Combinatorial Hopf algebras and dual graded graphs

Andr´es Rodr´ıguez Rey

Supervised by: C´esar Galindo

Being the generalization of the notion of group, Hopf algebras are an important algebraic structure which appears across many branches of mathematics. Moreover, their relationship with combinatorics is apparent with the notion of combinatorial Hopf algebras, which lets us study many combinatorial objects through symmetric and quasisymmetric functions [1]. On the other hand, the dual graded graphs introduced by Fomin in [6] generalize the Robinson-Schensted correspondence which relate permutations with Young tableaux. These two structures, which encode combinatorial information in different ways, became clearly related in [3], where Bergeron, Lam and Li introduced a construction of dual graded graphs from a pair of graded dual Hopf algebras. In this thesis, we study this construction and compute some examples which explicitly show some of its “restrictions”.

Siendo la generalizaci ´on de la noci ´on de grupo, las ´algebras de Hopf son una estructura algebraica de gran importancia que aparece en varias ´areas de las matem´aticas. Su relaci ´on con la combinatoria se vuelve aparente por medio de la noci ´on de ´algebras combinatorias de Hopf; que permiten estudiar una gran var-iedad de objetos puramente combinatorios por medio de t´ecnicas algebraicas. Ejemplos de esto son las funciones sim´etricas y las funciones quasisim´etricas. Por otro lado, los grafos duales graduados fueron introducidos por Fomin en [6] para generalizar la correspondencia de Robinson-Schensted, que relaciona permutaciones con tableaux’s de Young. Estas dos estructuras, que representan informaci ´on de un mismo objeto combinatorio de distinta forma, poseen una relaci ´on que se vuelve aparente con el trabajo de Berg-eron, Lam y Li [3], en el que ellos introducen la construcci ´on de grafos duales graduados a partir de pares de ´algebras graduadas de Hopf. En esta tesis estudiamos dicha construcci ´on y calculamos ciertos ejemplos que muestran algunas restricciones sobre la misma.

In this thesis we present a brief survey on Combinatorial Hopf algebras and some results on this area from different sources. The first result is that of Aguiar, Bergeron and Sottile [1] which shows that any combina-torial Hopf algebra can be studied from the combinacombina-torial Hopf algebra of Quasisimmetric functionsQSym, and the second is a construction from Bergeron, Lam and Li [3] which shows that from any pair of dual graded Hopf algebras one can construct a pair of dual graded graphs. Additionally, we show some examples from the construction of Bergeron et-al., just to show in a more “maleable” manner what the construction was trying to state. Finally, we expect the thesis to be a proper introduction to the study of combinatorial Hopf algebras so that it presents the necessary requirements for people interested in the area.

The thesis consists of three chapters which are divided into sections. In the first chapter we present the preliminaries, where we give the necessary background to understand the following chapters. The preliminaries are divided into two sections: in the first section we discuss the basic definitions and properties of the Hopf algebras as described by Grinberg and Reiner in [7] where we used auxiliary material from Nastasescu et al. [8] and Dascalescu et al. [4]. In the second section, we describe many combinatorial objects ranging from generating functions to compositions. Here we used many authors given the variety of topics: for generating functions we relied on Flajolet and Sedgewick’sAnalytic Combinatorics[5], for partially ordered sets and graph theory we used Stanley’sAlgebraic combinatorics [12] andEnumerative Combinatorics

Vol.1 [10], and Fomin’s paper [6], and for partitions and compositions we used both volumes of Stanley’s

Enumerative Combinatorics[10] and [11]. In the second chapter we define what a combinatorial Hopf algebra is and give two important examples. The chapter ends with a theorem, which describes the importance of the given examples. For this subject we relied on the lecture notes of Federico Ardila for the classHopf algebras in combinatorics [2] as well as on Grinberg et al. [7], Aguiar et al. [1], and Stanley [11]. Finally, in the third chapter we explicitly state the construction of [3] and construct the dual graded graphs for the examples in Chapter2.

