arXiv:1607.07586v2 [] 1 Jan 2017

DISTINGUISHING FINITE GROUP CHARACTERS AND
REFINED LOCAL-GLOBAL PHENOMENA
KIMBALL MARTIN AND NAHID WALJI

Abstract. Serre obtained a sharp bound on how often two irreducible degree
n complex characters of a finite group can agree, which tells us how many local
factors determine an Artin L-function. We consider the more delicate question
of finding a sharp bound when these objects are primitive, and answer these
questions for n = 2, 3. This provides some insight on refined strong multiplicity
one phenomena for automorphic representations of GL(n). For general n, we
also answer the character question for the families PSL(2, q) and SL(2, q).

1. Introduction
In this paper, we consider two questions about seemingly different topics:
(1) How often can two characters of a finite group agree?
(2) How many local Euler factors determine an L-function?
The first question is just about characters of finite groups, and the second is
a refined local-global principle in number theory. However, it has been observed,
notably by Serre, that being able to say something about (1) allows one to say
something about (2), which is our primary motivation, though both are natural
questions. Our main results about the first question are for comparing primitive
characters of degree ≤ 3 and characters of PSL(2, q) or SL(2, q). This will yield
sharp bounds on how many Euler factors one needs to distinguish primitive 2- or
3-dimensional L-functions of Galois representations. We address them in turn.
1.1. Distinguishing group characters. Let G be a finite group, and ρ, ρ′ be two
complex representations of G with characters χ, χ′ . We will study the quantities
δ(ρ, ρ′ ) = δ(χ, χ′ ) =

|{g ∈ G : χ(g) 6= χ′ (g)}|
.
|G|

Specifically, let δn (G) be the minimum of δ(ρ, ρ′ ) as ρ, ρ′ range over pairs of
inequivalent irreducible n-dimensional representations of G, with the convention
that δn (G) = 1 if there are no such pairs ρ, ρ′ . Note that δn (G) tells us what fraction
of elements of G we must check to distinguish irreducible degree n characters. Put
dn = inf G {δn (G)}.
An elementary consequence of orthogonality relations is
Proposition 1.1. We have dn ≥

1
2n2 .

Buzzard, Edixhoven and Taylor constructed examples to show this bound is
sharp when n is a power of 2, which Serre generalized this to arbitrary n (see
[Ram94b]).
Date: January 3, 2017.
1

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KIMBALL MARTIN AND NAHID WALJI

Theorem 1.2 (Serre). For any n, there exists G such that δn (G) =
1
2n2 .

1
2n2 ,

so dn =

In particular, the infimum in dn is a minimum. We will recall the proof of
Proposition 1.1 and Serre’s construction in Section 2.2. For now, the main points
to note are that Serre’s examples must be solvable and the representations are
induced.
In this paper, we consider two kinds of refinements of determining dn . The first
refinement is about restricting to primitive representations and the second is about
restricting to certain families of groups.
Define δn♮ (G) to be the infimum of δ(ρ, ρ′ ) where ρ, ρ′ range over pairs of inequivalent irreducible primitive n-dimensional complex representations of G. Let
d♮n = inf G {δn♮ (G)}. From Serre’s theorem, we get a trivial bound d♮n ≥ dn = 2n1 2 .
Our first result is to determine d♮n for n ≤ 3.
Theorem 1.3. We have d♮1 = 21 , d♮2 = 14 and d♮3 = 72 . Furthermore, δ2♮ (G) = 14
if and only if G is an extension of H ×C2 C2m where m ∈ N and H = [48, 28] or
H = [48, 29]. Also, δ3♮ (G) = 27 if and only if G is an extension of PSL(2, 7).
Here G being an extension of H by some N ⊳ G means G/N ≃ H. The groups
[48, 28] and [48, 29] are the two groups of order 48 which are extensions of S4 by
the cyclic group C2 and contain SL(2, 3).
The n = 1 case is already contained in Proposition 1.1 as d1 = d♮1 . For n = 2, 3,
1
from
these bounds are much better than the trivial bounds d♮2 ≥ 18 and d♮3 ≥ 18
Proposition 1.1. For n = 2, related results were previously obtained by the second
author in [Wal14] and will be discussed below.
Note that while dn is a strictly decreasing sequence for n ≥ 1, our result says
this is not the case for d♮n .
In a slightly different direction, one can look for stronger lower bounds than 2n1 2
for certain families of groups. We do not begin a serious investigation of this here,
but just treat two basic families of finite groups of Lie type which are related to
the calculations for δ2♮ (G) and δ3♮ (G).
Theorem 1.4. We compute δn (G) and δn♮ (G) where G = PSL(2, q) and G =
SL(2, q); for n not listed explicitly below, δn (G) = δn♮ (G) = 1.
For G = SL(2, q) with q arbitrary or for G = PSL(2, q) with q even,

= 2q if n = q±1
2 and q is odd,
δn (G) = δn♮ (G)
1
≥ 6 if n = q − 1,
♮
and δq+1 (G) ≥ 61 whereas δq+1
(G) = 1.
For G = PSL(2, q) and q odd,

q−1
2
 = q if n = 2 and q ≡ 3 mod 4,
δn (G) = δn♮ (G)
= 2q if n = q+1
2 and q ≡ 1 mod 4,

≥ 16 if n = q − 1,

and δq+1 (G) ≥

1
6

♮
(G) = 1.
whereas δq+1

We remark that we completely determine δq±1 (G) for G = SL(2, q) and PSL(2, q)
in Section 6, but the exact formulas are a bit complicated and depend on divisibility

DISTINGUISHING FINITE GROUP CHARACTERS

3

conditions of q ∓ 1. In particular, δq±1 (SL(2, q)) = 16 if and only if 12|(q ∓ 1), and
δq±1 (PSL(2, q)) = 16 if and only if 24|(q ∓ 1).
The values for SL(2, q) immediately give the following bounds.
Corollary 1.5. d♮(q±1)/2 ≤

2
q

for q any odd prime power greater than 3.

Note the upper bound in the corollary for q = 7 is the exact value of d♮3 .
Even though Theorem 1.3 implies d♮n is not a decreasing sequence for n ≥ 1, this
corollary at least suggests that d♮n → 0 as n → ∞.
The proof of Theorem 1.3 relies on consideration of various cases according to the
possible finite primitive subgroups of GL2 (C) and GL3 (C) which are “minimal lifts”,
and about half of these are of the form PSL(2, q) or SL(2, q) for q ∈ {3, 5, 7, 9}. Thus
Theorem 1.4 is a generalization of one of the ingredients for Theorem 1.3. However,
most of the work involved in the proof of Theorem 1.3 is the determination of and
reduction to these minimal lifts, as described in Section 3.
1.2. Distinguishing L-functions. Let F be a number field, and consider an Lfunction L(s), which is a meromorphic function of a complex variable
s satisfying
Q
certain properties, principally having an Euler product L(s) =
Lv (s) where v
runs over all primes of F for s in some right half-plane. For almost all (all but
finitely many) v, we should have Lv (s) = (pv (qv−s ))−1 where qv is the size of the
residue field of Fv and pv a polynomial of a fixed degree n, which is the degree of
the L-function.
Q
Prototypical L-functions of degree n are L-functions L(s, ρ) = L(s, ρv ) of ndimensional Galois representations
ρ : Gal(F̄ /F ) → GLn (C) (or into GLn (Qp )) and
Q
L-functions L(s, π) = L(s, πv ) of automorphic representations π of GLn (AF ). In
fact it is conjectured that all (nice) L-functions are automorphic. These L-functions
are local-global objects, and one can ask how many local factors Lv (s) determine
L(s).
First consider the automorphic case: suppose π, π ′ are irreducible cuspidal automorphic representations of GLn (AF ), S is a set of places of F and we know that
L(s, πv ) = L(s, πv′ ) for all v 6∈ S. Strong multiplicity one says that if S is finite,
then L(s, π) = L(s, π ′ ) (in fact, π ≃ π ′ ). Ramakrishnan [Ram94b] conjectured that
if S has density < 2n1 2 , then L(s, π) = L(s, π ′ ), and this density bound would be
sharp. This is true when n = 1, and Ramakrishnan also showed it when n = 2
[Ram94a].
Recently, in [Wal14] the second author showed that when n = 2 one can in fact
obtain stronger bounds under various assumptions, e.g., the density bound 18 from
[Ram94a] may be replaced by 14 if one restricts to non-dihedral representations
(i.e., not induced from quadratic extensions) or by 92 if the representations are not
twist-equivalent.
Our motivation for this project was to try to understand an analogue of [Wal14]
for larger n. However the analytic tools known for GL(2) that are used in [Wal14]
are not known for larger n. Moreover, the classification of GL(2) cuspidal representations into dihedral, tetrahedral, octahedral and icosahedral types has no
known nice generalization to GL(n). So, as a proxy, we consider the case of Galois
(specifically Artin) representations. The strong Artin conjecture says that all Artin
representations all automorphic, and Langlands’ principle of functoriality says that
whatever is true for Galois representations should be true (roughly) for automorphic
representations as well.

