elements of the commute. Hence
K((12)(34); v)) = 0 so this module has zero eigenvalue.
(2) For the 3-cycle conjugacy class in S4 we similarly nd
CC(3;1) =    ;
with explicit embeddings of the Specht modules S(3;1); S(2;1;1) and S(1
4)
given by applying CSn to the non-zero vectors
c 1 2 3
4

(123) = 6 ((123) + (132) (234) (243))
c 1 4
2
3

(123) = 3(123) 3(132) + (124) (142) + (143) (134) + (234) (243)
c 1
2
3
4

(123) = 3 ((123) (132) (234) + (243) + (134) (143) (124) + (142)) :
Next, all products of 2-cycles are either e, a 3-cycle or a 2-2-cycle and
we compute the number of xed points (the character of the conjugation
representation) as 8; 2; 0 respectively. Hence the Killing form in basis order

26 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
(123); (132); (142); (124); (134); (143); (243); (234) comes out as
K(3;1) =
0
B
B
B
B
B
B
B
B
B
B
@
2 8 2 0 2 0 2 0
8 2 0 2 0 2 0 2
2 0 2 8 2 0 2 0
0 2 8 2 0 2 0 2
2 0 2 0 2 8 2 0
0 2 0 2 8 2 0 2
2 0 2 0 2 0 2 8
0 2 0 2 0 2 8 2
1
C
C
C
C
C
C
C
C
C
C
A
:
Denoting by v the stated eigenvector taken without any overall factor, we
compute the eigenvalue from the coecient of (123) in K((123); v). For
example, in the rst case K((123); v) = 2 + 8 0 2 = 8.
(3) For the 4-cycle conjugacy class in S4 we similarly nd
CC(4) =   ;
with the explicit embeddings of the Specht modules S2;2 and S2;1;1 given
by applying CSn to the non-zero vectors
c 1 2
3 4

(1234) = 2 ((1342) + (1243) (1423) (1324))
c 1 4
2
3

(1234) = 2 ((1234) + (1243) + (1423) (1342) (1324) (1432)) :
This time products of 4-cycles are either e, 2-2-cycles or 3-cycles with number of
xed points or character values 6,2,0 respectively. Hence the Killing form in basis
order (1234); (1243); (1324); (1342); (1423); (1432) is
K(4) =
0
B
B
B
B
B
B
@
2 0 0 0 0 6
0 2 0 6 0 0
0 0 2 0 6 0
0 6 0 2 0 0
0 0 6 0 2 0
6 0 0 0 0 2
1
C
C
C
C
C
C
A
:
We use the eigenvectors v above without the overall factors and compute the eigen-
value as the coecient of (1243) in K((1243); v). For example, in the rst case
K((1243); v) = 6 + 2 0 0 = 6.
Looking as the tables we see that for S3 the `expected' correspondence holds but
the eigenvalues of K are degenerate in the case of 2-cycles (as above). For S4
there are also some eigenvalue degeneracies and the 4-cycle representation does not
contain the `expected' Specht module 1. On the other hand, we see that a dierent
assignment of irreps to conjugacy classes is possible in such a way as to give a 1-1
correspondence between irreps and conjugacy classes.
We end with a couple of more concrete conjectures for Sn based on limited computer
verication to n  8. They remain a topic for further work.
Conjecture 5.6. For Sn, n > 4 the conjugacy classes with reducible K are pre-
cisely the n12 -fold 2-cycles for n odd. In this case the maximal eigenvalue has
eigenspace decomposition 1 (n 1), where n 1 is the standard representation.

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 27
C  S3 jCj Decomposition (eigenvalue) preferred irrep
(12) 3 1(3) 2(3) 2
(123) 2 1(9) 1(0) 1
C  S4 jCj Decomposition (eigenvalue) preferred irrep
(12) 6 1(8) 2(8) 3(4) 3
(12)(34) 3 1(9) 2(0) 2
(123) 8 1(16) 3(8) 1(0) 3(8) 1
(1234) 6 1(8) 2(8) 3(4) 3
Table 3. Decomposition of span of conjugacy classes into irreps
(eigenvalue of Killing form in brackets), for S3; S4. Here irreps are
labelled by their dimensions with a bar when tensored with the
sign representation.
Conjecture 5.7. For all Sn, n > 4 all conjugacy classes have non-degenerate K
and of these precisely [n2 ] are positive denite, namely the classes of m-fold 2-cycles
class for m = 1; 2;