1 p r e l i m i na r i e s 9

1.1 Hopf Algebras 9

1.1.1 Algebras, coalgebras and duality. 9

1.1.2 Graded Hopf algebras 16

1.1.3 Free objects 20

1.2 Combinatorics 21

1.2.1 Generating functions 21 1.2.2 Partially ordered sets 22

1.2.3 Graphs 23

1.2.4 Partitions 25

1.2.5 Compositions 28

2 c o m b i nat o r i a l h o p f a l g e b r a s 31

2.1 The Combinatorial Hopf Algebra of Symmetric functions 31

2.2 The Combinatorial Hopf Algebra of Quasisymmetric functions 38

2.2.1 Sym as a subspace of QSym 41

2.2.2 The graded dual of QSym 42

2.3 The terminal object of the Category of Combinatorial Hopf Algebras 45 3 f r o m g r a d e d h o p f a l g e b r a s t o d ua l g r a d e d g r a p h s 47

3.1 The construction 47

3.2 The pair (Sym, Sym) 49

3.2.1 Monomials and Homogeneous 49

3.2.2 Schur functions 51

3.3 The pair (QSym, NSym) 52

1

P R E L I M I N A R I E S

In this chapter we will give the necessary background and establish coherent notation and terminology that will be used throughout the thesis. We will assume that the reader has basic knowledge of linear algebra, abstract algebra and category theory. Unless otherwise specified, k denotes a commutative ring. Here a ring will always be assumed to have a multiplicative identity and it will be denoted as 1 unless specified otherwise.

1.1 h o p f a l g e b r a s

Here we define different algebraic structures, give some examples, and prove some of the basic results that we will need to understand the next chapters.

1.1.1 Algebras, coalgebras and duality.

Definition1.1.1.

1. A gradedk-module M is ak-module such thatM =

M

n≥0

Mn, where each Mn is a submodule of M. If

x ∈ Mn, then x is homogeneous of degree n and we denote the degree as deg(x) = n. Additionally,

everym∈ Mcan be uniquely represented as a summ=

∑

n≥0

mnwithmn∈ Mn, where there are finitely

many nonzeromn’s.

2. If MandN are gradedk-modules, then M⊗N=

M

n≥0

(M⊗N)nwhere(M⊗N)n = M

i+j=n

Mi⊗Nj.

3. Ak-linear map_φ: M →Nbetween two gradedk-modules is called graded ifφ(M_n)⊂ N_nfor all n.

4. If Mis a gradedk-module such thatM₀=k, then Mis said to be connected.

5. IfV is a vector space over k we denote its dual by V∗. Additionally, in the case whereV is a graded

k-vector space we define the graded dualV◦ =M

n≥0

V_n∗.

Remark. The reader should note that the dual of a graded vector space V does necesarilly have a graded structure, that is the main reason to consider the graded dual for this thesis. Additionally, we note that we do not necessarily have thatV◦ =V∗ but we do have that the graded dual is contained in the dual.

Definition1.1.2. A graded associativek-algebra Ais a gradedk-module with a gradedk-linear associative mapm: A⊗A→ Aand a gradedk-linear unitu:k→ A, whereusends 1kto the two sided multiplicative

identity element 1Ain A. This is equivalent to requiring that the following equations hold:

1. m(a⊗m(b⊗c)) =m(m(a⊗b)⊗c)fora,b,c∈ A.

2. a =m(1_A⊗a) =m(a⊗1_A)for all a∈ A.

For a more intuitive approach to the latter definition you can think of a graded algebraAas a commutative ring with a graded vector space structure, it is easy to show that the definition used here is equivalent to this intuitive understanding. Throughout the text m(a,b)will be simply denoted as ab to avoid harsh notation. Let us see some useful examples:

Example.

1. Thegroup algebra: LetGbe a finite group. We define the group algebra

kG={

n

∑

i=1

λigi|λi ∈ k,gi ∈G}with multiplicationm(g⊗h) = ghand unitu(1) =e.

Heremanduare extended by linearity andeis the identity ofG. Let’s see thatkGis in fact an algebra: Letg,h,l∈ G, clearlym(m(g⊗h)⊗l) =m(gh⊗l) = (gh)l= g(hl) =m(g⊗hl) =m(g⊗m(h⊗l))by the associativity ofG, and ifπkGis the respective projection we haveg= ge= m(g⊗e) =m(g⊗u(1)) and g = eg = m(e⊗g) = m(u(1)⊗g). Since these is true for all the elements of G then its true, by linearity, on allkG. By definition of algebra we conclude this is an algebra. HerekG= M

n≥0

An where

An=Ø for alln>1 and A0=kG.

2. Algebra of Polynomials: Letk[x]be a polynomial ring on one variable, then we can define the following operations:

• m(xi⊗xj) =xi+j foriandjnatural numbers.

• u(1) =x0 =1.

Heremanduare extended by linearity. Let’s see thatk[x]is in fact an algebra, for this we can just see that it holds for the basis and then it will follow by linearity: The base ofk[x]is the set of all monomials {1,x,x2, ...}, then fori,j,k ∈Nwe have that

m(m(xi⊗xj)⊗xk) =m(xi+j⊗xk) =x(i+j)+k = xi+(j+k)= m(xi⊗xj+k) =m(xi⊗m(xj⊗xk)),

by associativity of N. Additionally and if π_k[x] is the respective projection we have xi = xi(1) =

m(xi⊗1) and xi = (1)xi = m(1⊗xi). We conclude that k[x] is an algebra. Here the graduation is given by the degree of the monomials, that isk[x] =M

n≥0

Anwhere An=kxn= {axn|a∈k}.

