Table Of ContentRaising operators, recurrences, and the
Littlewood–Richardson polynomials
Alex Fun
School of Mathematics and Statistics
University of Sydney
[email protected]
A thesis submitted in fulfillment of
the requirements for the degree of Doctor of Philosophy.
September 26, 2012
This was made possible by the following people:
Alex, my supervisor
Eve, my wife
My friends
My family
Acknowledgements
The following people contributed to the creation of this thesis. I would like to thank
AnthonyHenderson, who supervised my Honoursthesisand set me on thepath on re-
search by encouraging me to start a PhD, and also for his latex code. I would also like
tothankDavidEasdownforgivingmetheopportunitytostartaPhDtopicwithhim.
Most of all, to Alex Molev: I still vividly remember the 2nd year introduction to
modern algebra exam you set, which caused me to take a one year hiatus from pure
mathematics. No one would believe me when I told them I had started doing a PhD
with you. But the truth is this. You taught symmetric functions with such vigour
and enthusiasm, and I was hooked immediately. Thank you for your dedication and
hard work in supervising me, and for getting me out of tough spots when the chips
were down. Thank you also for your willingness to share what you know and for
listening to what I had to say. Without your advice none of this would have been
possible.
I am grateful to have had the opportunity to spend the last four years in research.
During this time, I was able to develop my interests in coffee, cooking, bicycles, and
travel. I am grateful for the companionship of my family and friends, and for the
experiences we shared together. To my wife, thank you for being my constant accom-
plice throughout all of this. Really, I don’t know how I would have done it without
your love and support. To usydgroup, likewise. To Olivia, Thom, Ben, et al, I am
glad Eve met Olivia in China. To my family, thank you for putting up with me. I
swear I will go do something “real” now, although you might lose out on the daytime
visiting hours. xxoo.
The proofreading achievement award is given to the following people: Eve Slavich,
Steph Betz, Daniel Yardley, Nicole Mealing, Poker Chen, Jon Cohen, Saritha Man-
ickam, James Haggerty, Alex Feigin. I am blessed to know such intelligent people
who can give me constructive commentary in a field even I sometimes find obtuse.
Furthermore, a special mention must go to Alex Molev who patiently corrected all of
my stupid mistakes and taught me how to write in a mathematical style, and to my
examiners who gave me heartwarming feedback.
Last but not least, I acknowledge the support of the School of Mathematics and
Statistics in the University of Sydney, for advice, financial support, and the opportu-
nity to tutor and lecture maths courses. This thesis was also funded by an Australian
Postgraduate Award.
Contents
Introduction i
1 Littlewood–Richardson Coefficients 1
1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Reformulating the Pieri rule . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Calculating the Littlewood–Richardson coefficients . . . . . . . . . . . 7
2 The symmetric functions 12
2.1 The ring of double symmetric polynomials . . . . . . . . . . . . . . . 12
2.2 Littlewood–Richardson polynomials . . . . . . . . . . . . . . . . . . . 19
3 Raising operators and the Littlewood–Richardson polynomials 23
3.1 Raising operators, Macdonald’s 9th Variation, and the Pieri rule . . . 24
3.2 Pieri rule: after specialisation . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Using tableaux to calculate dσ′(a) . . . . . . . . . . . . . . . . . . . . 36
3.4 Weights of tableaux express the coefficients dσ′(a) . . . . . . . . . . . 38
3.5 Pieri rule: statement and proof . . . . . . . . . . . . . . . . . . . . . 44
3.6 Littlewood–Richardson polynomials . . . . . . . . . . . . . . . . . . . 48
3.7 ψ : tail swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1
3.8 ψ : reorder barred entries . . . . . . . . . . . . . . . . . . . . . . . . 54
2
3.9 Weights of T and T are almost equal . . . . . . . . . . . . . . . . . . 55
3.10 Example of the involution . . . . . . . . . . . . . . . . . . . . . . . . 60
e
4 The dual Littlewood–Richardson polynomials 65
5 A Pieri rule for the generalised Frobenius–Schur functions 75
5.1 Generalised Frobenius–Schur functions . . . . . . . . . . . . . . . . . 76
5.2 Vanishing property for the generalised Frobenius–Schur functions . . 79
5.3 The recurrence relation . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 The difference formula . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.5 Barred tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.6 Word of a tableau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.7 Weight of tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.8 Dual tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.9 Calculating kλ (a) when λ/π contains one box . . . . . . . . . . . . . 98
θπ
5.10 Calculating kλ (a) when λ/π contains two boxes . . . . . . . . . . . . 101
θπ
5.11 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
(k+1) (k−1)
5.12 If ρ /ρ obeys Cond. 1 . . . . . . . . . . . . . . . . . . . . . . 133
+ +
(k+1) (k−1)
5.13 If ρ /ρ obeys Cond. 2 . . . . . . . . . . . . . . . . . . . . . . 134
+ +
(k+1) (k−1)
5.14 If ρ /ρ obeys Cond. 3 . . . . . . . . . . . . . . . . . . . . . . 134
+ +
5.15 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6 Conclusion 137
Bibliography 139
Introduction
The aim of this thesis is to calculate two types of Littlewood–Richardson polynomi-
als. These are structure coefficients in the ring of double symmetric functions Λ(x||a)
which has a distinguished basis consisting of the double Schur functions s (x||a). The
λ
first type of Littlewood–Richardson polynomials arises when we consider the product
of two double Schur functions, the second when the comultiplication operation in the
ring Λ(x||a) is applied to a double Schur function. When the ring Λ(x||a) is specialised
to the ring of symmetric functions Λ(x), we recover the Littlewood–Richardson coeffi-
cients. Apartfromtheirapplicationsinthecombinatoricsofsymmetricfunctions, the
Littlewood–Richardson polynomials are important for the following reasons. They
are applied in geometry and representation theory. The first type of polynomials de-
scribe a multiplication rule for equivariant Schubert classes, and also a multiplication
rule for virtual quantum immanants and higher Capelli operators. The second type
is relevant to describing equivariant cohomology of infinite grassmanians.