; [n2 ]. Here [ ] denotes integer part. All the other classes have
evenly split signature.
Note that of the positive-denite classes, the 2-cycles remain the more natural one
for the `compact real form' of the braided Lie algebra of Sn, with the others viewed
as dening higher order dierential calculi.
5.1. The sign representation in the conjugation representation. Not much
is known about decomposition of the conjugation representation of Sn into irre-
ducible representations other than Roth's property, that every irreducible represen-
tation occurs [3, 17] if n > 2, and some asymptotic estimates for the multiplicities
[14]. The more detailed question of which conjugacy classes a given irreducible
representation occurs in looks even more complicated in general. However for the
sign representation we can give a complete answer, which we do here.
First we explain brie
y the overall multiplicity of the sign representation in the
conjugation representation CSn. By character theory this multiplicity is precisely
the number of conjugacy classes consisting of even permutations minus the number
of conjugacy classes of odd permutations (the row sum in the character table, for
the sign representation). In analogy with Euler's product formula for the number
of all partitions, one can write a generating function for these multiplicities.
Namely, if s(n) is the multiplicity of the sign representation in CSn, then the above
description of s(n) implies
(5.3) 1 + t+
1X
n=2
s(n)tn =
1Y
k=1

1
1 + (t)k

:
Clearly, multiplying out the right-hand side of the formula above, we get a contri-
bution 1 to tn for every partition of n. Namely the contribution is +1 for every
partition of n with an even number of even parts and 1 for every partition of
n with an odd number of even parts. The former are in bijection with conjugacy

28 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
classes of even elements, and the latter with conjugacy classes of odd elements.
This proves the identity (5.3).
By the classical Euler identity which reads (after replacing the usual variable by
t and inverting),
(5.4)
1Y
k=1

1
1 + (t)k

=
1Y
k=1
(1 + t2k1);
it follows that the multiplicity of the sign representation in the conjugation rep-
resentation CSn is equal to the number of partitions of n into distinct odd parts.
We rene this observation in the following proposition, which could alternatively
be viewed as giving a representation theoretic proof of the Euler identity (5.4).
Proposition 5.8. The sign representation of Sn appears as a subrepresentation of
the conjugation representation CC if and only if  is a partition of n into distinct
odd parts. If it appears in CC, then it has multiplicity one.
Proof. Since the sign representation has multiplicity one in in the left-regular rep-
resentation CSn and the conjugation representation CC is a cyclic CSn-module, it
is clear that the sign representation can have multiplicity at most 1 in CC.
Let us now write 
0 = 01 for the conjugation action. Fix an element a in
the conjugacy class C. By Lemma 5.1, the sign representation appears in CC if
and only if the element
v =
X

(1)`()
a
in CC is nonzero. Moreover if it is nonzero then it spans the sign representation.
Now suppose v is nonzero and let  be an element of the centraliser Za . Then
we see that

v =
X

(1)`()(1)
a =
X

(1)`()1
a = v:
This implies that  is even, since v spans the sign representation. Therefore if the
sign representation occurs in CC then Za contains only even permutations.
The converse is true as well. If all elements in Z are even, then the coecient of a
in v comes out to be jZj, implying that v is nonzero, and the sign representation
occurs in CC.
It remains to prove that Za contains only even permutations, precisely if  is a
permutation of n into distinct odd parts.
Clearly, if  has an even part then there is a cycle of even length in a, which gives
an element of the centralizer that has odd parity. Also if  has two parts of size
k (we may assume k odd, by above), then there is an element of the centralizer
which exchanges the corresponding two k-cycles of a, which is a product of k many
2-cycles. So again there is an element of odd parity in Za . This shows that for
the sign representation to occur inside CC, we must have that  is a partition of
n into distinct, odd parts.
Conversely, if  is a partition of n into distinct odd parts, then the centraliser is
generated by the individual cycles in a, and these are all even permutations. 

