Explicit construction of universal sampling sets for finite abelian and symmetric groups

245 Pages·2014·1.24 MB·English

by  Lucia Morotti

Explicit construction of universal sampling sets for
finite abelian and symmetric groups
Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der
RWTH Aachen University zur Erlangung des akademischen Grades einer
Doktorin der Naturwissenschaften genehmigte Dissertation vorgelegt von
Master of science in Mathematics
Lucia Morotti
aus Bergamo, Italien
Berichter: Universitätsprofessor Dr. Hartmut Führ
Universitätsprofessor Dr. Gerhard Hiß
Professor Dr. Jørn B. Olsson
Tag der mündlichen Prüfung: 1. August 2014
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online
verfügbar.

Contents
Abstract 4
Introduction 5
1 Sampling Pairs and Universal Sampling Sets 8
1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Results over Arbitrary Finite Groups . . . . . . . . . . . . . . 9
2 Abelian Groups 16
2.1 Definitions and Basic Results . . . . . . . . . . . . . . . . . . 16
2.2 Sampling Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Universal Sampling Sets . . . . . . . . . . . . . . . . . . . . . 21
2.4 Bounds on the Size of Universal Sampling Sets over Fr . . . . 64
p
2.5 Stable Universal Sampling Sets . . . . . . . . . . . . . . . . . 71
3 Symmetric Groups 94
3.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.2 Irreducible Characters of S . . . . . . . . . . . . . . . . . . . 111
n
3.3 Universal Sampling Sets for 2 for S . . . . . . . . . . . . . . . 115
n
3.4 Universal Sampling Sets for 3 for S . . . . . . . . . . . . . . . 137
n
4 Alternating Groups 185
4.1 Conjugacy Classes and Irreducible Characters of A . . . . . . 185
n
4.2 Universal Sampling Sets for 2 for A . . . . . . . . . . . . . . 187
n
Bibliography 244
3
4
Abstract
In this thesis I study sampling pairs and universal sampling sets for finite
groups G. Sampling pairs are pairs (Ω,Γ), where Ω is a subset of the irre-
ducible characters of G and Γ is a subset of the conjugacy classes of G, such
that the only linear combination of elements of Ω vanishing on Γ is the zero
function. A universal sampling set for t is a subset Γ of the conjugacy classes
of G such that, for every subset Ω of the irreducible characters of G with at
most t elements, (Ω,Γ) is a sampling pair.
I will first consider the case where G is abelian. Here t can be chosen
arbitrarily and I will construct explicit universal sampling sets for t. In the
special case of elementary abelian p-groups I will also give some algorithms
that allow to reconstruct a linear combination of at most t irreducible char-
acters from its restriction to Γ.
I will then study the case where G is a symmetric or an alternating group.
Here I will construct explicit universal sampling sets for t small (t ∈ {2,3}
for symmetric groups and t = 2 for alternating groups).
In der vorliegenden Arbeit untersuche ich Abtast-Paare und universelle Ab-
tast-Mengen endlicher Gruppen G. Abtast-Paare sind Paare (Ω,Γ), wobei
Ω eine Teilmenge der irreduziblen Charaktere von G und Γ eine Teilmenge
der Konjugiertenklassen von G ist, welche die Bedingung erfüllen, dass die
einzige Linearkombination von Elementen von Ω, die auf Γ verschwindet, die
Nullfunktion ist. Eine universelleAbtast-Mengefür tist eineTeilmengeΓder
Konjugiertenklassen von G, so dass für jede höchstens t-elementige Menge
irreduzibler Charaktere Ω das Tupel (Ω,Γ) ein Abtast-Paar ist.
Zuerst untersuche ich den Fall, dass G zusätzlich abelsch ist. Hier kann
t beliebig gewählt werden, und ich konstruiere einige explizite universelle
Abtast-Mengenfürt.FürdenSonderfallelementarabelscherp-Gruppengebe
ich außerdem Algorithmen an, die es ermöglichen, eine Linearkombination
von höchstens t irreduziblen Charaktere aus ihrer Einschränkung auf Γ zu
rekonstruieren.
Weiterhin konstruiere ich universelle Abtast-Mengen für den Fall, dass
G eine symmetrische oder alternierende Gruppe und t hinreichend klein ist
(t ∈ {2,3} für symmetrische Gruppen und t = 2 für alternierende Gruppen).