Remark. The last example can be extended to any number of variables, where the multiplication is the usual one for polynomials of more than one variable and the unit remains the same.

Definition1.1.3. Let AandBbe gradedk-algebras. A morphism of graded algebrasφ: A→ Bis a graded

k-module homomorphism that makes the following diagrams commute:

A φ //B A φ //B

A⊗A

mA

O

O

φ⊗φ _//

B⊗B mB O O k uA ^ ^ uB @ @

Definition 1.1.4. Let A and B be graded k-algebras. The tensor product A⊗B also becomes a graded

k-algebra defining the multiplication and unit by:

m:(A⊗B)⊗(A⊗B)→ A⊗B u:k→A⊗B.

where m(a⊗b,a0⊗b0) = aa0⊗bb0 andu(1) = 1A⊗1B. This is equivalent to saying the following diagrams commute:

A⊗B⊗A⊗B

idA⊗T⊗idB

mA⊗B

'

'

k

uA⊗B

$

$

A⊗A⊗B⊗B mA⊗mB

/

/ _A⊗B k⊗k

uA⊗uB

/

/_A⊗B

Remark. The grading of A⊗Bis that described inDefinition1.1.1(2).

Example. In the last example we showed that for a ringkwe have thatk[x]is an algebra. Considerx andy

two different indeterminates, and the respective algebrasQ[x],Q[y]andQ[x,y].

1. ByDefinition1.1.4 we can take the tensor productQ[x]⊗Q[y]and get an algebra where the multipli-cation and unit are defined by:

• m(xi⊗yj,xh⊗yl) =xi+h⊗yj+l, wherexi,xh ∈Q[x]andyj,yl ∈Q[y].

• u(1) =1⊗1.

2. Define _φ : Q[x]⊗Q[y] → Q[x,y] where φ(xi⊗yj) = xiyj for all xi ∈ Q[x] and yj ∈ Q[y]. This is a morphism of algebras since the morphism preserves units and:

φ((xi⊗yj)(xh⊗yl)) =φ(xi+h⊗yj+l)

= xi+hyj+l

= xixjyjyl

= xiyjxhyl

=φ(xi⊗yj)φ(xh⊗yl).

For allxi,xh ∈Q[x]andyj,yl ∈Q[y].

Now we define the “dual” of an algebra, a coalgebra:

Definition1.1.5. A co-associative gradedk-coalgebraCis a gradedk-moduleCwith a comultiplication, that is, a gradedk-linear map∆:C→C⊗C, and a gradedk-linear counite:C→ksuch that for allc∈C:

1. (∆⊗Id)◦∆= (Id⊗∆)◦∆. 2.

∑

(c)

c1e(c2) =

∑

(c)

e(c1)c2=cfor allc∈C.

where ∆(c) =

∑

(c)

c1⊗c2 for allc∈C

The notation for∆(c)in the last definition is called the Sweedler notation. To see why this notation makes “sense” consider the following examples:

Example.

1. The group coalgebra: Let G be a finite group. We define the group coalgebra kG with the following operations:

• ∆(g) =g⊗gfor allg ∈G.

• e(g) =1 for allg∈ G.

Here∆andeare extended by linearity. Let’s see thatkGis a coalgebra: Forg∈ Gwe have that

(∆⊗Id)(∆(g)) = (∆⊗Id)(g⊗g) =∆(g)⊗g= (g⊗g)⊗g=g⊗(g⊗g) = (Id⊗∆)(g⊗g)

= (Id⊗∆)(∆(g))

by associativity of the tensor product, the comultiplication is associative. To show that the counit is compatible with ∆ note that g(1) = g = (1)g. Since this holds for all the elements of G then, by linearity, it holds on allkG. By definition of coalgebra we conclude this is a coalgebra.

2. The coalgebra of polynomials: Let k[x] be the polynomial ring on one variable, then we can define the following operations:

• ∆(xn) =

n

∑

k=0

n k

•

e(xn) = (

1, ifn=0, 0, otherwise.

