Table Of ContentREPRESENTATIONS OF THE SYMMETRIC GROUPS AND
COMBINATORICS OF THE FROBENIUS-YOUNG CORRESPONDENCE
By
MATTHEW JOHN HALL
A Thesis Submitted to The Honors College
In Partial Fulfillment of the Bachelor’s degree
With Honors in
Mathematics
THE UNIVERSITY OF ARIZONA
MAY 2008
Approved by:
Christopher Ryan Vinroot
Department of Mathematics
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Abstract
After introducing the concepts of partitions, Young diagrams, represen-
tation theory, and characters of representations, the 2006 paper by A. M.
Vershik entitled “A New Approach to the Representation Theory of the
Symmetric Group, III: Induced Representations and the Frobenius-Young
Correspondence” is discussed. In tracing through Vershik’s line of reason-
ing, a flaw emerges in his attempt to prove one of the key lemmas. The
attempted proof is an induction argument which, if valid, would lead to a
purelycombinatorialproofoftheFrobenius-Youngcorrespondence. Vershik
asserts that two statements are equivalent, while in fact the implication is
true in only one direction. Counterexamples are then given to Vershik’s
combinatorial statement and are then translated into counterexamples in
representation theory. Finally, possible directions for further study are
noted.
CONTENTS 4
Contents
1 Introduction 5
2 Partitions 6
2.1 Conjugate Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Diagrams and Tableaux . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Dominance Ordering on Conjugate Partitions . . . . . . . . . . . . 9
3 Representation Theory 11
3.1 Group Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 CG-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . 12
3.4 Decomposition of Representations . . . . . . . . . . . . . . . . . . . 13
3.5 Restricted and Induced Representations . . . . . . . . . . . . . . . 13
3.6 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.7 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.8 Frobenius Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.9 The Number of Irreducible Representations . . . . . . . . . . . . . . 16
3.10 Double Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.11 Mackey’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Representations of the Symmetric Groups 18
4.1 The Irreducible Representations of S . . . . . . . . . . . . . . . . . 19
n
4.2 Branching Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Young Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4 Frobenius-Young Correspondence . . . . . . . . . . . . . . . . . . . 21
4.5 Mackey’s Theorem Applied to S . . . . . . . . . . . . . . . . . . . 21
n
4.6 The Snapper Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 23
4.6.1 Combinatorial Statement . . . . . . . . . . . . . . . . . . . . 23
4.6.2 Combinatorial Counterexamples . . . . . . . . . . . . . . . . 25
4.6.3 Representation Theory Counterexamples . . . . . . . . . . . 26
4.7 Frobenius-Young Correspondence, Revisited . . . . . . . . . . . . . 28
5 Further Questions 28
1 INTRODUCTION 5
Prelude
The theory of representations of finite groups focuses on homomorphisms from a
finite group to the group of invertible linear transformations of a vector space.
After spending a semester actively studying these areas with the help of the excel-
lent text Representations and Characters of Groups by Gordon James and Martin
Liebeck, I was encouraged by my thesis adviser, Professor Ryan Vinroot, to ex-
plore a recent journal article by the mathematician A. M. Vershik entitled “A
New Approach to the Representation Theory of the Symmetric Groups, III: In-
duced Representations and the Frobenius-Young Correspondence,” since this arti-
cle served to illustrate and apply many of the concepts I had been learning. While
making my way through this dense work, I attempted to prove for myself some
of the theorems Vershik gave, so as to better understand his reasoning. However,
after a rather lengthy struggle to prove a central lemma of the paper, I realized
that there were in fact counterexamples, and hence the proof of the lemma was
not valid. After discussing the problem with Ryan, who agreed that my coun-
terexamples held, I began to investigate the ramifications of this error, whether
or not the paper’s main argument could still stand, and whether the error could
be corrected. This thesis is the culmination of these efforts; through it, I hope to
give to the reader familiar with the basic concepts of group theory a mathemat-
ical background sufficient to understand Vershik’s paper, and then to review his
argument and the counterexamples.