The structure of this thesis is as follows. In Chapter 1, we introduce well known
definitions associated with the ring of symmetric functions Λ(x). Using the Pieri rule
and Jacobi–Trudi identity, we then present a proof of a rule used to calculate the
Littlewood–Richardson coefficients. This is Theorem 1.8. This proof we present is a
simplified version of our main result in Chapter 3.
In Chapter 2, we introduce the ring of double symmetric functions Λ(x||a), which
is a generalisation of the classical ring Λ(x) depending on an extra set of infinite
variables a = (a ) . We introduce the basis of double Schur functions, and then
i i∈Z
explain how the two types of Littlewood–Richardson polynomials arise as structure
coefficients involving the double Schur functions. We also discuss the significance of
these structure coefficients in combinatorics, representation theory, and geometry.
In Chapter 3, we present one of the main results of this thesis using raising
operators. This is a new proof of Theorem 3.33, a known formula which calculates
the Littlewood–Richardson polynomials arising between the product of two double
Schur functions. Our proof relies on two things: first, we introduce a Jacobi–Trudi
identity for the double Schur functions. Second, we derive a Pieri rule for the ring
Λ(x||a). This Pieri rule is in turn a specialisation of a more general rule which we also
introduce for the ring A generated by the indeterminates h from the 9th Variation
r,s
of Macdonald [14].
In Chapter 4, we discuss the dual Littlewood–Richardson polynomials which arise
when comultiplication is applied to the double Schur functions. We also discuss
the dual Schur functions and skew double Schur functions. The dual Littlewood–
Richardson polynomials then give combinatorial identities involving these functions.
In the conclusion of Chapter 4, we present another main result of this thesis. This
i
is Theorem 4.3, which provides a stable formula to calculate the dual Littlewood–
Richardson polynomials.
In Chapter 5, we introduce the ring of generalised supersymmetric functions
Λ(x/y||a), which has a distinguished basis consisting of generalised Frobenius–Schur
functions s (x/y||a). Using a recurrence relation, we produce another main result of
λ
this thesis. This is a Pieri rule which gives the structure coefficients arising out of
the product between the functions s (x/y||a) and s (x/y||a), where λ is an arbitrary
θ λ
partition and θ is a skew partition not containg a 2 × 2 subdiagram; this is Theo-
rem 5.34. A specialisation of this theorem then lets us evaluate some of the dual
Littlewood–Richardson coefficients.
ii
1 Littlewood–Richardson Coefficients
This thesis is about the combinatorics which arise when attempting to solve problems
in the ring of double symmetric functions. We would like to give such an example in
this chapter, using a simplified version of a rule we use in Chapter 3. First, we will
state the Pieri rule and Jacobi–Trudi identity without proof. Then, the focus is to
use both of them to give a version of the Littlewood–Richardson rule (Theorem 1.8).
The proof of Theorem 1.8 illustrates the type of combinatorics of tableaux which we
use throughout the thesis. This is our main focus and in this chapter we will try to
keep definitions to a minimum; in the next chapter a proper introduction of the ring
of double symmetric functions will be given.
1.1 Definitions
A composition α of length l is a sequence of nonnegative integers (α ,α ,...,α ) such
1 2 l
that α is positive. The length of α is denoted by l(α). We will also write α as the
l
sequence (α ,...,α ,0,...); that is we will add as many zeroes as we like to the end
1 l
of the sequence α. The empty composition ∅ is an infinite sequence of zeroes, with
length equal to 0 by definition. We say a composition α contains the composition
β if α > β , for all i > 0, and denote this by α ⊇ β. If α and β are compositions
i i
the sum α +β is the composition defined as the pointwise sum of α and β; that is
α+β = (α +β ,α +β ,...), extending the sequence α or β with as many zeroes
1 1 2 2
as necessary to make their lengths equal. A partition λ is a composition such that
λ > λ > ....