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KIMBALL MARTIN AND NAHID WALJI

Consider ρ, ρ′ be irreducible n-dimensional Artin representations for F , i.e., irreducible n-dimensional continuous complex representations of the absolute Galois
group Gal(F̄ /F ) of F . For almost all places v of F , we can associate a well-defined
Frobenius conjugacy class Frv of Gal(F̄ /F ), and L(s, ρv ) determines the eigenvalues of ρ(Frv ), and thus tr ρ(Frv ). Let S be a set of places of F , and suppose
L(s, ρv ) = L(s, ρ′v ), or even just tr ρ(Frv ) = tr ρ′ (Frv ), for all v 6∈ S.
Continuity means that ρ and ρ′ factor through a common finite quotient G =
Gal(K/F ) of Gal(F̄ /F ), for some finite normal extension K/F . View ρ, ρ′ as irreducible n-dimensional representations of the finite group G. The Chebotarev
density theorem tells us that if C is a conjugacy class in G, then the image of Frv
|C|
. This implies that if
in Gal(K/F ) lies in C for a set of primes v of density |G|
♮
′
the density of S is < δn (G) (or < δn (G) if ρ, ρ are primitive), then ρ ≃ ρ′ , i.e.,
L(s, ρ) = L(s, ρ′ ). Moreover, this bound on the density of S is sharp.
Consequently, Proposition 1.1 tells us that if the density of S is < 2n1 2 , then
L(s, ρ) = L(s, ρ′ ), and Serre’s result implies this bound is sharp. (See Rajan [Raj98]
for an analogous result on ℓ-adic Galois representations.) In fact, this application
to Galois representations was Serre’s motivation, and it also motivated the bound
in Ramakrishnan’s conjecture. For us, the Chebotarev density theorem together
with Theorem 1.3 yields
Corollary 1.6. Let ρ, ρ′ be irreducible primitive n-dimensional Artin representations for F . Suppose tr ρ(Frv ) = tr ρ′ (Frv ) for a set of primes v of F of density
c.
(1) If n = 2 and c > 43 , then ρ ≃ ρ′ .
(2) If n = 3 and c > 75 , then ρ ≃ ρ′ .
When n = 2, if ρ and ρ′ are automorphic, i.e., satisfy the strong Artin conjecture,
then the above result already follows by [Wal14]. When n = 2, the strong Artin
conjecture for ρ is known in many cases—for instance, if ρ has solvable image by
Langlands [Lan80] and Tunnell [Tun81], or if F = Q and ρ is “odd” via Serre’s conjecture by Khare-Wintenberger [KW09]. We remark that the methods of [Wal14]
are quite different than ours here.
The above corollary suggests the following statement may be true: if π, π ′ are
cuspidal automorphic representations of GL3 (AF ) which are not induced from characters and L(s, πv ) = L(s, πv′ ) for a set of primes v of density > 57 , then π ≃ π ′ .
Since not all cuspidal π, π ′ come from Artin representations, the 57 bound is not
even conjecturally sufficient for general π, π ′ . However, it seems reasonable to think
that coincidences of a large fraction of Euler factors only happen for essentially algebraic reasons, so the density bounds are likely to be the same in both the Artin
and automorphic cases.
Acknowledgements. We thank a referee for pointing out an error in an earlier
version. The first author was partially supported by a Simons Collaboration Grant.
The second author was supported by Forschungskredit grant K-71116-01-01 of the
University of Zürich and partially supported by grant SNF PP00P2-138906 of the
Swiss National Foundation. This work began when the second author visited the
first at the University of Oklahoma. The second author would like to thank the
first author as well as the mathematics department of the University of Oklahoma
for their hospitality.

DISTINGUISHING FINITE GROUP CHARACTERS

5

2. Notation and Background
Throughout, G, H and A will denote finite groups, and A will be abelian. Denote
by Z(G) the center of G.
If G and N are groups, by a (group) extension of G by N we mean a group H
with a normal subgroup N such that H/N ≃ G. The extension is called central or
cyclic if N is a central or cyclic subgroup of H.
If G, H, and Z are groups such that Z ⊂ Z(G) ∩ Z(H), then the central product
G×Z H of G and H with respect to Z is defined to be direct product G× H modulo
the central subgroup {(z, z) : z ∈ Z}.
P
If χ1 , χ2 are characters of G, their inner product is (χ1 , χ2 ) = |G|−1 G χ1 (g)χ2 (g).
We denote a cyclic group of order m by Cm .
2.1. Finite subgroups of GLn (C). Next we recall some definitions and facts
about finite subgroups of GLn (C).
Let G be a finite subgroup of GLn (C), so one has the standard representation
of G on V = Cn . We say G is reducible if there exists a nonzero proper subspace
W ⊂ V which is fixed by G.
Suppose G is irreducible. Schur’s lemma implies that Z(G) ⊂ Z(GLn (C)). In
particular, Z(G) is cyclic. If there exists a nontrivial decomposition V = W1 ⊕ · · · ⊕
Wk such that G acts transitively on the Wj , then we say G is imprimitive. In this
case, each Wj has the same dimension, and the standard representation is induced
from a representation on W1 . Otherwise, call G primitive.
Let A 7→ Ā denote the quotient map from GLn (C) to PGLn (C). Similarly, if
G ⊂ GLn (C), let Ḡ be the image of G under this map. We call the projective image
Ḡ irreducible or primitive if G is. Finite subgroups of PGLn (C) have been classified
for small n, and we can use this to describe the finite subgroups of GLn (C).
Namely, suppose G ⊂ GLn (C) is irreducible. Then Z(G) is a cyclic subgroup
of scalar matrices, and Ḡ = G/Z(G). Hence the irreducible finite subgroups of
GLn (C), up to isomorphism, are a subset of the set of finite cyclic central extensions
of the irreducible subgroups Ḡ of PGLn (C).
Let H be an irreducible subgroup of PGLn (C). Given one cyclic central extension G of H which embeds (irreducibly) in GLn (C), note that the central product
G ×Z(G) Cm also does for any cyclic group Cm ⊃ Z(G), and has the same projective image as G. (Inside GLn (C), this central product just corresponds to adjoining
more scalar matrices to G.) Conversely, if G ×Z(G) Cm is an irreducible subgroup of
GLn (C), so is G. We say G is a minimal lift of H to GLn (C) if G is an irreducible
subgroup of GLn (C) with Ḡ ≃ H such that G is not isomorphic to G0 ×Z(G0 ) Cm
for any proper subgroup G0 of G.
2.2. Serre’s construction. Here we explain the proof of Proposition 1.1 and describe Serre’s construction.
Suppose χ1 and χ2 are two distinct irreducible degree n characters of a finite
group G. Let Y be the set of elements of G such that χ1 (g) 6= χ2 (g). Then we have
X
|G|((χ1 , χ1 ) − (χ1 , χ2 )) =
χ1 (g)(χ1 (g) − χ2 (g)).
Y

Using the bound |χi (g)| ≤ n for i = 1, 2 and orthogonality relations, we see
|G| = |G|((χ1 , χ1 ) − (χ1 , χ2 )) ≤ 2n2 |Y |.

