Schur-Positivity of Differences of Augmented Staircase Diagrams

157 Pages·2010·2.856 MB·English

by  Matthew Morin

Schur-Positivity of Differences of
Augmented Staircase Diagrams
by
Matthew Morin
B.Sc., Simon Fraser University, 2003
M.Sc., University of British Columbia, 2005
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
Doctor of Philosophy
in
The Faculty of Graduate Studies
(Mathematics)
The University of British Columbia
(Vancouver)
April 2010
© Matthew Morin 2010
Abstract
The Schur functions {s} and ubiquitous Littlewood-Richardson coefficients
c are instrumental in describing representation theory, symmetric functions,
and even certain areas of algebraic geometry. Determining when two skew
diagrams D1, D2 have the same skew Schur function or determining when
the difference of two such skew Schur functions is Schur-positive
S1D 5D2
—
reveals information about the structures corresponding to these functions.
By defining a set of staircase diagrams that we can augment with other
(skew) diagrams, we discover collections ofskew diagrams for which the ques
tion of Schur-positivity among each difference can be resolved. Furthermore,
for certain Schur-positive differences we give explicit formulas for computing
the coefficients of the Schur functions in the difference.
We extend from simple staircases to fat staircases, and carry on to dia
grams called sums of fat staircases. These sums of fat staircases can also be
augmented with other (skew) diagrams to obtain many instances of Schur
positivity.
We note that several of our Schur-positive differences become equalities
ofskew Schur functions when the number ofvariables is reduced. Finally, we
give a factoring identity which allows one to obtain many of the non-trivial
finite-variable equalities of skew Schur functions.
11
Contents
Abstract ii
Table of Contents iii
Acknowledgements v
1 Introduction 1
1.1 Partitions, Compositions, Ferrers Diagrams, and Skew Diagrams 1
1.2 Combining and Decomposing Diagrams 4
1.3 Tableaux 5
1.4 Symmetric Functions 6
1.5 The Littlewood-Richardson Rule 9
1.6 Some Useful Identities for Schur Functions 10
1.7 A Brief History of Schur Functions 12
1.8 Overview of Schur-Positivity Results 15
2 Staircases with Bad Foundations 23
2.1 Regular Staircases 23
2.2 Fat Staircases 27
2.3 Sums of Fat Staircases 33
3 Hook Foundations 43
3.1 Schur Comparability / Incomparability 43
.
3.2 Expressions for Schur-Positive Differences 60
3.3 Hasse Diagrams for k> 1 73
4 Differences of Transposed Foundations 91
4.1 Single Part Partitions 91
4.2 Two Part Partitions 95
111
5 Complements in a Rectangle 111
.
5.1 Preliminaries 111
5.2 Staircases with Hook Complement Foundations 116
5.3 Sums of Fat Staircases 123
5.4 Miscellaneous Results 128
6 Reduction to Finite Variables 133
6.1 Differences of Transposed Foundations 133
6.2 Factoring in Finite Variables 136
7 Conclusion 144
Bibliography 146
iv
Acknowledgements
I would like to thank my supervisor, Stephanie van Willigenburg, for her con
tinual support, her remarkable ability to neatly turn a problem on its head,
and her constant and vigilant attention to detail. Without her guidance, this
thesis would be naught but a work-in-progress.
I would also like to show my gratitude to all members of my thesis com
mittee for accepting the responsibility ofjudging my dissertation. The will
ingness to spend the time and effort for this endeavor is much appreciated.
Financial credit must be given to NSERC for funding the past three years
of my doctoral program. Life would have been much more difficult without
the scholarship they generously awarded me.
More thanks are due to UBC, for their support throughout my entire
years of study and the many opportunities they have granted. The entire
staff of the Mathematics department deserves praise for its ability to keep
everything running smoothly and efficiently not just in the department as a
whole, but down through each and every individual in the department.
Finally, I must thank all my friends, family, and loved-ones for their non
academic support. I could not be here now without your unwavering belief
in myself and my abilities. Although you may not fully understand what it
is I do, you have always shown pride in my desire to do it.
