Applications of Jeu de Taquin to Representation Theory and Schubert Calculus

38 Pages·2019·0.42 MB·English

by  Daniel S. Hono II

Applications of Jeu de Taquin to Representation Theory
and Schubert Calculus
by
Daniel S. Hono II
A Thesis
Submitted to the University at Albany, State University of New York
in Partial Fulfillment of
the Requirements of the Degree of
Master of Arts
College of Arts and Sciences
Department of Mathematics and Statistics
2019
(cid:3)
(cid:3)
(cid:3)
(cid:3)
ProQuest Number:13881177
(cid:3)
(cid:3)
(cid:3)
(cid:3)
All rights reserved
(cid:3)
INFORMATION TO ALL USERS
Thequality of this reproduction is dependent upon the qualityof the copy submitted.
(cid:3)
In the unlikely event that the authordid not send a complete manuscript
and there are missing pages,these will be noted. Also, if material had to be removed,
a notewill indicate the deletion.
(cid:3)
(cid:3)
(cid:3)
(cid:3)
(cid:3)
ProQuest 13881177
(cid:3)
Published by ProQuest LLC ( 2019). Copyrightof the Dissertation is held by the Author.
(cid:3)
(cid:3)
All rights reserved.
This work is protected against unauthorized copying under Title 17, United States Code
Microform Edition © ProQuest LLC.
(cid:3)
(cid:3)
ProQuest LLC.
789 East Eisenhower Parkway
P.O. Box 1346
Ann Arbor, MI 48106 - 1346
Abstract
We describe some applications of the jeu de taquin algorithm on standard Young tableaux of skew
shape λ/µ for λ and µ partitions. We first briefly survey the relevant background on symmetric functions
with a focus on the Schur functions. We then introduce the Littlewood-Richardson coefficients in terms
of Schur functions and survey some of the applications of the corresponding Littlewood-Richardson rule
to representation theory and Schubert calculus on the Grassmannian. A reformulation of the Littlewood-
Richardsonruleintermsofthejeudetaquinalgorithmandgrowth diagrams isthensurveyed. Weillustrate
this formulation with some examples. Finally, we discuss some work by Hugh Thomas and Alexander Yong
in extending the Littlewood-Richardson rule to the more general setting of (co)minuscule flag varieties. In
this setting we describe another reformulation of growth diagrams in terms of chains in Bruhat order and
describe some examples of the generalized jeu de taquin for root systems of types A , B , C , and D .
n−1 n n n
ii
Contents
1 Title i
2 Abstract ii
3 Symmetric Functions 1
3.1 The Ring of Symmetric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
3.2 Schur Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4 Applications to Representation Theory and Schubert Calculus 7
4.1 The Representation Theory of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
n
4.2 Representation Theory of GL (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
n
4.3 Schubert Calculus on the Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5 Jeu de Taquin and the Littlewood-Richardson Rule 13
5.1 Jeu de Taquin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 The Littlewood-Richardson Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.3 Growth Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Beyond Type A Grassmannians 18
6.1 (Co)Minuscule Flag Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.2 A Jeu de Taquin Procedure for (Co)Minuscule Flag Varieties . . . . . . . . . . . . . . . . . . 20
6.3 Example: Root Systems of Type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
n−1
6.4 Example: Root Systems of Type B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
n
6.5 Example: Root Systems of Type C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
n
6.6 Example: Root Systems of Type D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
n
iii
Chapter 3
Symmetric Functions
WebeginourexpositionwiththedefinitionoftheringofsymmetricfunctionsΛ. Wewillthenexploresome
ofthepropertiesofthe functionsthatformabasis ofΛ. Thesefunctionsarethemonomial, elementary, and
complete homogeneous symmetric functions. Only a brief overview of their properties are given.
In the next section, we then define a remarkable basis for Λ known as the Schur functions. The Schur
functions play important roles in the representation theory of the symmetric group, the representation of
GL , and the computation of the cup product in the cohomology ring of the Grassmanian.
n
The Schur functions arise in many different areas and therefore can be defined in a variety of ways. We
shall define them in a combinatorial manner using semi-standard Young Tableaux (SSYT).
Recall, a symmetric polynomial (with integer coefficients) is a polynomial p ∈ Z[x ,...x ] such that,
1 n
for every σ ∈ S we have that p(x ,...,x ) = p(x ,...,x ). That is, permuting the indeterminates
n σ(1) σ(n) 1 n
x ,...x does not change the polynomial. Symmetric functions are then a generalization of symmetric
1 n
polynomials in which the number of indeterminates is infinite. We begin with symmetric functions and then
we will specialize to the case of symmetric polynomials wherever necessary.