1 Introduction
This paper begins with an overview of some broad notions from combinatorics, in
particular the concepts of partitions of positive integers, Young diagrams, Young
tableaux,andorderingsonthesetofYoungdiagrams. Thesetoolsarefoundational
to understand Vershik’s argument in [10], because he attempts a purely combi-
natorial proof of the Frobenius-Young correspondence (see Section 4.4). Next,
some basic concepts of the representation theory of finite groups are discussed,
including Maschke’s Theorem and its consequences, induced representations, the
Frobenius Reciprocity Theorem, the number of non-isomorphic irreducible rep-
resentations, and Mackey’s Theorem. These tools are necessary to understand
what the Frobenius-Young correspondence states, as well as to translate com-
binatorial counterexamples to one of Vershik’s statements into their equivalent
representation theoretic counterexamples. Finally, the representation theory of
the symmetric group S is explored, using both methods from both combinatorics
n
2 PARTITIONS 6
and representation theory of general finite groups. After a formal statement of
the Frobenius-Young correspondence, facts such as the branching rules and the
notion of Young subgroups are used to trace through Vershik’s argument. An
in-depth analysis of Vershik’s third lemma (see section 4.6) reveals that where
Vershik claims an equivalence between statements, there is in fact only an impli-
cation in one direction. This error is explored, an infinite family of combinatorial
counterexamples is given, and the equivalent counterexamples in representation
theory are also given. Ultimately, though Vershik’s attempted purely combina-
torial proof collapses, it is possible that there may be another method by which
to successfully complete such a proof, and this question is proposed for further
investigation.
2 Partitions
Partitions are an important tool in group theory, particularly in the theory of the
symmetric group. A partition λ of the integer n, written λ ‘ n, is a sequence of
positive integers λ = (λ ,λ ,...,λ ) such that λ ≥ λ for all integers 1 ≤ i ≤
1 2 k i i+1
(k −1) and Pk λ = n. Each element λ of the partition is called a part of the
i=1 i i
partition.
2.1 Conjugate Partitions
For each partition λ of n, there is a conjugate partition of n, denoted λ0 =
(λ0,λ0,...,λ0), where
1 2 l
λ0 = max{j |λ ≥ i}.
i j
The fact that λ0 is actually a partition of n may not be immediately apparent but
will become clearer with the introduction of Young diagrams.
2.2 Diagrams and Tableaux
One important way of visualizing partitions is through the use of Young diagrams.
For each partition λ = (λ ,λ ,...,λ ) the corresponding Young diagram, [λ],
1 2 k
is a left-justified diagram in which the ith row from the top contains λ boxes, or
i
cells. For example, the Young diagram of the partition λ = (6,4,1,1) of n = 12
2 PARTITIONS 7
is:
[λ] =
The conjugate partition, as defined above, yields a conjugate Young dia-
gram. InrelationtotheoriginalYoungdiagram[λ], theconjugateYoungdiagram
[λ0] turns out to be the diagram which interchanges the rows and columns of [λ];
that is, the diagram obtained by reflecting [λ] about the main diagonal. This helps
to justify the fact that the conjugate partition is in fact another partition of n,
since by considering the Young diagrams, one may realize that reflection about
the main diagonal will result in a new diagram with the same number of boxes,
namely n. In particular, if λ = (6,4,1,1) as above, the conjugate partition is
λ0 = (4,2,2,2,1,1) and the Young diagram associated with this partition is:
[λ0] =
Proposition 2.1. (λ0)0 = λ
Proof. Passing to the conjugate is equivalent to making the rows of the original
diagram into the columns of the conjugate diagram and vice versa. Thus, taking
the conjugate again will switch the rows and columns back, once again giving the
original diagram.
Sometimes, such as when working with the certain aspects of the symmetric
group S , it may be desirable to place the integers 1 through n, once each, in
n
the boxes of the Young diagram; the result of this is a Young tableau. Using
the aforementioned partition λ = (6,4,1,1) of n = 12, one Young tableau of this
partition is given as follows:
1 2 3 4 5 6
7 8 9 10
11
12
One other important term is the shape of a Young tableau, which refers to the
partition that corresponds to the underlying Young diagram. The Young tableau
2 PARTITIONS 8
above has shape λ = (6,4,1,1), for example.
2.3 Orderings
There are two important orderings on Young diagrams, and thus also on partitions
of n. The dominance ordering is a partial ordering on partitions of n. Given
any two partitions µ = (µ ,µ ,...,µ ) and λ = (λ ,λ ,...,λ ) of n, µ dominates
1 2 l 1 2 k
λ, denoted µ D λ or λ E µ, if and only if Pm µ ≥ Pm λ for all m between 1
i=1 i j=1 j
and max{k,l}, inclusive, defining µ = 0 if i > l and λ = 0 if j > k. In terms of
i j
Young diagrams, [µ] D [λ] if and only if for all m between 1 and max{k,l}, there
are more boxes in the first m rows of [µ] than there are in the first m rows of [λ].