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KIMBALL MARTIN AND NAHID WALJI

This proves Proposition 1.1.
We now recall Serre’s construction proving Theorem 1.2, which is briefly described in [Ram94b] using observations from [Ser81, Sec 6.5].
Let H be an irreducible subgroup of GLn (C), containing ζI for each n-th root
of unity ζ, such that H̄ has order n2 . This means that H is of “central type” with
cyclic center. Such H exist for all n. For instance, one can take H̄ = A × A, where
A is an abelian group of order n. Some nonabelian examples of such H̄ are given
by Iwahori and Matsumoto [IM64, Sec 5]. Iwahori and Matsumoto conjectured
that groups of central type are necessarily solvable and this was proved using the
classification of finite simple groups by Howlett and
P Isaacs2 [HI82].
Since |H| = n3 and |Z(H)| = n, the identity
| tr h| = |H| implies tr h = 0
for each h ∈ H \ Z(H), i.e., the set of h ∈ H such that tr h = 0 has cardinality
n3 − n = (1 − n12 )|H|.
Let G = H × {±1} and consider the representations of G given by ρ = τ ⊗ 1 and
ρ′ = τ ⊗ sgn, where τ is the standard representation of H and sgn is the nontrivial
character of {±1}. Then tr ρ(g) = tr ρ′ (g) = 0 for 2(n3 − n) = (1 − n12 )|G| elements
of G. On the remaining 2n elements of Z(G), tr ρ and tr ρ′ must differ on precisely
n elements, giving G with δn (G) = 2n1 2 as desired.
Finally that ρ and ρ′ so constructed are induced for n > 1. It suffices to
show τ is induced. Since H̄ is solvable, there is a subgroup of prime index p,
so there exists P
a subgroup KPof H of index p which contains Z =PZ(H). Put
2
2
2
χ
=
tr
τ
.
Now
K |χ(k)| ≤
Z |χ(k)| = |H|. On the other hand
K |χ(k)| =
Pr P
2
i=1
K |ψi (k)| = r|K|, where χ|K = ψ1 + · · · + ψr is the decomposition of χ|K
into irreducible characters of K. Thus r ≥ p and we must have equality, which
means τ is induced from a ψi . We note that, more generally, Christina Durfee
informed us of a proof that ρ, ρ′ must be induced if δ(ρ, ρ′ ) = 2n1 2 .
3. General Methods
3.1. Central extensions and minimal lifts. The first step in the proof of Theorem 1.3 is the determination of the minimal lifts of irreducible finite subgroups of
PGL2 (C) and PGL3 (C). Here we explain our method for this.
Let G be a group and A an additive abelian group, which we view as a G-module
with trivial action. Then a short exact sequence of groups
ι

(3.1)

π

0 → A → H → G → 1,

where ι and π are homomorphisms, such that ι(A) ⊂ Z(H) gives a central extension
H of G by A. Let M (G, A) be the set of such sequences. (Note these sequences are
often called central extensions, but for our purpose it makes sense to call the middle
term H the central extension.) We say two sequences in M (G, A) are equivalent if
there is a map φ that makes this diagram commute:
0

A

ι

(3.2)

H

π

G

1

G

1

φ

0

A

ι′

H′

π′

Let M̃ (G, A) be M (G, A) modulo equivalence.
If two sequences in M (G, A) as above are equivalent, then H ≃ H ′ . However
the converse is not true. E.g., taking G ≃ A ≃ Cp , then |M̃ (G, A)| = p but there

DISTINGUISHING FINITE GROUP CHARACTERS

7

are only two isomorphism classes of central extensions of Cp by itself, namely the
two abelian groups of order p2 .
Let Cent(G, A) be the set of isomorphism classes of central extensions of G by A.
Then the above discussion shows we have a surjective but not necessarily injective
map Φ : M̃ (G, A) → Cent(G, A) induced from sending a sequence as in (3.1) to the
isomorphism class of H.
Viewing A as a trivial G-module, we have a bijection between M̃ (G, A) and
H 2 (G, A), with the class 0 ∈ H 2 (G, A) corresponding to all split sequences in
M (G, A). We can use this to help determine minimal lifts of irreducible subgroups
of PGLn (C). We recall H1 (G, Z) is the abelianization of G, and H2 (G, Z) is the
Schur multiplier of G.
Proposition 3.1. Let G be an irreducible subgroup of PGLn (C). Then any minimal
lift of G to GLn (C) is a central extension of G by Cm for some divisor m of the
exponent of H1 (G, Z) × H2 (G, Z).
Proof. Any lift of G to an irreducible subgroup H ⊂ GLn (C) corresponds to an
element of Cent(G, A) where A = Cm for some m, and thus corresponds to at least
one element of H 2 (G, A). The universal coefficients theorem gives us the exact
sequence
(3.3)

0 → Ext(H1 (G, Z), A) → H 2 (G, A) → Hom(H2 (G, Z), A) → 0.

′
Let m
L be the gcd ofLm with the exponent of H1 (G, Z) × H2 (G, Z). Recall that
Ext( Z/ni Z, A) =
A/ni A, so Ext(H1 (G, Z), Cm ) = Ext(H1 (G, Z), Cm′ ). An
analogous statement is true for Hom(H2 (G, Z), −) so |H 2 (G, Cm )| = |H 2 (G, Cm′ )|.
Assume m 6= m′ . Consider a sequence as in (3.1) with A = Cm′ . This gives a
sequence
0 → Cm → H ×Cm′ Cm → G → 1
in M (G, Cm ) by extending ι : Cm′ → H to be the identity on Cm . Note if one has
an equivalence φ of two sequences in M (G, Cm ) constructed in this way, then commutativity implies φ(H) = H so restricting the isomorphism φ on the middle groups
to H yields and equivalence of the corresponding sequences in M (G, Cm′ ). Hence
all elements of M̃ (G, Cm ) arise from “central products” of sequences in M (G, Cm′ ),
and thus no elements of Cent(G, Cm ) can be minimal lifts.


When H1 (G, Z) × H2 (G, Z) ≃ 1, then H 2 (G, A) = 0 for any abelian group A,
which means all central extensions are split, i.e., Cent(G, A) = {G × A} for any A.
When H1 (G, Z) × H2 (G, Z) ≃ Z/2Z, then (3.3) tells us that |H 2 (G, Cm )| has size
1 or 2 according to whether m is odd or even, so there must be a unique nonsplit
extension G̃ ∈ Cent(G, C2 ). Then the argument in the proof tells us any cyclic
central extension of G is a central product of either G or G̃ with a cyclic group.
However, in general, knowing H1 (G, Z) and H2 (G, Z) is not enough to determine
the size of Cent(G, Cm ). When |Cent(G, Cm )| < |H 2 (G, Cm )|, we will sometimes
need a way to verify that the central extensions of G by Cm we exhibit exhaust all
of Cent(G, Cm ). For this, we will use a lower bound on the size of the fibers of Φ,
i.e., a lower bound on the number of classes in M̃ (G, A) a given central extension
H ∈ Cent(G, A) appears in.
The central automorphisms of a group H with center Z, denoted AutZ (H), are
the automorphisms σ of H which commute with the projection H → H/Z, i.e.,
satisfy σ(h)h−1 ∈ Z for all h ∈ H.

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KIMBALL MARTIN AND NAHID WALJI

Proposition 3.2. Let A be abelian and H ∈ Cent(G, A) such that A = Z := Z(H).
|Aut(Z)|
Then |Φ−1 (H)| ≥ |Aut
. Moreover, if H is perfect, then |Φ−1 (H)| ≥ |Aut(Z)|.
Z (H)|
Recall H being perfect means H equals its derived group, i.e., H1 (H, Z) = 0. In
particular, non-abelian simple groups are perfect. By (3.3), central extensions of
perfect groups are simpler to study. In fact a perfect group H possesses a universal
central extension by H2 (H, Z).
Proof. Consider a commuting diagram of sequences as in (3.2) with H ′ = H. Suppose π = π ′ , which forces φ ∈ AutZ (H) and ι′ (A) = ker π = ι(A). Fixing π and ι,
there are |Aut(Z)| choices for ι′ , which gives |Aut(Z)| elements of M (G, A). Each
different ι′ must induce a different central automorphism φ ∈ AutZ (H). Thus at
most |AutZ (H)| of these |Aut(Z)| bottom sequences can lie in the same equivalence
class, which proves the first statement.
Adney and Yen [AY65] showed |AutZ (H)| = |Hom(H, Z)| when H has no abelian
direct factor. Consequently, AutZ (H) = 1 when H is perfect.

3.2. Reduction to minimal lifts. Let G be a finite group and ρ1 , ρ2 be two
inequivalent irreducible representations of G into GLn (C). Let Ni = ker ρi and
Gi = ρi (G) for i = 1, 2. We want to reduce the problem of finding lower bounds for
δ(ρ1 , ρ2 ) to the case where G1 and G2 are minimal lifts of Ḡ1 and Ḡ2 . Note that
δ(ρ1 , ρ2 ) is unchanged if we factor through the common kernel N1 ∩ N2 , so we may
assume N1 ∩ N2 = 1. Then N1 × N2 is a normal subgroup of G, N1 ≃ ρ2 (N1 ) ⊳ G2
and N2 ≃ ρ1 (N2 ) ⊳ G1 .
Write Gi = Hi ×Z(Hi ) Zi for i = 1, 2, where Hi is a minimal lift of Ḡi to GLn (C)
and Zi is a cyclic group containing Z(Hi ).
h6=0}|
as
For a subgroup H of GLn (C), let αn (H) be the minimum of |{h∈H:tr
|H|
one ranges over all embeddings (i.e., faithful n-dimensional representations) of H
in GLn (C).
Lemma 3.3. Let m = |ρ1 (N2 ) ∩ Z(G1 )|. Then δ(ρ1 , ρ2 ) ≥

m−1
m αn (H1 ).