Inthefollowing, letPar bethesetofallintegerpartitionsandletPar(n)denotethesetofallpartitions
of the positive integer n.
3.1 The Ring of Symmetric Functions
Let n∈N, then we call a sequence α=(α ,α ,...) a weak composition of n (cf [4]) if
1 2
∞
(cid:88)
α =n
i
i=1
1
where, α ≥0. Clearly, only finitely many of the α s can be non-zero, however the length of α is allowed
i i
to be infinite. Given a denumerable collection of indeterminates x=(x ,x ,...) and a weak composition α
1 2
of n, we then define the monomial xα as:
∞
(cid:89)
xα = xαi
i
i=1
The above product is finite as only finitely many of the elements of α are allowed to be nonzero.
Following [1] and [4] we begin by defining the ring of homogeneous symmetric functions of degree n. Let
R beacommutativeringwithidentityandletn∈N. Asymmetricfunctionf(x)=f(x ,x ,...)isaformal
1 2
power series:
(cid:88)
f(x ,x ,x ,...)= c xα
1 2 3 α
α
such that:
1. α ranges over weak compositions of n,
2. c ∈R, and
α
3. f(x ,x ,...)=f(x ,x ,...) for any permutation σ of the positive integers.
σ(1) σ(2) 1 2
Then, let Λn be the collection of all such formal power series in the indeterminates x ,x ,x ,..., of
R 1 2 3
degree n. It is clear that Λn is an R-module. We will write Λn when R is clear from the context. Note that
R
Λ0 =R.
R
We then define:
∞
(cid:77)
Λ=Λ = Λn
R
n=0
to be the ring of symmetric functions over the ring R. It is clear from the definition that Λ can be endowed
with the structure of a graded algebra over R. In the following discussion, we will take R = Z unless
otherwise noted.
TherearemanyremarkablebasesforΛZ andΛQ,andwementionafewofthemhereandreferthereader
to [4] for the full details. The problem of studying the transition matrices from one basis to another of Λ is
interesting in its own right. We will be concerned mainly with the basis of Schur functions, which will be
the focus of the next section.
We begin with the monomial symmetric functions.
2
Definition 3.1. Let λ=(λ ≥λ ≥...) be a partition of n. Then,
1 2
(cid:88)
m (x)= xµ
λ
µ:µ=σλ
where σ is some permutation of the positive integers, σλ = (λ ,λ ,...) and the sum is over all such
σ(1) σ(2)
distinct µ.
The following result is well known:
Theorem 3.1. (c.f. [4]) {m (x)|λ∈Par } forms a basis of Λ as a module over Z.
λ
Likewise, we can also define the elementary symmetric functions (e ) and the complete homogeneous
λ
symmetric functions (h ):
λ
1. Let e =m , then e =e e ...,
n (1n) λ λ1 λ2
(cid:80)
2. Let h = m , then h =h h ...
n λ λ λ1 λ2
|λ|=n
It is also well known that {h |λ∈Par } and {e |λ∈Par } are both basis of Λ as a Z-module. We
λ λ
refer the interested reader to [4].
Finally, thereisaninnerproductthatcanbeplacedonΛdefinedbytherelation: (cid:104)h ,m (cid:105)=δ where
λ µ λ,µ
δ istheKroneckerdelta. ThisinnerproductisknownastheHallinnerproductonΛ. Thedefiningrelation
λ,µ
for the inner product implies that (cid:104)s ,s (cid:105)=δ where s is the Schur function indexed by the partition λ.
λ µ λ,µ λ
In particular, this means that the Schur functions introduced in the next section form an orthonormal basis
of Λ.
3.2 Schur Functions
We can distinguish an important subset of Λ, called the Schur functions, indexed by partitions of n ∈ N.
The Schur functions will also form a basis of Λ, however it is not obvious from the definition that the Schur
functions are even symmetric.
First we will need the notion of a semi-standard Young tableau (SSYT).
Definition 3.2. Let λ(cid:96)n such that λ=(λ ,λ ,...,λ ). Then a Young diagram of shape λ is a collection
1 2 k
of left justified boxes such that there are λ boxes in the first row, λ boxes in the second row, and so on to
1 2
λ boxes in the kth row.
k
We illustrate the above definition with the following example.
3
Example 3.1. Let λ=(5,5,2,1,1). Then |λ|=14 and the corresponding Young diagram is:
Given a Young diagram of shape λ, a filling of the shape is an assignment of numbers to boxes of the
diagram. There are particular types of fillings that we need to identify.