Here∆andeare extended by linearity. Let’s see thatk[x]is in fact a coalgebra, for this we can just see that the equations hold for the basis and then it will follow by linearity. Letn ∈ _Nandxn _{∈k[x], we} have that:

∆⊗Id(∆(xn)) =∆⊗Id(

n

∑

k=0

n k

xk⊗xn−k)

=

n

∑

k=0

n k

∆(xk)⊗xn−k

=

n

∑

k=0

n k ( k

∑

i=0

k i

xi⊗xk−i)⊗xn−k

=

n

∑

k=0

k

∑

i=0

n k k i

xi⊗xk−i⊗xn−k.

And,

Id⊗_∆(∆(xn)) =Id⊗_∆(

n

∑

k=0

n k

xk⊗xn−k)

=

n

∑

k=0

n n−k

xk⊗∆(xn−k)

=

n

∑

k=0

n n−k

xk⊗(

n−k

∑

j=0

n−k j

xj⊗xn−k−j)

=

n

∑

k=0

n−k

∑

j=0

n n−k

n−k j

xk⊗xj⊗xn−k−j.

Now, if we make the change of variable u = n−k we will get that the last equation will just end up being

Id⊗∆(∆(xn)) =

n

∑

u=0

u

∑

j=0

n u u j

xn−u⊗xj⊗xu−j

=

n

∑

u=0

u

∑

j=0

n u u j

xj⊗xu−j⊗xn−u.

So(∆⊗Id)(∆(xn)) = Id⊗∆(∆(xn))for all n∈N, giving us coassociativity. It comes straightforward from the definition thateis the counit for the product. We conclude thatk[x]is a coalgebra.

Analogously for coalgebras we have the following definitions:

Definition1.1.6. LetCandDbe two gradedk-coalgebras. A morphism of graded coalgebrasψ:C→ Dis

a gradedk-module homomorphism that makes the following diagrams commute:

C φ //

∆C D ∆D

C φ //

eC

D

eD

C⊗C

φ⊗φ

/

/_D⊗D k

Definition 1.1.7. Let C and D be graded k-coalgebras. The tensor product C⊗D also becomes a graded

k-coalgebra with comultiplication∆:C⊗D→ (C⊗D)⊗(C⊗D), and counite:C⊗D→kdefined by:

∆(c⊗d) =

∑

(c⊗d)

and

e(c⊗d) =eC(c)eD(d)

where c∈Candd∈ D. Which is equivalent to requiring that the following diagrams commute:

C⊗D

∆C⊗∆D

∆C⊗D

)

)

C⊗D

eC⊗eD

eC⊗D

"

"

C⊗C⊗D⊗D

idC⊗T⊗idD

/

/_{C⊗D⊗C⊗D k⊗k} //_k

Remark. The grading ofC⊗Dcomes naturally from the grading of bothCandD, as was defined in Defini-tion1.1.1(2).

Consider the following proposition:

Proposition1.1.8. When A is both a gradedk-algebra and a gradedk-coalgebra the following are equivalent:

1. (∆,e)are morphisms for the graded algebra structure(m,u).

2. (m,u)are morphisms for the graded coalgebra structure(∆,e).

3. the following4diagrams commute:

A⊗A e⊗e // m

k⊗k

m

k u //

∆k A ∆

k id //

u

k

A e //

k k⊗k u⊗u//A⊗A A

e

@

@

A⊗A

∆⊗∆ u u m + +

A⊗A⊗A⊗A

id⊗T⊗id ₎₎

A

∆

{

{

A⊗A⊗A⊗A

m⊗m //A⊗A

We call such A a bialgebra.

Proof. First consider the following two diagrams

A⊗A

∆⊗∆ u u m + +

eA⊗A _//

k

A⊗A⊗A⊗A

mA⊗A

,

,

id⊗T⊗id ₎₎

A ∆ { { e O O

A⊗A⊗A⊗A

m⊗m //A⊗A

A⊗A

∆A⊗A

∆⊗∆ u u m + +

A⊗A⊗A⊗A

id⊗T⊗id ₎₎

A

∆

{

{

A⊗A⊗A⊗A _m⊗m // A⊗A k

u

O

O

uA⊗A

o

The small diagrams on the definition commute if and only ifuis a morphism of algebras andeis a morphism of coalgebras. Moreover, m is a morphism of coalgebras if and only if the first diagram commutes (by definition of morphism of coalgebras) but the first diagram commutes if and only if the diagrams in the proposition commute. Analogously for ∆, ∆ is a morphism of algebras if and only if the second diagram commutes (by definition of morphism of algebras) but the second diagram commutes if and only if the diagrams in the proposition commute. The theorem follows.

Example. For this example is enough to work only on the bases of both spaces.