Proof. Let K = N2 ∩ ρ1−1 (Z(G1 )), so ρ1 (K) is a cyclic subgroup of Z(G1 ) of order
m and ρ2 (K) = 1. Fix any g ∈ G. Then as k ranges over K, tr ρ1 (gk) ranges
over the values ζ tr ρ(g), where ζ runs through all m-th roots of 1 in C, attaining
each value equally often. On the other hand, tr ρ2 (gk) = tr ρ2 (g) for all k ∈ K.
1
|K| values on the
So provided tr ρ1 (g) 6= 0, tr ρ1 and tr ρ2 can agree on at most m
coset gK. Then note that the fraction of elements g ∈ G for which tr ρ1 (g) 6= 0 is
the same as the fraction of elements in h ∈ H1 for which tr h 6= 0.

We say a subgroup H0 of a group H is Z(H)-free if H0 6= 1 and H0 ∩ Z(H) = 1.
The above lemma implies that if G1 has no Z(G1 )-free normal subgroups, then
1)
δ(ρ1 , ρ2 ) ≥ αn (H
or N2 = 1 (as the K in the proof must be nontrivial). This
2
will often allow us to reduce to the case where N2 = 1, and similarly N1 = 1, i.e.,
G = G1 = G2 , when we can check this property for G1 and G2 . The following
allows us to simply check it for H1 and H2 .
Lemma 3.4. If H1 has no Z(H1 )-free normal subgroups, then G1 has no Z(G1 )free normal subgroups.
Proof. Suppose H1 has no Z(H1 )-free normal subgroups, but that N is a Z(G1 )free normal subgroup of G1 . Let N ′ = {n ∈ H1 : (n, z) ∈ G1 = H1 ×Z(H1 )

DISTINGUISHING FINITE GROUP CHARACTERS

9

Z1 for some z ∈ Z1 }. Then N ′ ⊳ H1 . If N ′ = 1, then N ⊂ Z1 = Z(G1 ), contradicting N being Z(G1 )-free. Hence N ′ 6= 1 and must contain a nontrivial a ∈ Z(H1 ).
But then (a, z) ∈ N ∩ Z(G1 ) for some z ∈ Z1 , which also contradicts N being
Z(G1 )-free.

This will often allow us to reduce to the case where G = H ×Z(H) A for some
cyclic group A ⊃ Z(H), where we can use the following.
Lemma 3.5. Let H be a finite group, A ⊃ Z(H) an abelian group and G =
H ×Z(H) A. Then δn♮ (G) ≥ min{ 21 αn (H), δn♮ (H)}.
Proof. We may assume m = |A| > 1. Let ρ1 , ρ2 : G → GLn (C) be distinct primitive
representations of G. They pull back to H×A, so for i = 1, 2 we can view ρi = τi ⊗χi
where τi : H → GLn (C) is primitive and χi : A → C× . By a similar argument
to the proof of Lemma 3.3, we have that δ(ρ1 , ρ2 ) ≥ m−1
m αn (H) if χ1 6= χ2 . If
χ1 = χ2 , it is easy to see δ(ρ1 , ρ2 ) = δ(τ1 , τ2 ).

In the simplest situation, this method gives the following.
Corollary 3.6. Let H be the set of minimal lifts of Ḡ1 and Ḡ2 to GLn (C). Suppose
that H has no Z(H)-free normal subgroups for all H ∈ H. Then
1
δ(ρ1 , ρ2 ) ≥ min{ αn (H), δn♮ (H) : H ∈ H}.
2
This corollary will address most but not all cases of our proof of Theorem 1.3.
Namely, when n = 3, it can happen that Ḡ1 has a lift H ≃ Ḡ1 which is simple,
so H is a Z(H)-free normal subgroup of itself. So we will need to augment this
approach when H1 or H2 is simple.
4. Primitive degree 2 characters
In this section we will prove the n = 2 case of Theorem 1.3.
We used the computer package GAP 4 [GG] for explicit group and character
calculations in this section and the next. We use the notation [n, m] for the m-th
group of order n in the Small Groups Library, which is accessible by the command
SmallGroup(n,m) in GAP. We can enumerate all (central or not) extensions of G
by N in GAP if |G||N | ≤ 2000 and |G||N | 6= 1024 as all groups of these orders
are in the Small Groups Library. We can also compute homology groups Hn (G, Z)
using the HAP package in GAP.
4.1. Finite subgroups of GL2 (C). Recall the classification of finite subgroups of
PGL2 (C) ≃ SO3 (C). Any finite subgroup of PGL2 (C) is of one of the following
types:
(A) cyclic
(B) dihedral
(C) tetrahedral (A4 ≃ PSL(2, 3))
(D) octahedral (S4 )
(E) icosahedral (A5 ≃ PSL(2, 5) ≃ PSL(2, 4) ≃ SL(2, 4))
Now suppose G is a subgroup of GL2 (C) with projective image Ḡ in PGL2 (C).
If Ḡ is cyclic, G is reducible. If Ḡ is dihedral, then G is not primitive.
Assume Ḡ is primitive. Then we have the following possibilities.

10

KIMBALL MARTIN AND NAHID WALJI

(C) Suppose Ḡ = A4 ≃ PSL(2, 3). Here H1 (A4 , Z) ≃ Z/3Z and H2 (A4 , Z) ≃ Z/2Z.
There is one nonsplit element of Cent(A4 , C2 ), namely SL(2, 3); one nonsplit
element of Cent(A4 , C3 ), namely [36, 3]; and one element of Cent(A4 , C6 )
which is not a central product with a smaller extension, namely [72, 3]. Of
these central extensions (and the trivial extension A4 ), only SL(2, 3) and [72, 3]
have irreducible faithful 2-dimensional representations.
Thus SL(2, 3) and [72, 3] are the only minimal lifts of A4 to GL2 (C). We
check that neither H = SL(2, 3) nor H = [72, 3] has Z(H)-free normal subgroups. In both cases, we have α2 (H) = 34 , and δ2♮ (H) = 23 .
(D) Next suppose Ḡ = S4 . Note H1 (S4 , Z) ≃ H2 (S4 , Z) ≃ Z/2Z. There are 3
nonsplit central extensions of S4 by C2 : [48, 28], [48, 29], [48, 30]. Neither S4
nor [48, 30] have faithful irreducible 2-dimensional representations, but both
[48, 28] and [48, 29] do.
Thus H = [48, 28] and H = [48, 29] are the minimal lifts of S4 to GL2 (C).
Neither of them have Z(H)-free normal subgroups. In both cases we compute
α2 (H) = 58 and δ2♮ (H) = 14 .
(E) Last, suppose Ḡ = A5 = PSL(2, 5). This group is perfect and H2 (A5 , Z) ≃
Z/2Z, with SL(2, 5) being the nontrivial central extension by C2 (the universal
central extension). Note A5 has no irreducible 2-dimensional representations.
Hence there is only one minimal lift of A5 to GL2 (C), H = SL(2, 5). We can
check that SL(2, 5) has no Z(SL(2, 5))-free normal subgroups, α2 (SL(2, 5)) = 34
and δ2♮ (SL(2, 5)) = 52 (cf. Theorem 1.4).
4.2. Comparing characters. Let ρ1 , ρ2 : G → GL2 (C) be inequivalent primitive
representations. By Corollary 3.6,
 
 

1 3 5 3
1
2 1 2
δ(ρ1 , ρ2 ) ≥ min
∪
= .
, ,
, ,
2 4 8 4
3 4 5
4
This shows d♮2 ≥ 14 . Furthermore, we can only have δ(ρ1 , ρ2 ) = 41 if Ḡ1 or Ḡ2 is S4 ,
which implies G1 or G2 is of the form H ×C2 C2m for some m with H = [48, 28] or
H = [48, 29]. Thus we can only have δ2♮ (G) = 41 if G is an extension of H ×C2 C2m
where m ∈ N and H = [48, 28] or H = [48, 29]. Moreover, if G is such an extension
δ2♮ (G) equals 41 because δ2♮ (H) does.
This completes the proof of Theorem 1.3 when n = 2.
5. Primitive degree 3 characters
Here we prove the n = 3 case of Theorem 1.3.
5.1. Finite subgroups of GL3 (C). First we review the classification of finite subgroups GL3 (C). The classification can be found in Blichfeldt [Bli17] or Miller–
Blichfeldt–Dickson [MBD61]. We follow the classification system therein. The
description involves 3 not-well-known groups, G36 = [36, 9], G72 = [72, 41], and
G216 = [216, 153]. Explicit matrix presentations for preimages in GL3 (C) are given
in [Mar04, Sec 8.1].
Any finite subgroup G of GL3 (C) with projective image Ḡ is one of the following
types, up to conjugacy:
(A) abelian
(B) a nonabelian subgroup of GL1 (C) × GL2 (C)