Introduction
This thesis investigates sampling pairs and universal sampling pairs of fi-
nite groups. To define these concepts , we will briefly introduce some basic
notation on representations of finite groups. For the remainder of this intro-
duction let G be a finite group.
A representation of G is a homomorphism ρ : G → GL(V) where V is
a vector space of finite dimension over a field K. We also say that ρ is a
representation of G on V. The character χ of ρ is the function of G with
ρ
values in K defined by
χ (g) := Tr(ρ(g)).
ρ
If g,h ∈ G are conjugate in G then χ (g) = χ (h), that is χ is a class
ρ ρ ρ
functionofG. Therepresentationρiscalledirreducibleifthereisnosubspace
0 (cid:54)= W (cid:40) V which is invariant under ρ(g) for all g ∈ G. If ρ is an irreducible
representation of G we say that χ is an irreducible character of G. Two
ρ
representations ϕ and ψ of G on V and W respectively are called equivalent
if, with the right choices of basis of V and W, the matrices corresponding to
ϕ(g) and ψ(g) are equal for every g ∈ G (the basis for V and W are fixed
and do not vary while varying through g ∈ G).
In this thesis we will only consider ordinary representations, that is we
will assume K = C. It can be proven that when working over C (or any
other algebraically closed field of characteristic 0) two representations are
equivalent if and only if their characters are the same. Also, the number of
irreducible representations up to equivalence (or of irreducible characters) of
GisexactlyequaltothenumberofconjugacyclassesofGandtheirreducible
characters of G form a basis of the set of class functions of G. The character
table of G is a matrix with rows indexed by the irreducible characters and
columns labeled by conjugacy classes, where the (χ,C)-entry is given by
χ(C) := χ(g) for any g ∈ C. From the previous remarks it follows easily that
the character table is a square matrix which is invertible.
Wewillnowgivethedefinitionofasamplingpairandauniversalsampling
set. Let Ω be a subset of the irreducible characters of G and let Γ be a subset
of the conjugacy classes of G. We say that (Ω,Γ) is a sampling pair if the
5
6 Introduction
only linear combination of the elements of Ω which vanishes on every element
of Γ is the zero function. Let t ∈ N. A subset Γ of the conjugacy classes
of G is called a universal sampling set for t for G if and only if (Ω,Γ) is a
universal sampling pair for every subset Ω of the irreducible characters of G
with |Ω| ≤ t.
We first consider in Section 2 the case where G is a finite abelian group.
In this case, if G = Z/n Z×···×Z/n Z, the irreducible characters of G can
1 k
be labeled by (a ,...,a ) ∈ Zn with 0 ≤ a < n such that
1 k i i
χ (x +Z/n Z,...,x +Z/n Z) = ωa1x1···ωakxk,
(a1,...,ak) 1 1 k k 1 k
where ω is a primitive n -th root of unity. In this case there is previous work
i i
byotherauthorsontheexistenceandconstructionofuniversalsamplingsets.
(cid:80) (cid:80)
Let f = a χ and g = b χ with χ running through the irreducible
χ χ χ χ
characters of G in both sums and with a ,b ∈ C. Assume that there exist
χ χ
(cid:80) (cid:80)
at most t non-zero coefficients a and that |b | (cid:28) |a |. Most of the
χ χ χ χ χ
previous work had been done in connection with approximating f from the
restriction of f + g to some universal sampling set. For example in [15] it
was proved that, for cyclic groups of prime order, if Γ is any subset of G
then Γ is a universal sampling set for t as long as t ≤ |Γ|; this is the best
possible bound. We will prove in Section 2.3 that if G is any finite cyclic
group, there is a universal sampling set for t of size min{|G|,t}, even if in
general not every subset of G of size min{|G|,t} is a universal sampling set
for t. Most of the previous work on finite abelian groups was concerned with
lower bounds for universal sampling sets for t, even if such sets were not
given explicitly. In [14] it was proved that there exist universal sampling sets
for t of size O(tlog4(|G|)). In [4], the infinite group H := {z ∈ C : |z| = 1}
was considered, whose irreducible characters can be labeled by the elements
of Z. For some fixed n ∈ N it was considered the problem of approximating
f from the f +g. Here f and g are as above, with the extra assumption that
a = 0 if χ is not labeled by some element of {−n,...,n}. Explicit universal
χ
sampling sets and algorithms are given which allow to approximate f. The
universal sampling sets constructed are subsets of Z/cnZ for some c ∈ N
depending on n and t and have size O(t2log2(n)). These methods cannot
however be applied to the finite cyclic group Z/nZ since in general the set
constructed is not a subset of Z/nZ but only of Z/cnZ.