V
Chapter 1
Introduction
1.1 Partitions, Compositions, Ferrers Diagrams,
and Skew Diagrams
A partition A of a positive integer n, written A H n, is a sequence of weakly
decreasing positive integers A (1,A A2, A,) with A n. \1Ve call
= , =
each A2 a part of A, and if A has exactly k parts we say A is of length k and
write 1(A) k. When A I— n we will also write IA n and say that the size
=
of A is n.
We shall use jT to denote the sequence j,j j consisting ofr j’s. Under
this notation, we shall write A = (1,k k — 17’k—1 . . , U’) for the partition
which has r1 parts of size one, r2 parts of size two, and rk parts of size k.
...,
Given two partitions A = (1,A . . . , A,) and p = (..h,. . . , ,Lim), we let A U ,u
denote the partition that consists of the parts A1, . , A, /1,J. , ltm placed
in weakly decreasing order.
We say a (1,a a2,.. is a composition of n, written a n, if each
= . , ak) =
aj is a positive integer and a n. As with partitions, we call each a2
=
a part of a, write a n for the size of a, and if a has exactly k parts we
=
say a is of length k and write 1(a) k. If we relax the conditions to consider
=
sums of non-negative integers, that is, allowing some of the a, to be zero,
then we obtain the concept of a weak composition of n. If a = n we obtain
a partition of n by reordering the a into weakly decreasing order.
We may sometimes find it useful to treat partitions and weak composi
tions as vectors with non-negative integer entries. VThen we write the vector
z = (1,z z2, z3,. z) we shall mean the infinite vector
. . ,
z1,2,3,...,z=,Ozz,O(,O,z...).
We shall only consider vectors with finitely many non-zero entries. Hence we
1
_
CHAPTER 1. INTRODUCTION
shall oniy display vectors with finite length. In this manner we may unam
biguously add vectors of different lengths. Thus, we have defined addition
among partitions and weak-compositions. Further, given a positive integer
i, we shall let e denote the i-th standard basis vector. That is, the vector
that has its i-th entry equal to 1 and all remaining entries equal to 0.
Given a partition A, we can represent it via the diagram of left justified
rows of boxes whose i-th row contains A boxes. The diagrams of these type
are called Ferrers diagrams, or just diagrams for short. We shall use the
symbol A when refering to both the partition and its Ferrers diagram.
Whenever we find a diagram D’ contained in a diagram D as a subset
of boxes, we say that D’ is a subdiagram of D. Suppose partitions A
(1,2,..AA.,A) F-n and fL (1,2,i.ti. .jtk) F-rn, with m < n, k <j, and
=
A for each i = 1, 2,.. . , k are given. Then t is a subdiagram of A, and
a particular copy of is found at the top-left corner of A. We can form the
skew diagram A/,u by removing that copy of p from A. Henceforth, when we
say that D is a diagram, it is assumed that D is either the Ferrers diagram
of some partition A, or D is the skew diagram A/j for some partitions A, ,u.
Example Here we consider the partitions A (4, 4, 2) and (3, 1), and
= =
form the skew diagram A/si.
[=iJr
H
For two boxes b1 and b2 in a diagram D, we define a path from b1 to b2
in D to be a sequence of steps either up, down, left, or right that begins at
b, ends at b2, and at no time leaves the diagram D. We say that a diagram
D is connected if for any two boxes b1 and b2 of D there is a path from b to
b2 in D. If D is not connected we say it is disconnected.
Given any diagram D, the 1800 rotation of a diagram D is denoted by
D°. Further, the conjugate or transpose diagram Dt is obtained by simply
transposing the array of boxes along the main diagonal. When D is the
Ferrers diagram of a partition A then the conjugate diagram Dt is also the
Ferrers diagram of a partition called the conjugate partition and denoted
At. It must be noted that most authors use the notation D’ and A’ for the
conjugate diagram of D and conjugate partition of A. We hope our notation
causes no confusion.