DISTINGUISHING FINITE GROUP CHARACTERS



(C) a group generated by a diagonal subgroup and 


(D) a group generated by a diagonal subgroup, 
1


a
b
form 
c
(E) Ḡ ≃ G36
(F) Ḡ ≃ G72
(G) Ḡ ≃ G216
(H) Ḡ ≃ A5 ≃ PSL(2, 5) ≃ PSL(2, 4) ≃ SL(2, 4)
(I) Ḡ ≃ A6 ≃ PSL(2, 9)
(J) Ḡ ≃ PSL(2, 7)

1
1
1

11



1



1 and a matrix of the

Of these types, (A), (B) are reducible, (C), (D) are imprimitive, and the remaining types are primitive. The first 3 primitive groups, (E), (F) and (G), have
non-simple projective images, whereas the latter 3, (H), (I) and (J), have simple
projective images.
Now we describe the minimal lifts to GL3 (C) of Ḡ for cases (E)–(J).
(E) We have H1 (G36 , Z) ≃ Z/4Z and H2 (G36 , Z) ≃ Z/3Z. The nonsplit extension
of G36 by C2 is [72, 19]. There is one non split extension of G36 by C4 which is
not a central product, [144, 51]. However, G36 , [72, 19] and [144, 51] all have
no irreducible 3-dimensional representations.
There is 1 nonsplit central extension of G36 by C3 , [108, 15]; there is one
by C6 which is not a central product, [216, 25]; there is one by C12 which is
not a central product, [432, 57]. All of these groups have faithful irreducible
3-dimensional representations.
Hence any minimal lift of G36 to GL3 (C) is H = [108, 15], H = [216, 25] or
H = [432, 57]. In all of these cases, H has no Z(H)-free normal subgroups,
α3 (H) = 97 and δ3♮ (H) = 12 .
(F) We have H1 (G72 , Z) ≃ Z/2Z×Z/2Z and H2 (G72 , Z) ≃ Z/3Z. There is a unique
nonsplit central extension of G36 by C2 , [144, 120]; a unique central extension
of G by C3 , [216, 88]; and a unique central extension of G by C6 which is not
a central product, [432, 239]. Of these extensions (including G72 ), only the
latter two groups have faithful irreducible 3-dimensional representations.
Thus there are two minimal lifts of G72 to GL3 (C), H = [216, 88] and H =
[432, 239]. In both cases, H has no Z(H)-free normal subgroups, α3 (H) = 89
and δ3♮ (H) = 21 .
(G) We have H1 (G216 , Z) ≃ H2 (G216 , Z) ≃ Z/3Z. There are 4 nonsplit central extensions of G216 by C3 : [648, 531], [648, 532], [648, 533], and [648, 534]. Neither
G216 nor [648, 534] has irreducible faithful 3-dimensional representations.
Thus there are three minimal lifts of G216 to GL3 (C), H = [648, 531],
H = [648, 532], and H = [648, 533]. In all cases H has no Z(H)-free nor♮
4
mal subgroups, α3 (H) = 20
27 and δ3 (H) = 9 .
(H) As mentioned in the n = 2 case, A5 ≃ PSL(2, 5) is perfect and we have
H2 (A5 , Z) ≃ Z/2Z. The nontrivial extension by C2 (the universal central

12

KIMBALL MARTIN AND NAHID WALJI
√
√
√
3
1 0 1±2 5
2
3
Ḡ
3
2
1
G36
36
4
9
1
7
1
G72
72
8
9
1
5
7
1
G216
216
8
27
9
1
1
1
2
A5
60
4
3
5
1
3
2
2
A6
360
8
9
5
1
3
1
2
PSL(2, 7) 168
8
3
7
Table 1. Fraction of group elements with primitive degree 3 characters having given absolute value

extension) is SL(2, 5), but SL(2, 5) has no faithful irreducible 3-dimensional
representations.
Thus the only minimal lift of A5 to GL3 (C) is A5 itself. We have α3 (A5 ) = 23
and δ3♮ (A5 ) = δ3♮ (PSL(2, 5)) = 52 (cf. Theorem 1.4).
(I) The group A6 is also perfect, but (along with A7 ) exceptional among alternating groups in that H2 (A6 , Z) ≃ Z/6Z. Neither A6 ≃ PSL(2, 9), nor its
double cover SL(2, 9), has irreducible 3-dimensional representations. There is
a unique nonsplit central extension of A6 by C3 , sometimes called the Valentiner group, which we denote V1080 = [1080, 260] and is also a perfect group. It
is known (by Valentiner) that V1080 has an irreducible faithful 3-dimensional
representation.
To complete the determination of minimal lifts of A6 to GL3 (C), we need
to determine the central extensions of A6 by C6 . Here we cannot (easily)
proceed naively as in the other cases of testing all groups of the appropriate
order because we do not have a library of all groups of order 2160. We have
|M̃ (A6 , C6 )| = 6, with one class accounted for by the split extension and one
by SL(2, 9) ×C2 C6 . Since V1080 must correspond to two classes in M̃ (A6 , C3 ),
V1080 ×C3 C6 corresponds to two classes in M̃ (A6 , C6 ) by the proof of Proposition 3.1. Since A6 is perfect, it has a universal central extension by C6 , which
we denote Ã6 . By Proposition 3.2, Ã6 accounts for the remaining 2 classes
in M̃ (A6 , C6 ), and thus we have described all elements of Cent(A6 , C6 ). The
group Ã6 is the unique perfect group of order 2160 and can be accessed by
the command PerfectGroup(2160) in GAP, and we can check that it has no
faithful irreducible 3-dimensional representations.
Hence V1080 is the unique minimal lift of A6 to GL3 (C). We note H = V1080
has no Z(H)-free normal subgroups, α3 (H) = 97 , and δ3♮ (H) = 25 .
(J) The group PSL(2, 7) is perfect and H2 (PSL(2, 7), Z) ≃ Z/2Z. Since SL(2, 7)
has no faithful irreducible 3-dimensional representations, any minimal lift of
PSL(2, 7) to GL3 (C) is just H = PSL(2, 7). Here α3 (H) = 23 and δ3♮ (H) = 72
by Theorem 1.4.
5.2. Comparing characters. Let G be a finite group and ρ1 , ρ2 : G → GL3 (C)
be two inequivalent primitive representations. Let Gi , Ni , Hi , Zi be as in Section
3.2. As before, we may assume N1 ∩ N2 = 1, so G contains a normal subgroup
isomorphic to N1 × N2 whose image in G1 is N2 and image in G2 is N1 .