1. The group bialgebra: By the last proposition is enough to show that the four diagrams commute, by definition of eand u the first three diagrams commute. So we are only missing the last diagram, for this notice that:

g⊗h ∆⊗∆ v v m + +

g⊗g⊗h⊗h

id⊗T⊗id ₍₍

gh

∆

{

{

g⊗h⊗g⊗h _m⊗m //gh⊗gh

holds for every g⊗hwhere g,h∈G.

2. Polynomials: By the last proposition is enough to show that the four diagrams commute, it comes straightforward that the first three diagrams commute. For the last diagram to commute we need to prove that for every m,n∈_Nwe have that

(m⊗m)(id⊗T⊗id)((∆⊗∆)(xn⊗xm)) =∆(xn+m).

Note that:

m⊗m(id⊗T⊗id(∆⊗∆))(xn⊗xm) =m⊗m(id⊗T⊗id(

n

∑

k=0

m

∑

j=0

n k m j

xk⊗xn−k⊗xj⊗xm−j))

=m⊗m(

n

∑

k=0

m

∑

j=0

n k m j

xk⊗xj⊗xn−k⊗xm−j)

=

n

∑

k=0

m

∑

j=0

n k m j

xk+j⊗xn+m−(k+j)

=

n+m

∑

h=0 k+

∑

j=h

n k m j

xk+j⊗xn+m−(k+j)

! ,

and that:

∆(xn+m) =

n+m

∑

h=0

n+m h

xh⊗xn+m−h.

Equality holds because of the combinatorial identity

∑

k+j=h

n k m j =

m+n h

. To prove the

combi-natorial identity think of a deck of colored cards where there are n red cards and mblue cards. The number of ways to gethcards from the deck is(m+_hn), but you can count this number with a different method; you can split the deck into the two colors and then look at all the possible ways of gettingk

red cards andjblue cards whereh =j+k. Since the number of ways to do this is

∑

k+j=h

n k m j , the

proof of the combinatorial identity is done.

Proposition1.1.9. Let C be a gradedk-coalgebra and A a gradedk-algebra. Consider the space of linear transforma-tions from C to A denoted by Hom(C,A), and define the convolution product as:

(f∗g)(c) =

∑

(c)

f(c1)g(c2),

for f,g ∈Hom(C,A)and c∈C with∆(c) =

∑

(c)

c1⊗c2. Then we have the following:

• The convolution product is associative.

• u◦eis the identity for the convolution product.

In other words, Hom(C,A)is an algebra under the convolution product.

Proof. We will show the things that are itemized in the proposition:

• First note that the convolution product is ak-linear map because it is a composition of linear maps. To show associativy let f,g,h∈ Hom(C,A). We know that:

((f ∗g)∗h)(c) =

∑

(c)

(f∗g)(c1)h(c2)

=

∑

(c)

(f∗g)(c1)h(c2)

=

∑

(c)(

∑

c1)

f(c11)g(c12)h(c2),

and that:

(f ∗(g∗h))(c) =

∑

(c)

f(c1)(g∗h)(c2)

=

∑

(c)

f(c1)g(c21)h(c22)

=

∑

(c)(

∑

c2)

f(c1)g(c21)h(c22),

which are equal by coassociativity of∆.

• We know that for every c ∈ C we have that c = _∑c1e(c2) = ∑e(c1)c2. By linearity, if we apply any

f ∈ Hom(C,A)to cwe will get that f(c) =_∑f(c1)e(c2) =∑e(c1)f(c2), which in turn is equivalent to

saying that f = f ∗(u◦e) = (u◦e)∗ f.

Remark. The reader should note that the last proposition holds not only to Hom(C,A), the space of all

linear transformations, but to certain subspaces as well, such as Homgr(C,A), the space of allgradedlinear transformations. Moreover, we can even extend this to a more simple case such as the graded dualC◦ where the convolution product is well defined and is graded.

Before going any further we want to emphasize that we will only be interested on algebraic structures that are locally finite; that is, whendim(Ai)< ∞for alli, where A=

M

n≥0

An. Therefore, we will now discuss the

concept of duality, of great importance in the thesis:

Definition1.1.10. LetV andW two graded vector spaces overk:

1. Let T: V→ W be a graded linear transformation. Define: T∗ :W◦ →V◦, where T∗(f) = f◦Tfor all

f ∈W◦. NaturallyT∗ is linear, graded and well defined.

2. Defineh,i:V∗×V→k, to be the bilinear map wherehf,ai= f(a)for all f ∈V∗ anda∈V.

3. Let _ρ : V∗⊗W∗ → (V⊗W)∗, be a linear map such that for every f ⊗g ∈ V∗⊗W∗ we have that ρ(f⊗g)is an element such that:

hρ(f⊗g),a⊗bi= f(a)g(b)for all a⊗b∈V⊗W.