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 29
We note that since, as in the proof of the previous proposition, one can always give
a precise vector spanning a copy of the sign representation, one can in principle use
the methods from above to also nd its eigenvalue under K.
As a very special case, note that Proposition 5.8 implies in particular that if n is
even, then the sign representation does not occur in the class of n-cycles. This
means that, as already remarked, the naive guess at a bijection of conjugacy classes
with Specht modules contained within them (given by transposing the diagram)
does not work in general. Nevertheless, for the cases we have looked there is at
least one alternative assignment. Thus for Sn with n > 2 it is quite possible that
one should be able to nd each irrep in a distinct conjugacy class, but clearly some
further ideas are needed.
6. Computer verifications for simple groups
In order to provide empirical supporting evidence for our conjectures as well as
get a grip on the behaviour of the Killing forms associated to minimal calculi
for nite simple groups we have performed an extensive amount of computational
verications.
Most of our calculations have been performed using the open source computer al-
gebra system Sage in a linux workstation, relying heavily on GAP for some internal
procedures. The code of the actual implementation is available from the authors
upon request.
In the present section we list the computational methods chosen as well as the
obtained results.
6.1. Eective calculation of the Killing form. In order to compute the Killing
form K associated to a conjugacy class C = gG we take advantage of the ad-
invariance K(xaga1 ; xh) = K(xg; xa1ha). A sketch of the procedure goes as follow:
STEP 1: Pick a generator g of C,
STEP 2: Compute a section s : C ! G, such that h = s(h)gs(h)1 for all h 2 C,
STEP 3: Compute the function f(h) = jZ(gh) \ Cj and cache its values,
STEP 4: Compute Kab as f(s(a)1bs(a)).
In practical terms, the suggested method reduces the computation of the Killing
form to the computation of the rst row and the permutations that create all
the other rows from the rst one. The resulting algorithm provides a reasonably
quick computation of the Killing form, allowing to compute them for all conjugacy
classes of all nite simple groups of order up to 75000. The limiting factors of
the implementation are related to exhaustion of the computer memory rather than
computing times, with the rst conjugacy class out of our reach being the class 6B
of elements of order 6 with centralizer of size 6 in the next larger group, the Mathieu
group M12 of order 95040. The naming of the conjugacy classes is according to the
convention in the ATLAS [2].

30 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
6.2. Exploring nondegeneracy of K. Nondegeneracy of K is checked by using
Sage internal algorithms for computing the rank of K. Computationally, nding
the rank is quick enough and doesn't require substantially more resources than
computing the matrix itself. Up to order 75,000 the Killing form is nondegenerate
in all cases but the following exceptions:
 The (mutually inverse) conjugacy classes 7A and 7B of elements of order 7
in the alternating group A7,
 The (pairwise inverse) conjugacy classes 4A, 4B, 8A, 8B, 12A, 12B of ele-
ments of orders 4, 8 or 12 in the unitary group PSU3(3) = G2(2)0.
 The two (mutually inverse) conjugacy classes 7A and 7B of elements of
order 7 in PSL(3; 4).
As mentioned before, all the degenerate cases occur in conjugacy classes that are
not closed under inversion, so reality of the conjugacy class appears to be a sucient
condition (though by no means necessary) for the nondegeneracy of K.
6.3. Irreducibility of K. The irreducibility of K is tested by using standard
algorithms for checking connectedness of the graph GK with vertices indexed by
elements of C and containing an edge (a; b) if and only if Ka;b 6= 0. The size of
the graph GK is substantially smaller than the size of K and the connectedness
checking is also fast, so again this test does not produce much of an additional
overhead.
Generically, the tested Killing forms yield irreducible matrices, so that Perron-
Frobenius theorem applies and the eigenspace associated to the maximal eigenvalue
is 1-dimensional; the only observed exceptions to irreducibility are given by the
conjugacy classes of involutions in the linear groups PSL(2; 4) = A5, PSL(2; 8),
PSL(2; 16), PSL(2; 32), the exceptional Suzuki group Suz(8) and the unitary group
PSU(3; 4). The observed reducible cases suggest a particular behaviour for the
classes of involutions only when the group naturally appears as a group of matrices
with coecients on a eld of characteristic 2. However, not every such group and
class of involutions is reducible, as shown by the data for PSL(3; 4).
6.4. Eigenspaces and irrep decompositions. For this problem we nd some
additional limiting factors. The computation of the characteristic polynomial and
the determination of the eigenvalues gets slow and as the size of the conjugacy
classes increase. The eigenvalues with their corresponding multiplicity have been
computed for all the listed groups. That computation has revealed that the Killing
form appears to be positive denite whenever it comes from a conjugacy class
consisting of involutions, plus the mutually inverse classes 3A and 3B of elements
of order 3 and centralizer of size 648 in the unitary group PSU(4; 2). Thus to the
level visible in the data and for real conjugacy classes the classes of involutions are
precisely those where the Killing form is positive denite. Moreover, we see also that
for real conjugacy classes those which are not classes of involutions have precisely
an equal number of positive and negative eigenvalues taken with multiplicity, i.e.
zero signature.