We first consider in Section 2.3 general finite abelian groups G written in
cyclic decomposition as G = Z/n Z×···×Z/n Z. In this case we explicitly
1 k
construct a universal sampling set for t of size at most
t(1+log(n ))···(1+log(n )).
2 k
7 Introduction
This set depends on the factorisation of G. We will show that, among the
subsets obtained from the different factorisations of G, that of minimal size
is the one obtained from the factorisation which satisfies n | n | ... | n .
k k−1 1
For cyclic groups Z/nZ we also construct in Section 2.5.1 some other
explicit universal sampling sets for t of size at most
16t3log(n/2)log(2tlog(n/2)).
In Section 2.5.2 for elementary abelian p-groups (Z/pZ)r we construct
explicit universal sampling sets for t of size at most
2pt2r2.
Our construction uses techniques from [4]. Let f and g be as before. We
give some algorithms that allow to reconstruct f from its restriction to the
universal sampling sets constructed in this section for 2t. If applied to f +g
instead of f these algorithms return an approximation of f.
We then study in Sections 3 and 4 the case of symmetric and alternating
groups. Irreducible characters and conjugacy classes of the symmetric group
S can be labeled by partitions of n. Characters of the symmetric group
n
can be computed recursively using the Murnaghan-Nakayama formula (see
Theorem3.2.1), whichconnectsirreduciblecharactersandhookremovalfrom
partitions. Irreducible characters and conjugacy classes of the alternating
group A can be easily obtained from those of S , a well known fact recalled
n n
in Section 4.1.
Here the problem of constructing universal sampling sets has not been
studied yet and universal sampling sets will only be constructed for small
values of t, that is t = 2 or t = 3 for the symmetric group and t = 2 for the
alternating group. Invertible submatrices will be constructed for each pair
or triple of irreducible characters, and they will then be used to construct
explicit universal sampling sets.
Chapter 1
Sampling Pairs and Universal
Sampling Sets
In this section we will give the definitions of sampling pairs and universal
sampling sets over arbitrary finite groups and we will give some results about
them.
1.1 Definitions
If G is a finite group let Irr(G) denote the set of the irreducible characters
of G and C(G) the set of conjugacy classes of G. We will start with two
definitions.
Definition 1.1.1 (Sampling Pair). Let Ω ⊆ Irr(G) and Γ ⊆ C(G). We say
(cid:80)
that (Ω,Γ) is a sampling pair if whenever f = a χ and f = 0 then
χ∈Ω χ |Γ
f = 0.
It follows from the definition that (Ω,Γ) is a sampling pair if and only if
thesubmatrixofthecharactertableofGconsistingoftherowscorresponding
to elements of Ω and columns corresponding to elements of Γ has rank |Ω|.
An immediate consequence of this is that if (Ω,Γ) is a sampling pair then
|Ω| ≤ |Γ|.
Definition 1.1.2 (Universal Sampling Set). Let Γ ⊆ C(G). We say that Γ is
a universalsamplingset for t if (Ω,Γ) is a sampling pair for every Ω ⊆ Irr(G)
with |Ω| ≤ t.
As an easy consequence of the definition we have that if Γ is a universal
sampling set for t for G then |Γ| ≥ min{t,|G|}.
8
9 1.2. Results over Arbitrary Finite Groups
1.2 Results over Arbitrary Finite Groups
Assume that t ≥ 2. The next two theorems shows that how the problem
of constructing sampling pairs (Ω,Γ) with Ω ⊆ Irr(G) of size t or universal
sampling sets for t for G could be reduced to the same problem for 2, however
over the wreath product of G with A . Even if this reduction will not be used
t
in the following it is still written here to show that if we would be able to
solve the problem of constructing sampling pairs or universal sampling sets
for t = 2 for every group, then we would be able to solve it also for arbitrary
t. We will first give the definition of wreath product.
Definition 1.2.1 (Wreath Product). Let G be a group and H ≤ S . The
t
wreath product of G with H is given by
G(cid:111)H := {(g ,...,g ;h) : g ,...,g ∈ G, h ∈ H}.
1 t 1 t
The wreath product is a group with multiplication given by
(g ,...,g ;h)(g(cid:48),...,g(cid:48);h(cid:48)) = (g g(cid:48) ,...,g g(cid:48) ;hh(cid:48)).