Clearlyρis injective, and for the finite dimensional (locally finite) case is bijective.

With this, we are ready to define duality:

Proposition1.1.11. Let A be a graded k-algebra. Then, A∗has a coalgebra structure with comultiplication and counit:

• ∆=ρ−1◦m∗.

• e=φ◦u∗.

Whereφis the isomorphism between k and k∗. A∗ is called the dual graded coalgebra of A.

Proposition1.1.12. Let C be a graded k-coalgebra. Then, C∗has an algebra structure with multiplication and unit:

• m=∆∗◦ρ.

• u=e∗◦φ.

Whereφis the isomorphism between k and k∗. C∗ is called the graded dual algebra of C.

Proof. Substitte A for k in Proposition 1.1.9, noting thatm is just the convolution product the proposition follows.

The proof that A∗ is a coalgebra is similar and we omit it.

Remark. Although the last proposition is true for C∗ and A∗, this spaces may lack the graded structure that we have been emphazising all over this chapter. With some help of the remark inProposition1.1.9, is natural to choose as our “substite” of duality that ofC◦(A◦since the convolution product defines a gradedk-algebra structure onC◦ (k-coalgebra structure on A◦).

1.1.2 Graded Hopf algebras

Now we are ready to define what a Hopf algebra is:

Definition1.1.13. A graded bialgebraAis called a graded Hopf algebra if there is an elementS∈Endgr(A), called the antipode ofA, sucht that the following diagram commutes:

A⊗A S⊗idA //A⊗A m

#

#

A

∆ ;;

∆ _##

e //

k u //A

A⊗A

idA⊗S

/

/_A⊗A m

;

;

Equivalently, the following equation holds for every a∈ A:

u(e(a)) =

∑

(a)

S(a1)a2=

∑

(a)

a1S(a2), where∆(a) =

∑

(a)

a1⊗a2.

Remark. Endgr(A)is the space of gradedk-linear endomorphisms ofA. It is worth noting that the conditions stated above are just of that element whose inverse under the convolution product is the identity. In other words, the antipodeSis the only function inEndgr(A)such thatS∗Id= Id∗S=u◦e.

Proposition1.1.14. The antipode S in a Hopf algebra A has the following properties:

1. S◦u=u.

2. e◦S= e.

4. ∆S(a) =

∑

(a)

S(a2)⊗S(a1)for all a ∈ A.

Proof. Let a,b∈ A.

1. Since∆ is an algebra map we have that ∆(1) = 1⊗1, hence 1_A = u(e(1_A)) =S(1_A)1_A = 1_AS(1_A) =

S(1A). By linearity ofSit follows.

2. Sinceuis a coalgebra map we have thate(1_A) =1_k, and by 1 we get thate(S(1_A)) =e(1_A), by linearity it follows.

3. ConsiderA⊗Aas a coalgebra andAas an algebra. ByProposition1.1.9we know thatHom(A⊗A,A) is an associative algebra with a convolution product∗where(f∗g)(a⊗b) =

∑

(a⊗b)

f(a1⊗b1)g(a2⊗b2)

for all f,g∈ Hom(A⊗A,A)anda⊗b∈ A⊗A.

We also know that u◦e is the identity of the convolution product. If we take f,g and h such that

f(a⊗b) = ab,g(a⊗b) = S(b)S(a)andh(a⊗b) = S(ab)for alla⊗b∈ A⊗A, we will get that for all

a⊗b∈ A⊗A:

h∗f(a⊗b) =

∑

a⊗b

h(a1⊗b1)f(a2⊗b2)

=

∑

a⊗b

S(a1b1)(a2b2)

=uA◦eA(ab)

=uA◦eA⊗A(a⊗b).

Analogously f ∗g=uA◦eA⊗A. But,

h=h∗uA(eA⊗A)

=h∗(f ∗g)

= (h∗f)∗g

= (uA◦eA⊗A)∗g

= g.

We conclude thatS(ab) =S(b)S(a).

4. The proof is analogous to that of(3.).

Corollary1.1.15. Commutativity (cocommutativity) of A implies that the antipode is an involution.

Proof. For this we take the convolution product∗overEnd(A)defined by(f ∗g)(a⊗b) =

∑

(a)

f(a1)g(a2)for

f,g∈ End(A). By Proposition1.1.9we know that the identity of the convolution algerbra is u◦eand that

Idhas as inverseS. If we assume commutativity then:

S∗S2(a) =

∑

a

S(a1)S2(a2)

=S

∑

a

S(a2)a1 !

=S

∑

a

a1S(a2) !