1 2
The diagram of a composition α is a finite collection of boxes, left justified, with
rows numbered 1 to l, starting from the top and ending at the bottom, such that
there are α boxes in row i. Note that we will use the words diagram and composition
i
interchangeably.
Example 1.1. For example, for the composition α = (2,3,5) we have the diagram
α =
and for λ = (5,3,1) we have the diagram
λ =
1
If α and β are both compositions and α ⊇ β, we may define the skew composition
α/β as the set theoretic difference consisting of boxes in α but not in β. For example,
if α = (2,3,5) and β = (0,3,4), we have the skew composition:
α/β =
If µ ⊆ ν are a pair of partitions, then the skew diagram ν/µ is called a horizontal
strip if there is at most one box in every column of ν/µ.
For example, for the partitions ν = (5,4,3) and µ = (4,3,1) we have that
ν/µ =
is a horizontal strip.
Let A be the ring of polynomials in the variables x = (x ,...,x ) with coefficients
1 n
from Z. The symmetric group on n elements S acts on A by permuting the indices
n
of the variables x . The ring of symmetric polynomials Λ (x) is the subring of A
i n
consisting of all polynomials which are invariant under the action of all permutations
in the group S . It is generated over Z by the complete symmetric polynomials,
n
which we denote by h , for all p = 1,2,..., with
p
h = x x ...x .
p i1 i2 ip
n>i1>iX2>...>ip>1
This means that any element in Λ (x) is written as a polynomial P(h ,h ,...) with
n 1 2
coefficients from Z. For convenience, define h = 1 and h = 0 if p < 0. If κ is a
0 p
composition, we denote by h the product h h ...h .
κ κ1 κ2 κl(κ)
The ring Λ (x) has a basis over Z consisting of the family of Schur polynomials
n
{s }, indexed by all partitions Λ(x) with length at most n. We can define these
λ
polynomials in terms of the complete symmetric polynomials using the Jacobi–Trudi
identity. First we need to define the action of permutations on compositions, that
is, for the transposition σ = (i,i + 1) ∈ S , we have σ(α) is the composition equal
l
to α with its i-th and i + 1-th entry swapped. Let τ = (l − 1,l − 2,...,0) be a
l
composition. For a permutation ω ∈ S , define the composition κω = ω(λ+τ )−τ
l l l
andsgn(λω) = sgn(ω), theparityofthepermutationω. Wenowusethesedefinitions
to write the Schur polynomial s in terms of the complete symmetric polynomials h .
λ p
Let Λ(x) be a partition and l = l(λ). The Schur polynomial s may be expressed
λ
in terms of the complete symmetric polynomials via the Jacobi–Trudi identity:
s = det(h )
λ λi+j−i 16i,j6l
(1.1)
= sgn(κ)h ,
κ
κ
X
2
summed over compositions κ = λω, for all ω ∈ S . The second line of this equation
l
follows by expanding the determinant; each summand under this expansion is of the
form sgn(κ)h , for a κ = λω.
κ
Example 1.2. Let λ = (3,1). Then for the transposition (1,2) ∈ S , we have that
2
κ = λ(1,2) creates the composition κ = (0,4). Thus, using the Jacobi–Trudi identity
(1.1):
s = h −h
(3,1) (3,1) (0,4)
h h
= det 3 4
h h
(cid:18) 0 1(cid:19)
We consider the product s s . Since the Schur polynomials form a basis we may
λ µ
expand this product in terms of the Schur polynomials,
s s = cν (n)s , (1.2)
λ µ λµ ν
ν
X
summed over partitions ν. The structure coefficients cν (n) arising when we de-
λµ
compose the product of two Schur polynomials (1.2) are integers which are called
the Littlewood–Richardson coefficients, and the Littlewood–Richardson rule gives a
calculation of cν . These coefficients play an important role in the combinatorics
λµ
of the symmetric polynomials [15], representation theory [24], and geometry [4]. In
principle, the coefficient cν (n) depends on the number of variables x = (x ,...,x ).
λµ 1 n
However, this coefficient stabilises as n goes towards infinity. An equivalent way of
considering this is that the coefficients cν (n) do not depend on n when n is large
λµ
enough; henceforth we will simply write cν to denote the Littlewood–Richardson
λµ
coefficients.
1.2 Reformulating the Pieri rule
Our aim: In this chapter, we wish to calculate the Littlewood–Richardson coefficient
cν , where λ,µ,ν are partitions. For a composition κ, define the Kostka numbers Kν
λµ κµ
to be the structure coefficient occuring in the product between a complete symmetric
polynomial and a Schur polynomial:
h s = Kν s , (1.3)
κ µ κµ ν
ν
X
summedoverallpartitionsν. SincetheJacobi–Trudiidentity(1.1)givesthefollowing
expansion for s :
λ
s = sgn(κ)h ,
λ κ
κ
X
3