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 31
For the same groups, we have also computed the decomposition into irreps of the
adjoint representation on CC by means of character theory, looking for some correla-
tion between both decompositions. As the group get larger, the observed behaviour
is that the dimensions of the eigenspaces coincide with the dimensions of irreducible
representations, so as the group size increases we expect that each eigenspace con-
tains exactly one irrep. The obvious exceptions to this rule are the conjugacy classes
yielding reducible Killing forms mentioned in the previous paragraph.
We have tried to make the correlation between irreps and eigenspaces more pre-
cise, and implemented an algorithm that nds the irrep decomposition of each
eigenspace. However, the exact determination of the eigenspaces turns out to be
too slow and memory intensive to be of any practical use except for the few smallest
conjugacy classes. We hope we will be able to get around this issue in the near
future.
6.5. Data: Here we summarize some of the obtained data for all nite simple
groups up to order 75,000. We list whether the conjugacy class is real (closed for
inverses), whether the corresponding Killing form irreducible, and its signature.
The naming of the conjugacy classes follows the convention at the ATLAS, and
conjugacy classes of elements order with the same centralizer sizes have been amal-
gamated whenever they show identical behaviour. Listing the actual eigenspace
decomposition of the adjoint representation on CC would be too lengthy and not
particularly enlightening, so we shall omit that data here. Whenever the Killing
form is reducible we have included between brackets the number of irreducible com-
ponents in the corresponding column. The signature of the Killing form is expressed
as the triple (p; n; z) where p, n and z are respectively the number (counted with
multiplicities) of positive, negative and zero eigenvalues; in particular, nondegener-
acy is given by zero as the last number of this triple. In supplementary information
we list the maximal eigenvalue max of the Killing form, equal to the row sum.
For a real conjugacy class (max jCj)=jCj is a measure of the typical size of the
other entries of the Killing form matrix after the principal entry jCj in each row.
We also list the value C(C) of the character of the adjoint representation on a
typical element of C as a measure of the degree to which the braided Lie algebra is
nonabelian. It counts the number of elements in C that commute with any given
element of C.
A5, order 60
C jCj C(C) Real Irred max Signature
2A 15 3 True False (5) 21 (15, 0, 0)
3A 20 2 True True 34 (10, 10, 0)
5AB 12 2 True True 24 (6, 6, 0)
PSL(2; 7), order 168
C jCj C(C) Real Irred max Signature
2A 21 5 True True 49 (21, 0, 0)
3A 56 2 True True 94 (28, 28, 0)
4A 42 2 True True 76 (21, 21, 0)
8AB 24 3 False True 30 (16, 8, 0)

32 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
A6, order 360
C jCj C(C) Real Irred max Signature
2A 45 5 True True 73 (45, 0, 0)
3AB 40 4 True True 88 (20, 20, 0)
4A 90 2 True True 156 (45, 45, 0)
5AB 72 2 True True 134 (36, 36, 0)
PSL(2; 8), order 504
C jCj C(C) Real Irred max Signature
2A 63 7 True False (9) 105 (63, 0, 0)
3A 56 2 True True 112 (28, 28, 0)
7A C 72 2 True True 130 (36, 36, 0)
9A C 56 2 True True 112 (28, 28, 0)
PSL(2; 11), order 660
C jCj C(C) Real Irred max Signature
2A 55 7 True True 121 (55, 0, 0)
3A 110 2 True True 208 (55, 55, 0)
5AB 132 2 True True 234 (66, 66, 0)
6A 110 2 True True 208 (55, 55, 0)
11AB 60 5 False True 80 (36, 24, 0)
PSL(2; 13), order 1092
C jCj C(C) Real Irred max Signature
2A 91 7 True True 157 (91, 0, 0)
3A 182 2 True True 328 (91, 91, 0)
6A 182 2 True True 328 (91, 91, 0)
7A C 156 2 True True 298 (78, 78, 0)
13AB 84 6 True True 192 (42, 42, 0)
PSL(2; 17), order 2448
C jCj C(C) Real Irred max Signature
2A 153 9 True True 273 (153, 0, 0)
3A 272 2 True True 526 (136, 136, 0)
4A 306 2 True True 564 (153, 153, 0)
8AB 306 2 True True 564 (153, 153, 0)
9A C 272 2 True True 526 (136, 136, 0)
17AB 144 8 True True 336 (72, 72, 0)
A7, order 2520
C jCj C(C) Real Irred max Signature
2A 105 9 True True 273 (105, 0, 0)
3A 70 10 True True 256 (35, 35, 0)
3B 280 4 True True 616 (140, 140, 0)
4A 630 2 True True 1068 (315, 315, 0)
5A 504 4 True True 936 (252, 252, 0)
6A 210 6 True True 528 (105, 105, 0)
7AB 360 3 False True 324 (171, 140, 49)