=S(u(e(a)))

=u(e(a))

The equality holds by the definition of the antipode and parts 1 and 3 ofProposition1.1.14. BecauseShas a unique inverse under the convolution product the corollary follows. For the case of cocommutativity Athe proof is analogous.

The reader should note that the existence of the antipode is not always granted for a given bialgebra, we present an example that exposes this phenomenon.

Example.

1. So far we constructed a bialgebra structure on the group algebrakG. Truth is, the construction applies to weaker structures such as monoids. Consider A= kNas a bialgebra and suppose that there exists an antipode over A. We have, by definition of the antipode, thatS(n) +_Nn =0 for eachn, this means thatS(n) =−nwhich would lead to a contradiction since−nis not in A(We emphasize the fact that “+_N” is the addition operation of the natural numbers, different from “+” the sum opperation in A).

Lemma1.1.16. Let A be a graded connectedk-bialgebra with decomposition A= M

n≥0

An. Then the following holds:

1. If x∈ Anfor some n>0, then∆(x)∈x⊗1+A⊗I, where I = M

n>0

An.

2. If x∈ Anfor some n>0, thene(x) =0.

Proof.

1. Letx ∈ A_n. Since∆is graded we have that∆(x)∈

M

i+j=n

(Ai⊗Aj). We can reorder the terms so that

∆(x) =

∑

deg(xi)=n

xi⊗1+

∑

deg(xi)6=n

xi⊗xj.

By the definition of the counit we also know that

x=

∑

deg(xi)=n

xi+

∑

deg(xi)6=n

xie(xj).

Since x is homogeneous of degree n and A is decomposed into a direct sum there exists a unique

decomposition of x. Therefore, x =

∑

deg(xi)=n

xi and

∑

deg(xi)6=n

xie(xj) = 0. We conclude that ∆(x) ∈

x⊗1+A⊗I for every x∈ A.

2. Letx ∈ A_n, by the last result and the definition of counit we know that:

x =e(x) +

∑

deg(xi)6=n

xie(xj),

where ∆(x) =x⊗1+

∑

deg(xi)6=n

xi⊗xj. Because xis uniquely represented in the direct sum

decomposi-tion and Ais connected, we have that e(x)is homogeneous of degree 0 so thate(x) =0.

Proposition1.1.17. A graded connected bialgebra A has a unique antipode S, which endows A with a Hopf structure.

Proof. We will define Sin each homogeneous component by induction:

• Base case: Suppose k = 0. Since A is connected then A0 is isomorphic to k. Define S = Id for this

component.

• Inductive step: Suppose you have defined the firstkcomponents. Let’s defineSfor the k+1 homoge-neous component of A: Leta∈ A_k+1, by the last lemma we know that∆(a) =a⊗1+

∑

a1⊗a2 where

deg(a1)< k+1. BecauseSneeds to be the inverse of Idunder the convolution product, we will need

that:

But by the last lemma we have that for any homogeneous element ofAwe will have that(u◦e)(a) =0, then we must have that S(a) = −

∑

S(a1)a2. This definition of S works since S has been defined

inductively so that it is the inverse of Idunder the convolution product.

Example.

1. By construction we have thatkGis not connected, but this algebra has a quite natural antipode. Define

S(g) = g−1 for eachg ∈ Gand extend by linearity. To prove thatS is the antipode is enough to show it for the basis:

S(g)g= g−1g= e= gg−1= gS(g).

We conclude thatkGis a Hopf algebra.

2. In contrast to the last example we know thatk[x]is connected. We will prove by induction that:

S(xk) = (−1)kxk.

Proof.

• Base case: We know by connectedness ofk[x]thatS(1) =1= (−1)01.

• Inductive step: Suppose that for everyi<k you have thatS(xi) = (−1)ixi. We will show that the same holds fori= k. ByProposition1.1.8we know that:

S(xk) =

k−1

∑

i=0

n i

S(xi)xk−i =

k−1

∑

i=0

k i

(−1)ixk =xk k−1

∑

i=0

k i

(−1)i

! .

To finish we need to prove that k−1

∑

i=0

k i

(−1)i = (−1)k. For this, recall that(1−x)k ₌

∑

k i=0

k i

(−1)ixi,

when we replace x =1 we get k

∑

i=0

k i

(−1)i = 0, by solving for(−1)k we have what we wanted.

By induction we have thatk[x]is a Hopf algebra.

We finish this subsection with a result involving the relationship between bialgebra maps and Hopf algebra maps, for this we will use a lemma on the convolution algebra:

Proposition1.1.18. Let A, A0 be twok-algebras, C and C0 twok-coalgebras. Letα: A→ A0 be a map ofk-algebras,

andγ:C→C0be a map ofk-coalgebras. Then, the mapφ:Hom(C0,A)→ Hom(C,A0)defined byφ(f) =α◦f◦γ

for each f ∈ Hom(C0,A)is a convolution algebra morphism.