Definition 3.3. Let λ be a partition. Then a semi-standard Young tableau (SSYT) is a filling of the shape
λ with positive integers such that reading across rows form weakly increasing sequences and reading down
columns form strictly increasing sequences.
Example 3.2. Let λ be as in example 3.1. Then the following is a SSYT of shape λ:
3 7 8 8 8
4 9 10 10 11
5 15
8
9
WewillnextneedthenotionofaskewSSYT(orskewtableau). Thefollowingdefinitionwillbenecessary
towards this end.
Definition 3.4. Let λ=(λ ,λ ,...) and µ=(µ ,µ ,...) be partitions. Then µ⊆λ if and only if µ ≤λ
1 2 1 2 i i
for all i.
The above definition makes precise the idea of one Young diagram being contained in another. If µ⊆λ,
then we can remove µ from λ to produce a skew diagram.
Definition 3.5. Let µ and λ be partitions such that µ ⊆ λ, then λ/µ is defined to be the shape obtained
from λ by removing the boxes of µ.
Example 3.3. Let λ=(5,4,1,1,1) and µ=(3,2,1) then µ⊆λ and the resulting skew shape λ/µ is given
by the following diagram.
4
where the gray colored boxes indicate that they have been removed from the overall diagram.
Given a skew shape λ/µ we can define a semi-standard Young tableau of skew shape λ/µ in a way
analogous to the above.
Definition 3.6. Let λ/µ be a skew shape. Then a semi-standard Young tableau of shape λ/µ is a filling
of the boxes of λ/µ with positive integers such that every row forms a weakly increasing sequence and every
column forms a strictly increasing sequence.
Example 3.4. One example of a SSYT of the shape given in 3.3 is
5 5
3 7
6
7
Associated to any skew SSYT of shape λ/µ is the skew Schur function s . Let SSYT(λ/µ) denote the
λ/µ
set of all semi-standard Young tableaux of shape λ/µ (therefore, SSYT(λ/µ) is an infinite set).
Definition 3.7. Let λ,µ ∈ Par such that µ ⊆ λ and T ∈ SSYT(λ/µ). Define xT = xa1xa2... where a is
1 2 i
the number of times the number i appears in T. Then the skew Schur function is given by:
(cid:88)
s (x ,x ,...) = xT
λ/µ 1 2
T∈SSYT(λ/µ)
In the above definition, if µ=∅, the empty partition, then s is known as the Schur function indexed
λ/µ
by λ.
Example 3.5. Consider the partition λ = (3,1,1) and consider fillings of the shape λ with the numbers
{1,2,3}. Then the corresponding segment of the Schur function s is given by:
(3,1,1)
x3x x +x x2x2+x2x2x +x2x x2+x x3x +x x x3
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
and the corresponding SSYT are:
1 1 1 1 1 2 1 1 3 1 2 2 1 3 3 1 2 3
2 2 2 2 2 2
3 3 3 3 3 3
As noted above, it is not immediate from the definition that s is a symmetric function , but it turns
λ/µ
out that this is the case.
5
Theorem 3.2. (c.f. [4]) Let λ,µ∈Par such that µ⊆λ, then s ∈Λ.
λ/µ
In fact, the Schur functions (with µ=∅) form a basis of Λ as a Z-module.
Theorem 3.3. (c.f. [4]) {s |λ∈Par } is a basis of Λ as a Z-module.
λ
NowthatwehavethefactthattheSchurfunctionsformabasisofΛ,wecanstatethefollowingdefinition
of the Littlewood-Richardson coefficients:
Definition 3.8. The Littlewood-Richardson coefficients cν are defined in the following equivalent ways.
λ,µ
Let λ and µ be partitions such that µ⊆λ, then:
(cid:88)
s = cν s
ν/λ λ,µ µ
µ⊆ν
(cid:88)
s s = cν s
λ µ λ,µ ν
ν⊇µ,λ
where |ν|=|λ|+|µ|.
It is a remarkable fact that cν ∈ Z . Given that the Littlewood-Richardson coefficients are always
λ,µ ≥0
non-negative we then state the famous Littlewood-Richardson problem:
Problem 3.1. Find a combinatorial interpretation for the numbers cν .
λ,µ
Indeed, many combinatorial interpretations of the Littlewood-Richardson coefficients have been found.
We delay these descriptions until chapter 3 after surveying some applications to representation theory and
Schubert calculus.
6