2
CHAPTER 1. INTRODUCTION
Example Here we consider the partitions A = (4, 4, 2) and i = (3, 1), and
form the skew diagram A1i./
HH
The conjugates, At = (3,3,2,2), = (2,1,1) and (A/ji)t = At/itt are
shown below.
Finally, we see the skew diagrams (A/i)° and (t)(A°/j) = ((A/)0)t.
The number of boxes that appear in a given row or a given column of a
diagram is called the length of that row or column. A hook is the Ferrers
diagram corresponding to a partition A that satisfies A < 1 for all i > 1.
Hence a hook has at most one row of length larger than 1.
Example Here we see the hooks A = (4, 1, 1) and t = (5, 1).
Li
‘A’ It
3
_ __
CHAPTER 1. INTRODUCTION
1.2 Combining and Decomposing Diagrams
In this section we look at combining and decomposing skew diagrams in a
few simple ways.
Given two diagrams D1 and D2 the concatenation ofD1 and D2, denoted
• D2, is the skew diagram obtained by placing D1 and D2 so that the top-
right box of D1 is immediately below the bottom-left box of D2. Similarly,
the near-concatenation ofD1 and D2, denoted D1 0 D2, is the skew diagram
obtained by placing D1 and D2 so that the top-right box of D1 is immediately
left of the bottom-left box of D2. Finally, if the last column of D1 and first
column of D2 are both of length i, then the near-concatenation of depth
i of D1 and D2, denoted D1 0 D2, is the is the skew diagram obtained by
placing D1 and D2 so that the top-right box of D1 is one step left and i 1
—
steps up from the bottom-left box of D2. We note that 0 0.
=
Example Let D1 be the skew diagram (2, 2, 2)/(1, 1) and D2 be the Ferrers
diagram (4,4,2).
Here we show the diagrams D1 D2, D1 0 D2, D1 02 D2, and D1 0 D2.
I_
D1 0 D2 D1 02 D2 D1 03 D2
Given a diagram D, the connected components of D are the maximally
connected subdiagrams of D. Any skew diagram D decomposes into a finite
number of connected components D1, D2,. D,. Conversely, given con
. . ,
nected diagrams D1, D2,. . . , Dk, we define the diagram D = D, called
the direct sum of the D, to be the skew diagram with connected components
D1, D2,. Dk such that the top-right box of D is one step left and one step
. . ,
down from the bottom-left box of D1 for i = 1, 2,. . . , k — 1. We note that
=0o.
4
CHAPTER 1. INTRODUCTION
Example With D1 (2, 2, 2)/(1, 1) and D2 (4, 4, 2), as in the previous
= =
example, we now show the diagram D1 D2.
D1D2
1.3 Tableaux
If D is a diagram, then a tableau—plural tableaux—T of shape D is the
array obtained by filling the boxes of the D with the positive integers, where
repetition is allowed. A tableau is said to be a semistandard Young tableau
(or simply semistandard, for short) if each row gives a weakly increasing
sequence of integers and each column gives a strictly increasing sequence of
integers. We will often abbreviate “semistandard Young tableau” by SSYT
and “semistandard Young tableaux” by SSYTx.
When we wish to depict a certain tableau we will either show the under
lying diagram with the entries residing in the boxes of the diagram or we
may simply replace the boxes with the entries, so that the entries themselves
depict the underlying shape of the tableau.
The content of a tableau T is the weak composition given by
v(T) (1,2,..uv.),
=
where is the number of i’s that appear in T.
Vj
Example Here we consider a semistandard Young tableau 7Ej with shape
D1 (2, 2, 2)/(1, 1) and content (1,0, 1, 2), and asemistandardYoung tableau
=
7 of shape D2 = (4,4,2) and content (2,2,3,2, 1).
1124
3 2335
144
34
7 72
We may sometimes find it useful to take the transpose of a tableau T
so that we obtain a tableau of conjugate shape. In these cases we simply
transpose the entries of T along with the underlying shape D. In this way,
we obtain a tableau 7 of shape Dt with c(Tt) c(T).
=
5