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 33
PSL(2; 19), order 3420
C jCj C(C) Real Irred max Signature
2A 171 11 True True 361 (171, 0, 0)
3A 380 2 True True 706 (190, 190, 0)
5AB 342 2 True True 664 (171, 171, 0)
9A C 380 2 True True 664 (190, 190, 0)
10AB 342 2 True True 706 (171, 171, 0)
19AB 180 9 False True 252 (100, 80, 0)
PSL(2; 16), order 4080
C jCj C(C) Real Irred max Signature
2A 255 15 True False (17) 465 (255, 0, 0)
3A 272 2 True True 514 (136, 136, 0)
5AB 272 2 True True 514 (136, 136, 0)
15AD 272 2 True True 514 (136, 136, 0)
17AH 240 2 True True 480 (120, 120, 0)
PSL(3; 3), order 5616
C jCj C(C) Real Irred max Signature
2A 117 13 True True 489 (117, 0, 0)
3A 104 14 True True 412 (52, 52, 0)
3B 624 6 True True 1224 (312, 312, 0)
4A 702 2 True True 1356 (351, 351, 0)
6A 936 2 True True 1848 (468, 468, 0)
8AB 702 2 False True 600 (337, 365, 0)
13AB 432 3 False True 399 (224, 208, 0)
13C D 432 3 False True 399 (236, 196, 0)
PSU(3; 3) = G22
0
, order 6048
C jCj C(C) Real Irred max Signature
2A 63 7 True True 177 (63, 0, 0)
3A 56 2 True True 112 (28, 28, 0)
3B 672 6 True True 1332 (336, 336, 0)
4AB 63 7 False True 105 (22, 14, 27)
4C 378 6 True True 852 (189, 189, 0)
6A 504 2 True True 1104 (252, 252, 0)
7AB 864 3 False True 555 (436, 428, 0)
8AB 756 2 False True 752 (364, 365, 27)
12AB 504 2 False True 480 (238, 224, 42)
PSL(2; 23), order 6072
C jCj C(C) Real Irred max Signature
2A 253 13 True True 529 (253, 0, 0)
3A 506 2 True True 988 (253, 253, 0)
4A 506 2 True True 988 (253, 253, 0)
6A 506 2 True True 988 (253, 253, 0)
11A E 552 2 True True 1038 (276, 276, 0)
12AB 506 2 True True 988 (253, 253, 0)
23AB 264 11 False True 374 (144, 120, 0)

34 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
PSL(2; 25), order 7800
C jCj C(C) Real Irred max Signature
2A 325 13 True True 601 (325, 0, 0)
3A 650 2 True True 1228 (325, 325, 0)
4A 650 2 True True 1228 (325, 325, 0)
5AB 312 12 True True 744 (156, 156, 0)
6A 650 2 True True 1228 (325, 325, 0)
12AB 650 2 True True 1228 (325, 325, 0)
13A F 600 2 True True 1174 (300, 300, 0)
M11, order 7920
C jCj C(C) Real Irred max Signature
2A 165 13 True True 489 (165, 0, 0)
3A 440 8 True True 946 (220, 220, 0)
4A 990 2 True True 2108 (495, 495, 0)
5A 1584 4 True True 3096 (792, 792, 0)
6A 1320 2 True True 2568 (660, 660, 0)
8AB 990 2 False True 920 (515, 475, 0)
11AB 720 5 False True 575 (355, 365, 0)
PSL(2; 27), order 9828
C jCj C(C) Real Irred max Signature
2A 351 15 True True 729 (351, 0, 0)
3AB 364 13 False True 520 (196, 168, 0)
7A C 702 2 True True 1376 (351, 351, 0)
13A F 756 2 True True 1434 (378, 378, 0)
14A C 702 2 True True 1376 (351, 351, 0)
PSL(2; 29), order 12180
C jCj C(C) Real Irred max Signature
2A 435 15 True True 813 (435, 0, 0)
3A 812 2 True True 1594 (406, 406, 0)
5AB 812 2 True True 1594 (406, 406, 0)
7A C 870 2 True True 1656 (435, 435, 0)
14A C 870 2 True True 1656 (435, 435, 0)
15AD 812 2 True True 1594 (406, 406, 0)
29AB 420 14 True True 1008 (210, 210, 0)
PSL(2; 31), order 14880
C jCj C(C) Real Irred max Signature
2A 465 17 True True 961 (465, 0, 0)
3A 992 2 True True 1894 (496, 496, 0)
4A 930 2 True True 1828 (465, 465, 0)
5AB 992 2 True True 1894 (496, 496, 0)
8A 930 2 True True 1828 (465, 465, 0)
15AD 992 2 True True 1894 (496, 496, 0)
16A E 930 2 True True 1828 (465, 465, 0)
31AB 480 15 False True 690 (256, 224, 0)