For instance, given the partitions µ = (6,3,2,1) and λ = (5,3,3,1) of n = 12, one
can see that µ D λ, that is:
D
For n ≤ 5, the dominance ordering is more than a partial ordering; it is in fact
a total ordering. Consider, for instance, the first two values of n. When n = 1,
the only Young diagram is the unique diagram with one block, that is:
[λ ] =
1
Moving up to n = 2, there are two distinct Young diagrams: the one obtained
by adding a second block to the one row of [λ ] and the one obtained by adding
1
a second row of one block to [λ ]. It is clear that this is total ordering, since
1
D
ByconstructingthetreeofYoungdiagramswhichpartitionsuccessivelygreater
n, it can be checked that for n ≤ 5, any two diagrams partitioning n can be com-
pared by the dominance ordering; hence, for these n, the dominance ordering is
a total ordering. However, when n = 6, the partitions and corresponding dia-
grams of λ = (3,1,1,1) and µ = (2,2,2) cannot be compared with the dominance
ordering, and so the ordering becomes only a partial ordering.
The other important ordering is used to relate two Young diagrams which
correspond to partitions of integers n and n+1. If γ ‘ n and λ ‘ (n+1), then
2 PARTITIONS 9
one writes that γ ≺ λ or λ (cid:31) γ if and only if the diagram [λ] can be obtained
by adding one block to [γ] in such a way that the result is still a valid diagram,
or equivalently, [γ] can be obtained by removing a block from [λ] in such a way
that the result is still a valid diagram. Using the partition λ = (6,3,2,1) of 12,
and defining a partition γ = (5,3,2,1) of 11, one can see by looking at the Young
diagrams that γ ≺ λ:
≺
2.4 Dominance Ordering on Conjugate Partitions
In Vershik’s attempt to develop a purely combinatorial proof of the Frobenius-
Young correspondence (see Section 4.4), he seeks to build a connection between
two collections of Young diagrams. His goal in doing so is to eventually show the
uniqueness of a diagram that he claims is common to both collections. The proof
of Vershik’s second lemma is a corollary of another lemma, which follows.
Lemma 2.2. Let λ and µ be two partitions of n. Then µ D λ if and only if
λ0 D µ0.
Proof. (Following the proof of (1.11) in [4].) First, note that proving one implica-
tion also proves the other, because if (µ D λ) ⇒ (λ0 D µ0), then Proposition 2.1
implies that (λ0 D µ0) ⇒ (µ00 D λ00) ⇒ (µ D λ).
Define partitions λ = (λ ,λ ,...,λ ) and µ = (µ ,µ ,...,µ ) of the integer n.
1 2 a 1 2 b
Suppose λ0 4 µ0 and µ0 4 λ0 (this second condition is necessary; without it, one
could simply switch the roles of µ and λ). Then there exists an j ∈ N such that
p p
X X
λ0 ≥ µ0
i i
i=1 i=1
for all 1 ≤ p ≤ j −1, and
j j
X X
λ0 < µ0. (1)
i i
i=1 i=1
Taken together, these two equations imply that µ0 = m is greater than λ0 = l.
j j
Also, since the sum of all the parts of both µ0 and λ0 must equal n, equation (1)
2 PARTITIONS 10
implies that
a b
X X
λ0 > µ0. (2)
i i
i=(j+1) i=(j+1)
Now consider the Young diagrams [λ] and [µ] corresponding to λ and µ, re-
spectively. The sum Pa λ0 gives the number of boxes of [λ] which are to the
i=(j+1) i
right of column j, and similarly the sum Pb µ0 gives the number of boxes of
i=(j+1) i
[µ] which are to the right of column j. Hence,
a l b m
X X X X
λ0 = (λ −j) and µ0 = (µ −j).
i k i k
i=(j+1) k=1 i=(j+1) k=1
Thus, applying equation (2) and observing that m > l and µ ≥ j whenever
k
1 ≤ k ≤ m, one may see that
l m l
X X X
(λ −j) > (µ −j) ≥ (µ −j). (3)
k k k
k=1 k=1 k=1
Adding lj to both sides of the inequality of the leftmost and rightmost sum in
equation (3) yields
l l
X X
λ > µ
k k
k=1 k=1
and thus, µ 4 λ.
Therefore, (λ0 4 µ0) ⇒ (µ 4 λ), and the contrapositive of this statement shows
that (µ D λ) ⇒ (λ0 D µ0), which by the earlier argument also demonstrates that
(µ D λ) ⇔ (λ0 D µ0).
Corollary 2.3. (Lemma2in[10]) Let [λ] be an arbitrary diagram which partitions
n, and let [λ0] be its conjugate diagram. Define two collections of diagrams: A =
{[α]|α ‘ n,α D λ} and B = {[β0]|β0 ‘ n,β D λ0}. Then A∩B = {[λ]}.
Proof. By the above lemma, B is equivalent to the set of all diagrams [β0] for
which λ D β0 (since λ00 = λ by Proposition 2.1). So for a diagram [φ] to be in the
intersection of A and B, it must be true that both φ D λ (to be in A) and λ D φ
(to be in B). Since D is a partial ordering, this can only occur if φ = λ. Thus, [λ]
is the only diagram common to both sets.