1 t 1 t 1 h−1(1) t h−1(t)
Notice that Gt ∼= {(g ,...,g ;1) : g ,...,g ∈ G} ≤ G(cid:111)H. We will now
1 t 1 t
write the two theorems.
Theorem 1.2.2. Let G be a finite group and H ≤ S . Assume that χ ,...,χ
t 1 t
are pairwise different irreducible characters of G and that x = (g ,...,g ;h)
1 t
is an element of G(cid:111)H. Then ϕ := IndG(cid:111)At(χ ···χ ) is an irreducible character
Gt 1 t
of G(cid:111)H. Also if ϕ(x) (cid:54)= 0 then h = 1.
Proof. From Theorem 2.15 of [6] it follows that ϕ is irreducible. Since Gt is
a normal subgroup of G(cid:111)H we also have that if ϕ(x) (cid:54)= 0 then x ∈ Gt, that
is h = 0.
Theorem 1.2.3. Let t ≥ 2, Ω = {χ ,...,χ } consists of irreducible char-
1 t
acters of G and Γ = {C ,...,C } consists of conjugacy classes of G. Then
1 t
(Ω,Γ) is a sampling pair if and only if
(IndG(cid:111)At(χ χ χ ···χ ))(C ,...,C ;1)(cid:54)=(IndG(cid:111)At(χ χ χ ···χ ))(C ,...,C ;1)
Gt 1 2 3 t 1 t Gt 2 1 3 t 1 t
and this happens if and only if
({IndG(cid:111)At(χ χ χ ···χ ),IndG(cid:111)At(χ χ χ ···χ )},{1,(C ,...,C ;1)})
Gt 1 2 3 t Gt 2 1 3 t 1 t
is a sampling pair.
10 1.2. Results over Arbitrary Finite Groups
Proof. As |Ω| = |Γ| we have that (Ω,Γ) is a sampling pair if and only if the
corresponding submatrix of the character table of G has non-zero determi-
nant. We have that
t
(cid:88) (cid:89)
|(χ (C ))| = sign(π) χ (C )
i j π(i) i
π∈St i=1
t t
(cid:88) (cid:89) (cid:88) (cid:89)
= χ (C )− χ (C )
π(i) i (π(1,2))(i) i
π∈At i=1 π∈At i=1
(cid:88)
= (χ χ χ ···χ )(C ,...,C )
π(1) π(2) π(3) π(t) 1 t
π∈At
(cid:88)
− (χ χ χ ···χ )(C ,...,C )
π(2) π(1) π(3) π(t) 1 t
π∈At
(cid:88) (cid:0) (cid:1)
= (χ χ χ ···χ )(1,...,1;π) (C ,...,C )
1 2 3 t 1 t
π∈At
(cid:88) (cid:0) (cid:1)
− (χ χ χ ···χ )(1,...,1;π) (C ,...,C )
2 1 3 t 1 t
π∈At
= (IndG(cid:111)At(χ χ χ ···χ ))(C ,...,C ;1)
Gt 1 2 3 t 1 t
−(IndG(cid:111)At(χ χ χ ···χ ))(C ,...,C ;1).
Gt 2 1 3 t 1 t
Also
IndG(cid:111)At(χ χ χ ···χ ))(1) = deg(IndG(cid:111)At(χ χ χ ···χ )))
Gt 1 2 3 t Gt 1 2 3 t
|G(cid:111)A |
t
= deg(χ χ χ ···χ )
1 2 3 t
|G|
|G(cid:111)A |
t
= deg(χ )deg(χ )deg(χ )...deg(χ )
1 2 3 t
|G|
|G(cid:111)A |
t
= deg(χ χ χ ···χ )
2 1 3 t
|G|
= deg(IndG(cid:111)At(χ χ χ ···χ )))
Gt 2 1 3 t
= IndG(cid:111)At(χ χ χ ···χ ))(1).
Gt 2 1 3 t
Since (Ω(cid:48),Γ(cid:48)) is a sampling pair if and only if the correspond submatrix of
the character group has rank |Ω(cid:48)| the theorem follows.
In the next few theorems we will show how universal sampling sets for a
direct product of groups can be obtained from universal sampling sets for the
groups appearing in the product. Through these theorems for finite groups
G ,...,G we will identify Irr(G ×···×G ) with Irr(G )···Irr(G ). These
1 k 1 k 1 k