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 35
A8, order 20160
C jCj C(C) Real Irred max Signature
2A 105 25 True True 849 (105, 0, 0)
2B 210 18 True True 996 (210, 0, 0)
3A 112 22 True True 784 (56, 56, 0)
3B 1120 4 True True 3028 (560, 560, 0)
4A 1260 8 True True 3280 (630, 630, 0)
4B 2520 4 True True 4736 (1260, 1260, 0)
5A 1344 4 True True 2996 (672, 672, 0)
6A 1680 6 True True 3600 (840, 840, 0)
6B 3360 2 True True 6168 (1680, 1680, 0)
7AB 2880 3 False True 2466 (1375, 1505, 0)
15AB 1344 4 False True 1556 (597, 747, 0)
PSL(3; 4), order 20160
C jCj C(C) Real Irred max Signature
2A 315 27 True True 1305 (315, 0, 0)
3A 2240 8 True True 4888 (1120, 1120, 0)
4A C 1260 12 True True 3312 (630, 630, 0)
5AB 4032 2 True True 7284 (2016, 2016, 0)
7AB 2880 3 False True 2466 (1398, 1302, 180)
PSL(2; 37), order 25308
C jCj C(C) Real Irred max Signature
2A 703 19 True True 1333 (703, 0, 0)
3A 1406 2 True True 2704 (703, 703, 0)
6A 1406 2 True True 2704 (703, 703, 0)
9A C 1406 2 True True 2704 (703, 703, 0)
18A C 1406 2 True True 2704 (703, 703, 0)
19A I 1332 2 True True 2626 (666, 666, 0)
37AB 684 18 True True 1656 (342, 342, 0)
PSU(4; 2), order 25920
C jCj C(C) Real Irred max Signature
2A 45 13 True True 201 (45, 0, 0)
2B 270 22 True True 1188 (270, 0, 0)
3AB 40 13 False True 196 (40, 0, 0)
3C 240 6 True True 720 (120, 120, 0)
3D 480 12 True True 1548 (240, 240, 0)
4A 540 8 True True 1488 (270, 270, 0)
4B 3240 4 True True 5440 (1620, 1620, 0)
5A 5184 4 True True 9836 (2592, 2592, 0)
6AB 360 5 False True 708 (231, 129, 0)
6C D 720 4 False True 1272 (364, 356, 0)
6E 1440 2 True True 3336 (720, 720, 0)
6F 2160 2 True True 4176 (1080, 1080, 0)
9AB 2880 3 False True 2646 (1595, 1285, 0)
12AB 2160 2 False True 1824 (1035, 1125, 0)