DISTINGUISHING FINITE GROUP CHARACTERS

13

Proposition 5.1. Suppose at least one of Ḡ1 , Ḡ2 is simple. Then δ(ρ1 , ρ2 ) ≥ 27 ,
with equality only if Ḡ1 ≃ Ḡ2 ≃ PSL(2, 7).
Proof. Say Ḡ1 is simple. Then by above, H1 is isomorphic to one of A5 , V1080 and
PSL(2, 7).
Case I: Suppose Ḡ1 6≃ Ḡ2 . For i = 1, 2, the fraction of g ∈ G for which
| tr ρi (g)| = x is the same as the fraction of h ∈ Hi for which | tr h| = x. Calculations
show that the proportion of such g ∈ G (given x) depends neither on the minimal
lift Hi nor its embedding into GL3 (C), but just on Ḡi . These proportions are given
in Table 1.
If Ḡ1 ≃ PSL(2, 7), we see δ(ρ1 , ρ2 ) ≥ 72 just from considering elements with
√
absolute character value 2. Looking at other absolute character values shows this
inequality is strict.
If Ḡ1 ≃ A5 or A6 and Ḡ2 is not√isomorphic to A5 or A6 , then considering elements
with absolute character value 1±2 5 shows δ(ρ1 , ρ2 ) ≥ 25 .
So assume Ḡ1 ≃ A5 and Ḡ2 ≃ A6 . Then G1 = A5 × Cm and G2 ≃ V1080 ×C3 C3r
for some m, r ∈ N. Suppose δ(ρ1 , ρ2 ) < 31 . By Lemma 3.3, ρ1 (N2 ) and ρ2 (N1 )
are either Z(G1 )- and Z(G2 )-free normal subgroups of G1 and G2 or trivial. This
forces N1 = 1, so G ≃ G1 , but it is impossible for a quotient of G1 to be isomorphic
to G2 . Hence δ(ρ1 , ρ2 ) ≥ 13 > 72 in this case.
Case II: Suppose Ḡ1 ≃ Ḡ2 .
First suppose N1 or N2 is trivial, say N1 . Then G ≃ G1 . By Lemma 3.5, we
have δ3♮ (G) ≥ min{ 13 , δ3♮ (H1 )}. Thus δ3♮ (G) = 72 if and only if H1 = PSL(2, 7).
So assume N1 and N2 are nontrivial. By Lemma 3.3, we can assume ρ1 (N2 ) and
ρ(N1 ) are Z(G1 )- and Z(G2 )-free normal subgroups of G1 and G2 . This is only
possible if N1 ≃ N2 ≃ H1 ≃ H2 is isomorphic to A5 or PSL(2, 7).
Let N = ρ−1
1 (N2 ) ⊳ G and we identify N = N1 × N2 . Fix g ∈ G. Then for
any n1 ∈ N1 , tr ρ1 (g(n1 , 1)) = tr ρ1 (g) but tr ρ2 (g(n1 , 1)) = tr ρ2 (g(n1 , 1)). Since
ρ2 (g(N1 × 1)) = H2 × {z} for some z ∈ Z2 , the fraction of elements of g(N1 × 1)
(and thus of G) on which tr ρ1 and tr ρ2 can agree is at most the maximal fraction
of elements of H1 with a given trace. By Table 1 this is less than 21 for either
Ḡ1 ≃ A5 or Ḡ1 ≃ PSL(2, 7).

To complete the proof of Theorem 1.3 for n = 3, it suffices to show δ(ρ1 , ρ2 ) > 27
when Ḡ1 and Ḡ2 are each one of G36 , G72 and G216 . Using Corollary 3.6, in this
situation we see
 
 

1 7 8 20
1 1 4
10
δ(ρ1 , ρ2 ) ≥ min
, ,
, ,
.
∪
=
2 9 9 27
2 2 9
27

This finishes Theorem 1.3.

6. Families SL(2, q) and PSL(2, q)
We consider SL(2, q) and PSL(2, q), for even and odd prime powers q. We
separate these into three subsections: SL(2, q), q odd; SL(2, q) ≃ PSL(2, q), q even;
and PSL(2, q), q odd. We refer to, and mostly follow the notation of, Fulton–
Harris [Ful91] for the representations of these groups.
√
× 2
Choose an element ∆ ∈ F×
q − (Fq ) . Denote by E := Fq ( ∆) the unique
√
quadratic extension of Fq . We can write the elements of E as a+bδ, where δ := ∆.

14

KIMBALL MARTIN AND NAHID WALJI

2
2
The norm map N : E× → F×
q is then defined as N (a + bδ) = a − b ∆. We also
1
denote E to be the kernel of the norm map.

6.1. SL(2, q), for odd q. The order of SL(2, q) is (q + 1)q(q − 1). We begin by
describing the conjugacy classes for SL(2, q):
(A) I.
(B) −I.




ǫ γ
, where
ǫ
ǫ = ±1 and γ = 1 or ∆. So there are four conjugacy classes, each of size
(q 2 − 1)/2.


x
(D) Conjugacy classes of the form [c3 (x)], where c3 (x) =
with x 6=
x−1
−1
±1. Since the conjugacy classes c3 (x) and c3 (x ) are the same, we have
(q − 3)/2 different conjugacy classes, each of size q(q + 1).


x ∆y
(E) Conjugacy classes of the form [c4 (z)], where c4 (z) =
where z =
y x
x + δy ∈ E1 and z 6= ±1. Since c4 (z) = c4 (z̄) we have (q − 1)/2 conjugacy
classes, each of size q(q − 1).
(C) Conjugacy classes of the form [c2 (ǫ, γ)], where c2 (ǫ, γ) =

We give a brief description of the representations that appear in the character
table. The first set of representations, denoted Wα , are induced from the subgroup
×
B of upper triangular matrices. Given a character α ∈ Fc
q , we can extend this to a
character of B, which we then induce to a (q + 1)-dimensional representation Wα
of SL2 (Fq ). If α2 6= 1, then the induced representation is irreducible. If α = 1,
then W1 decomposes into its irreducible consituents: the trivial representation U
and the Steinberg representation V . If α2 = 1 and α 6= 1, then it decomposes into
two irreducible constituents denoted W + and W − .
For the remaining irreducible representations, we consider characters α and ϕ
of the diagonal subgroup A and the subgroup S := {c4 (z) | z ∈ E1 }, respectively,
where the characters agree when restricted to A ∩ S. Then we construct a virtual
G
character πϕ := IndG
A (α) − Wα − IndS (ϕ) (note that the virtual character will not
depend on the specific choice of α).
When ϕ = ϕ, πϕ decomposes into two distinct characters. In the case when ϕ is
trivial, π1 decomposes into the difference between the characters for the Steinberg
representation and the trivial representation. If ϕ is the unique (non-trivial) order 2
character of S, then πϕ decomposes into two distinct irreducible characters of equal
dimension; we will label the corresponding representations X + and X − . If ϕ 6= ϕ,
then πϕ corresponds to an irreducible representation, which we denote as Xϕ . Two
irreducibles Xϕ and Xϕ′ are equivalent if and only if ϕ = ϕ′ or ϕ = ϕ′ . We note
that out of all the irreducible representations, the imprimitive representations are
exactly all the Wα (for α2 6= 1).
We define some notation that will appear in the character table for SL(2, q).
×
with α 6= ±1, and ϕ a character of E1 with ϕ2 6= 1. Fix τ to be the
Let α ∈ Fc
q

DISTINGUISHING FINITE GROUP CHARACTERS

15

× 2
\
non-trivial element of F×
q /(Fq ) , and let

p
1
(τ (ǫ) ± τ (ǫγ) τ (−1)q),
2
p
1
±
u (ǫ, γ) = ǫ(−τ (ǫ) ± τ (ǫγ) τ (−1)q).
2
s± (ǫ, γ) =

Lastly, we define ψ to be the non-trivial element of E1\
/(E1 )2 . The character table
is:
[I]
[−I]
[c2 (ǫ, γ)]
[c3 (x)]
[c4 (z)]
q2 −1
Size:
1
1
q(q
+
1)
q(q
− 1)
2
Rep
#
U
1
1
1
1
1
1
q−1
q−1
±
·
ψ(−1)
X±
2
u
(ǫ,
γ)
0
−ψ(z)
2
2
q+1
q+1
2
s± (ǫ, γ)
τ (x)
0
W±
2
2 · τ (−1)
q−1
Xϕ
q
−
1
(q
−
1)ϕ(−1)
−ϕ(ǫ)
0
−ϕ(z)
−
ϕ(z −1 )
2
1
q
q
0
1
−1
V
−1
Wα q−3
q
+
1
(q
+
1)α(−1)
α(ǫ)
α(x)
+
α(x
)
0
2
The pair of representations X ± : The two (q −1)/2-dimensional representations
X + and X − have the same trace character values for exactly all group elements
outside of [c2 (ǫ, γ)], so we have δ(X + , X − ) = 2/q.
The pair of representations W ± : The two (q+1)/2-dimensional representations
W + and W − have the same trace character values exactly for all group elements
outside of the [c2 (ǫ, γ)] conjugacy classes. So again we have δ(W + , W − ) = 2/q.
(q − 1)-dimensional representations: There are (q − 1)/2 such representations,
c1 , for ϕ2 6= 1. Note that |E1 | = q + 1.
denoted Xϕ , where ϕ ∈ E
In order to determine δ(Xϕ , Xϕ′ ), we need to find the number of z ∈ E1 for
which ϕ(z) + ϕ(z −1 ) = ϕ′ (z) + ϕ′ (z −1 ), and whether ϕ(−1) = ϕ′ (−1).
We begin with the first equation. Note that Im(ϕ), Im(ϕ′ ) ⊂ µq+1 , where µn
denotes the nth roots of unity. Then ϕ(z) + ϕ(z −1 ) is of the form ζ a + ζ −a , where ζ
is the primitive (q + 1)th root of unity e2πi/(q+1) and a is a non-negative integer less
than q + 1. Now ζ a + ζ −a = ζ b + ζ −b for some 0 ≤ a, b < q + 1 implies that a = b or
(q + 1) − b. So ϕ(z) + ϕ(z −1 ) = ϕ′ (z) + ϕ′ (z −1 ) iff ϕ(z) = ϕ′ (z) or ϕ(z) = ϕ′ (z −1 ).
If ϕ(z) = ϕ′ (z), then this is equivalent to (ϕ′ )−1 ϕ(z) = 1, and the number of z
for which this holds is |ker (ϕ′ )−1 ϕ|. The number of z for which ϕ(z) = ϕ′ (z −1 ) is
|ker ϕ′ ϕ|. Thus the number of z ∈ E1 for which ϕ(z) + ϕ(z −1 ) = ϕ′ (z) + ϕ′ (z −1 ) is
|ker (ϕ′ )−1 ϕ| + |ker ϕ′ ϕ| − |ker ϕ′ ϕ ∩ ker (ϕ′ )−1 ϕ|.

c1 can
Now E1 is a cyclic group, so we can fix a generator g. The elements of E
m
then be denoted as {ϕ0 , ϕ1 , ϕ2 , . . . , ϕq }, where ϕm is defined via ϕm (g) = ζ . Note
that |ker ϕm | = (m, q + 1). Define
(m + m′ , k) + (m − m′ , k) − (m + m′ , m − m′ , k) − 1 − tm,m′
,
2
= 1 if both k and m + m′ are even, and 0 otherwise.