36 J LOPEZ PE~NA, S MAJID, AND K RIETSCH
Suz8, order 29120
C jCj C(C) Real Irred max Signature
2A 455 7 True False (65) 497 (455, 0, 0)
4AB 1820 4 False True 2768 (755, 1065, 0)
5A 5824 4 True True 9796 (2912, 2912, 0)
7A C 4160 2 True True 7690 (2080, 2080, 0)
13A C 2240 4 True True 4748 (1120, 1120, 0)
PSL(2; 32), order 32736
C jCj C(C) Real Irred max Signature
2A 1023 31 True False (33) 1953 (1023, 0, 0)
3A 992 2 True True 1984 (496, 496, 0)
11A E 992 2 True True 1984 (496, 496, 0)
31AO 1056 2 True True 2050 (528, 528, 0)
33A J 992 2 True True 1984 (496, 496, 0)
PSL(2; 41), order 34440
C jCj C(C) Real Irred max Signature
2A 861 21 True True 1641 (861, 0, 0)
3A 1640 2 True True 3238 (820, 820, 0)
4A 1722 2 True True 3324 (861, 861, 0)
5AB 1722 2 True True 3324 (861, 861, 0)
7A C 1640 2 True True 3238 (820, 820, 0)
10AB 1722 2 True True 3324 (861, 861, 0)
20AD 1722 2 True True 3324 (861, 861, 0)
21A F 1640 2 True True 3238 (820, 820, 0)
41AB 840 20 True True 2040 (420, 420, 0)
PSL(2; 43), order 39732
C jCj C(C) Real Irred max Signature
2A 903 23 True True 1849 (903, 0, 0)
3A 1892 2 True True 3658 (946, 946, 0)
7A C 1892 2 True True 3658 (946, 946, 0)
11A E 1806 2 True True 3568 (903, 903, 0)
21A F 1892 2 True True 3658 (946, 946, 0)
22A E 1806 2 True True 3568 (903, 903, 0)
43AB 924 21 False True 1344 (484, 440, 0)
PSL(2; 47), order 51888
C jCj C(C) Real Irred max Signature
2A 1081 25 True True 2209 (1081, 0, 0)
3A 2162 2 True True 4276 (1081, 1081, 0)
4A 2162 2 True True 4276 (1081, 1081, 0)
6A 2162 2 True True 4276 (1081, 1081, 0)
8AB 2162 2 True True 4276 (1081, 1081, 0)
12AB 2162 2 True True 4276 (1081, 1081, 0)
23AK 2256 2 True True 4374 (1128, 1128, 0)
24AD 2162 2 True True 4276 (1081, 1081, 0)
47AB 1104 23 False True 1610 (576, 528, 0)

LIE THEORY OF FINITE SIMPLE GROUPS AND ROTH CONJECTURE 37
PSL(2; 49), order 58800
C jCj C(C) Real Irred max Signature
2A 1225 25 True True 2353 (1225, 0, 0)
3A 2450 2 True True 4756 (1225, 1225, 0)
4A 2450 2 True True 4756 (1225, 1225, 0)
5AB 2352 2 True True 4654 (1176, 1176, 0)
6A 2450 2 True True 4756 (1225, 1225, 0)
7AB 1200 24 True True 2928 (600, 600, 0)
8AB 2450 2 True True 4756 (1225, 1225, 0)
12AB 2450 2 True True 4756 (1225, 1225, 0)
24AD 2450 2 True True 4756 (1225, 1225, 0)
25A J 2352 2 True True 4654 (1176, 1176, 0)
PSU(3; 4), order 62400
C jCj C(C) Real Irred max Signature
2A 195 3 True False (65) 201 (195, 0, 0)
3A 4160 2 True True 8134 (2080, 2080, 0)
4A 3900 12 True True 7824 (1950, 1950, 0)
5AD 208 13 False True 484 (79, 129, 0)
5E F 2496 6 True True 5436 (1248, 1248, 0)
10AD 3120 3 False True 3756 (1586, 1534, 0)
13AD 4800 3 False True 3948 (2310, 2490, 0)
15AD 4160 2 False True 4054 (2041, 2119, 0)
PSL(2; 53), order 74412
C jCj C(C) Real Irred max Signature
2 1431 27 True True 2757 (1431, 0, 0)
3 2756 2 True True 5458 (1378, 1378, 0)
9A C 2756 2 True True 5458 (1378, 1378, 0)
13A F 2862 2 True True 5568 (1431, 1431, 0)
26A F 2862 2 True True 5568 (1431, 1431, 0)
27A I 2756 2 True True 5458 (1378, 1378, 0)
53AB 1404 26 True True 3432 (702, 702, 0)
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Queen Mary University of London, School of Mathematical Sciences, Mile End Rd,
London E1 4NS, UK
E-mail address: j.lopez@qmul.ac.uk
Queen Mary University of London, School of Mathematical Sciences, Mile End Rd,
London E1 4NS, UK
E-mail address: s.majid@qmul.ac.uk
Kings College London, Department of Mathematics, The Strand, London, UK
E-mail address: konstanze.rietsch@kcl.ac.uk