Mk (m, m′ ) :=
where tm,m′
Then:

16

KIMBALL MARTIN AND NAHID WALJI

Lemma 6.1. For distinct integers 0 ≤ m, m′ < q + 1, we have
|{[c4 (z)] : ϕm (z) + ϕm (z −1 ) = ϕm′ (z) + ϕm′ (z −1 )}| = Mq+1 (m, m′ ).
If m and m′ have the same parity, then ϕm (−1) = ϕm′ (−1) so


q−1
1
′
(6.1)
− Mq+1 (m, m ) .
δ(Xϕm , Xϕm′ ) =
q+1
2
If m and m′ have different parity, then
 2

q +1
1
(6.2)
− Mq+1 (m, m′ )(q − 1) .
δ(Xϕm , Xϕm′ ) = 2
q −1
2
To determine the minimum possible value of δ above, we consider the maximum
possible size of Mk (m, m′ ).
Lemma 6.2. Suppose k = 2j ≥ 8. Then
max Mk (m, m′ ) = 2j−2 − 1 =

k
− 1,
4

where m, m′ run over distinct classes in Z/kZ \ {0, k2 } with m 6≡ ±m′ .
Suppose k ∈ 2N is not a power of 2 and let p be the smallest odd prime dividing
k. Then

( 
k
1
1
+
p − 1 k ≡ 0 (mod 4)
max Mk (m, m′ ) = 4
k−2
k ≡ 2 (mod 4),
4
where m, m′ range as before.
In all cases above, the maximum occurs with m, m′ of the same parity if and only
if 4|k.
Proof. Let d = (m + m′ , k) and d′ = (m − m′ , k), so our restrictions on m, m′ imply
that d, d′ are proper divisors of k of the same parity. Note that any pair of such
d, d′ arise from some m, m′ if d 6= d′ , and the case d = d′ = k2 does not occur. Then
Mk (m, m′ ) = 21 (d + d′ − (d, d′ ) − 1 − tm,m′ ), and m, m′ have the same parity if and
only if d, d′ are both even.
The case k = 2j has a maximum with d = k2 and d′ = k4 .
Suppose k = 2pk ′ as in the second case. Then note d + d′ − (d, d′ ) is maximised
when d = k2 and d′ = kp , which is an admissible pair if k ′ is even. Otherwise, we
k
get a maximum when d = k2 and d′ = 2p
.

In all cases we have
(6.3)

max Mk (m, m′ ) ≤

k
− 1,
3

and equality is obtained if and only if 12|k for suitable m, m′ of the same parity.
This leads to an exact formula for δq−1 (SL(2, q)) with q > 3 odd by combining
with (6.1) and (6.2). We do not write down the final expression, but just note the
consequence that δq−1 (SL(2, q)) ≥ 61 with equality if and only 12|(q + 1).

DISTINGUISHING FINITE GROUP CHARACTERS

17

×
(q + 1)-dimensional representations: Consider Wα , Wα′ , where α, α′ ∈ Fc
q −
{±1} and α 6= α′ . Since |F×
|
=
q
−
1,
we
know
that
Im(α)
<
µ
.
So,
given
a
q−1
q
c
×
m
generator g of the cyclic group F×
q , we define the elements of Fq as: αm (g) = ζ ,
2πi/(q−1)
where ζ := e
, and 0 ≤ m ≤ q − 2.

Using similar arguments to the (q − 1)-dimensional case above, we have:
Lemma 6.3. For distinct integers 0 ≤ m, m′ < q − 1, we have

{[c3 (x)] : αm (x) + αm (x−1 ) = αm′ (x) + αm′ (x−1 )} = Mq−1 (m, m′ ).

Given that the value of αm (−1) is +1 if m is even and −1 if m is odd, we obtain
that if m and m′ have the same parity, then


q−3
1
− Mq−1 (m, m′ ) .
δ(Wαm , Wαm′ ) =
q−1
2

Whereas if m and m′ have different parity, then
 2

1
q −3
′
δ(Wαm , Wαm′ ) = 2
− Mq−1 (m, m )(q + 1) .
q −1
2

Combining these with Lemma 6.2 for q > 5 gives a formula for δq+1 (SL(2, q)).
In particular, (6.3) gives δq+1 (SL(2, q)) ≥ 61 , with equality if and only if 12|(q − 1).
6.2. SL(2, q), for even q. We keep the notation from the previous section. The
order of SL(2, q) is again q(q + 1)(q − 1). The conjugacy classes for SL(2, q), q even,
are as follows:
(A) I.


1 1
(B) [N ] =
. This conjugacy class is of size q 2 − 1.
0 1


x
(C) [c3 (x)], where c3 (x) =
, with x 6= 1. We note that [c3 (x)] =
x−1
[c3 (x−1 )], so there are (q − 2)/2 such conjugacy classes. Each one is of size
q(q + 1).


x ∆y
(D) [c4 (z)], where c4 (z) =
for z = x + δy ∈ E1 with z 6= 1. Since
y x
c4 (z) = c4 (z̄), there are q/2 such conjugacy classes, each of size q(q − 1).
The representations for q even are constructed similarly to the case of q odd,
with a couple of differences: Since, for q even, the subgroup S has odd order, it
does not have characters of order two, and so the irreducible representations X ± do
not arise. Similarly, the character α cannot be of order two, and so the irreducible
representations W ± do not occur. The character table is:
[I]
[N ]
[c3 (x)]
[c4 (z)]
Size:
1
q2 − 1
q(q + 1)
q(q − 1)
Rep
#
U
1
1
1
1
1
q/2
q−1
−1
0
−ϕ(z) − ϕ(z −1 )
Xϕ
V
1
q
0
1
−1
1
α(x) + α(x−1 )
0
Wα (q − 2)/2 q + 1

18

KIMBALL MARTIN AND NAHID WALJI

Representations of dimension q − 1: The analysis here is similar to that in
Section 6.1, which gives us:

1 q
δ(Xϕm , Xϕm′ ) =
− Mq+1 (m, m′ ) .
q+1 2
Analogous to Lemma 6.2, we have when k ≥ 3 is odd,
 

1 k − 1
2 p

(6.4)
max Mk (m, m′ ) =
k
1
(p
+
p
−
1)
−
1
1
2
2 p1 p2

k = pj
k = p1 p2 k ′

where m, m′ run over all nonzero classes of Z/kZ such that m 6≡ ±m′ and in the
latter case are the two smallest distinct primes dividing k. The above two equations
give an exact expression for δq−1 (SL(2, q)), q ≥ 4. For k odd, note
(6.5)

max Mk (m, m′ ) ≤

7k − 15
,
30

with equality if and only if 15|k. Thus δq−1 (SL(2, q)) ≥
only if 15|(q + 1).

4
15

with equality if and

Representations of dimension q + 1: A similar analysis to that in Section 6.1
gives


1
q−2
′
δ(Wαm , Wαm′ ) =
− Mq−1 (m, m ) .
q−1
2

Combining this with (6.4) gives an exact formula for δq+1 (SL(2, q)) for q ≥ 8, and
4
from (6.5), we again get δq+1 (SL(2, q)) ≥ 15
with equality if and only if 15|(q − 1).
6.3. PSL(2, q), for odd q. The order of PSL(2, q) is
conjugacy classes are as follows:
(A) I.



1
2
2 q(q

− 1) if q is odd. The


γ
(B) [c2 (γ)], where c2 (γ) = c2 (1, γ) =
for γ ∈ {1, ∆}.
1
(C) [c3 (x)], (x 6= ±1), where c3 (x) is as in the previous two sections. Since c3 (x) =
c3 (−x) = c3 (1/x) = c3 (−1/x), the number of such conjugacy classes when
q ≡ 3 (mod 4) is (q − 3)/4. In this case, all of the c3 (x) conjugacy classes have
size q(q + 1).
If q ≡ 1 (mod
√ 4), then −1 is a square in Fq and there is a conjugacy class
denoted by c3 ( −1) which has size q(q + 1)/2; the remaining c3 (x) conjugacy
classes (there are (q − 5)/4 such classes) have size q(q + 1).
(D) [c4 (z)], for z ∈ E1 , z 6= ±1, where c4 (z) is defined as in the previous two
sections. Since c4 (z) = c4 (z̄) = c4 (−z) = c4 (−z̄), when q ≡ 1 (mod 4), the
number of such conjugacy classes is (q − 1)/4, and they are all of size q(q − 1).
When q ≡ 3 (mod 4), we can choose ∆ to be −1 (since it is not a square),
and so we see that δ ∈ E1 . The conjugacy class associated to c4 (δ) has size
q(q − 1)/2, whereas the rest of the c4 (z) conjugacy classes (of which there are
(q − 3)/4 such classes) have size q(q − 1).
1

The representations of PSL(2, q) are the representations of SL(2, q) which are
trivial on −I; this depends on the congruence class of q modulo 4.

DISTINGUISHING FINITE GROUP CHARACTERS

19

6.3.1. q ≡ 1 (mod 4).
For the character table below, the notation is the same as in previous subsections.
√
[I]
[c2 (γ)] [c3 ( −1)]
[c3 (x)]
[c4 (z)]
2
q(q+1)
q −1
Size:
1
q(q
+
1)
q(q
− 1)
2
2
q−5
q−1
Rep
#
1
2
1
4
4
U
1
1
1
1
1
1
√
q+1
W±
2
s± (1, γ) τ ( −1)
τ (x)
0
2
q
−
1
−1
0
0
−ϕ(z)
−
ϕ(z −1 )
Xϕ q−1
4
V
1
q
0
1
1
−1
√
−1
Wα q−5
q
+
1
1
2α(
−1)
α(x)
+
α(x
)
0
4
Representations W ± : The trace characters of these (q + 1)/2-dimensional representations agree everywhere but for the conjugacy classes [c2 (γ)]. This gives us
δ(W + , W − ) = 2/q.
Representations of dimension q − 1: Assume q ≥ 9. Any two representations
Xϕ , Xϕ′ have trace characters that may differ only for the conjugacy classes [c4 (z)].
We may view ϕ as a map into µ q+1 and parameterize the ϕ by ϕm for nonzero
2

m ∈ Z/ q+1
2 Z similar to before. Analogously, we obtain


q−1
1
− 2M q+1 (m, m′ ) .
δ(Xϕm , Xϕm′ ) =
2
q+1
2
From (6.5), this gives δq−1 (PSL(2, q)) ≥

4
15 ,

with equality if and only if 30|(q + 1).

Representations of dimension q + 1: Assume q ≥ 13. The analysis follows in
a similar manner to that in previous sections. View α : F×
q /{±1} → µ q−1 , and we
2

can parametrize such α by m ∈ Z/ q−1
2 Z as before. One difference is that we must
√
consider the case when x = −1. Note that this is the only
√ conjugacy√class of the
form [c3 (x)] that has size q(q + 1)/2. We find that αm ( −1) = αm′ ( −1) if and
only if m, m′ have the same parity. Overall we get


q−5
1
− 2M q−1 (m, m′ ) + 1 − tm,m′ .
δ(Wαm , Wαm′ ) =
2
q−1
2
From (6.3) we get δq+1 (PSL(2, q)) ≥
6.3.2. q ≡ 3 (mod 4).
[I]
Size:
1
Rep Number
1
U
1
1
q−1
X±
2
2
q−3
q−1
Xϕ
4
V
1
q
q−3
Wα
q
+
1
4

[c2 (γ)]

1
6

with equality if and only if 24|(q − 1).

q2 −1
2

[c3 (x)]
q(q + 1)

2
1
u± (1, γ)
−1
0
1

1
0
0
1
α(x) + α(x−1 )

q−3
4

where u± (1, γ) and ψ are defined as before.

[c4 (z)]
q(q − 1)
q−3
4

1
−ψ(z)
−ϕ(z) − ϕ(z −1 )
−1
0

[c4 (δ)]
q(q−1)
2

1
1
−ψ(δ)
−2ϕ(δ)
1
0

20

KIMBALL MARTIN AND NAHID WALJI

Representations X ± : For W ± , the characters of the representations X ± agree
everywhere but for the conjugacy classes [c2 (γ)], so: δ(X + , X − ) = 2/q.
Representations of dimension q − 1: Assume q ≥ 11. Any two representations
Xϕ , Xϕ′ have trace characters that may differ only for the conjugacy classes [c4 (z)].
In the case of the conjugacy class [c4 (δ)], we note that δ has order 2 in E1 /{±1}.
Parametrize the nontrivial maps ϕ : E1 /{±1} → µ q+1 by 1 ≤ m ≤ q−3
4 as before.
2
Then ϕm (δ) = ϕm′ (δ) if and only if m, m′ have the same parity. We obtain


1
q−3
′
′
δ(Xϕm , Xϕm′ ) =
− 2M q+1 (m, m ) + 1 − tm,m .
2
q+1
2
By (6.3), we get δq−1 (PSL(2, q)) ≥

1
6

with equality if and only if 24|(q + 1).

Representations of dimension q + 1: Assume q ≥ 11. We obtain


1
q−3
′
δ(Wαm , Wαm′ ) =
− 2M q−1 (m, m ) .
2
q−1
2
By (6.5), we get δq+1 (PSL(2, q)) ≥

4
15 ,

with equality if and only if 30|(q − 1).

References
[AY65] J. E. Adney and Ti Yen, Automorphisms of a p-group, Illinois J. Math. 9 (1965),
137–143. MR0171845
[Bli17] HF Blichfeldt, Finite collineation groups, Chicago, 1917.
[Ful91] William and Harris Fulton Joe, Representation theory, Graduate Texts in Mathematics,
vol. 129, Springer-Verlag, New York, 1991. A first course, Readings in Mathematics.
[GG] GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.8.3.
[HI82] Robert B. Howlett and I. Martin Isaacs, On groups of central type, Math. Z. 179 (1982),
no. 4, 555–569.
[IM64] Nagayoshi Iwahori and Hideya Matsumoto, Several remarks on projective representations of finite groups, J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), 129–146 (1964).
[KW09] Chandrashekhar Khare and Jean-Pierre Wintenberger, Serre’s modularity conjecture.
I, Invent. Math. 178 (2009), no. 3, 485–504.
[Lan80] Robert P. Langlands, Base change for GL(2), Annals of Mathematics Studies, vol. 96,
Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980.
[Mar04] Kimball Martin, Four-dimensional Galois representations of solvable type and automorphic forms, ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)–California Institute of Technology.
[MBD61] G. A. Miller, H. F. Blichfeldt, and L. E. Dickson, Theory and applications of finite
groups, Dover Publications, Inc., New York, 1961.
[Raj98] C. S. Rajan, On strong multiplicity one for l-adic representations, Internat. Math. Res.
Notices 3 (1998), 161–172.
[Ram94a] Dinakar Ramakrishnan, A refinement of the strong multiplicity one theorem for GL(2).
Appendix to: “l-adic representations associated to modular forms over imaginary quadratic fields. II” [Invent. Math. 116 (1994), no. 1-3, 619–643] by R. Taylor, Invent.
Math. 116 (1994), no. 1-3, 645–649.
, Pure motives and automorphic forms, Motives (Seattle, WA, 1991), Proc.
[Ram94b]
Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 411–446.
[Ser81] Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst.
Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR644559
[Tun81] Jerrold Tunnell, Artin’s conjecture for representations of octahedral type, Bull. Amer.
Math. Soc. (N.S.) 5 (1981), no. 2, 173–175.
[Wal14] Nahid Walji, Further refinement of strong multiplicity one for GL(2), Trans. Amer.
Math. Soc. 366 (2014), no. 9, 4987–5007.

DISTINGUISHING FINITE GROUP CHARACTERS

Department of Mathematics, University of Oklahoma, Norman, OK 73019 USA
Department of Mathematics, Occidental College, Los Angeles, CA 90041 USA

21