Proof. For this we need to show thatφis an algebra morphism, that is:

• φ(uA◦eC0) =u_A0◦_eC:

φ(uA◦e_C0) =α◦(uA◦e_C0)◦γ

=uA0◦eC.

The last equality follows from the fact that αis an algebra homomorphism andγis a coalgebra homo-morphism.

• φ(f ∗g) =φ(f)∗φ(G):

φ(f∗g) =α◦(f ∗g)◦γ

=α◦(m◦(f⊗g)◦∆)◦γ

=α◦m◦(f⊗g)◦∆◦γ

=m◦(α⊗α)◦(f⊗g)◦(γ⊗γ)◦∆

=m◦((α◦ f◦γ)⊗(α◦g◦γ))◦∆

=m◦(φ(f)⊗φ(g))◦∆

=φ(f)∗φ(g).

The fourth equality holds because αis an algebra morphism andγis a morphism of coalgebras.

It follows that the mapφis a map of convolution algebras.

Corollary 1.1.19. Let H and H0 be two Hopf algebras with antipodes S and S0 respectively. If β : H → H0 is a

bialgebra homomorphism, then the mapβis a Hopf homomorphism ( i.e. S0◦β= β◦S).

Proof. Consider the map φ1 from the last proposition such that H = A0 = C0 = C, H0 = A, α = β, and γ= IdH. Then we have the following:

β◦S=φ1(S) =φ1(Id∗−H 1) = (φ1(IdH))∗−1 = (β◦IdH)∗−1 =β∗−1,

where∗−1is the inverse under convolution product. On the other hand, if we consider the mapφ2from the

proposition whereH=C0, H0 = A= A0 =C,α= IdH0, and_γ=_βwe have that:

S0◦β=φ2(S0) =φ2(Id∗−_H01) = (φ2(IdH0))∗−1= (IdH0◦β)∗−1 =β∗−1.

Hence, β◦S=S0◦βas we wanted.

1.1.3 Free objects

To finish this section we want to discuss what a free object in a given category is. Here we are assuming basic concepts of category theory such as: what a category is, what a faithfull functor is, whatSetmeans, etc. To those readers who do not have the necessary background we recommend [4]. First, suppose that we have two categoriesC andDwith an objectXinD. Moreover, assume you have a forgetful functorF:C→D.

Definition1.1.20. AfreeC-objecton Xis an object AinCtogether with an inclusioni: X→F(A)inDsuch that for all objects Bin C and f : X → F(B)in D there exists a unique morphism g : A → Bsuch that the following diagram commutes:

X

f

i

'

'

F(A)

F(g)

w

w

F(B)

Remark. For the purpose of this thesis the categoryDwill be the categorySet.

Example.

1. In the category of commutative algebras we have that k[x₁, ...x_n] is a free object for each n. Clearly those algebras isomorphic to an algebra of polynomials will be free too.

2. We can extend the previous example to the category of locally finite graded algebras, for which we have a free objectk[x1,x2, ...].

3. In the category of associative algebras we have thatkhx

1, ...,xniis a free object. Here, unlike the other examples, the indeterminates do not commute with each other.

4. We can extend the last example to the category of graded associative algebras, which has as a free objectkhx1,x2, ...i.

1.2 c o m b i nat o r i c s

In this section we will discuss many combinatorial objects that will help us understand two of the most important examples of combinatorial Hopf algebras; this objects range over generating functions to compo-sitions.

1.2.1 Generating functions

Here we will define what a generating function is, state what the benefits of working with them are and provide few examples. We start with the most classical of the generating functions, which appears vastly on probability and combinatorics:

Definition1.2.1. The ordinary generating function (OGF) of a sequence(An)n∈Nis the formal power series

A(z) =

∑

n≥0

Anzn.

The generating functions helps us to codify the information of a given combinatorial object as the co-efficients of a “big polynomial”, giving us tools for understanding how two objects may interact by un-derstanding how their generating functions relate to each other. It’s worth saying that the usual sum and multiplication of polynomials, without caring about convergence, will make the set of generating functions a ring. We finish this small introduction with two examples:

Example.

1. For the combinatorial object “ number of subsets of a finite set” we have the following generating function:

∑

n≥0

2nzn=

∑

n≥0 (2z)n

= 1

1−2z.

2. From the combinatorial object “ Fibonacci numbers ” we have the following generating function:

F(z) =

∑

n≥0

fnzn

=

∑

n≥2

(fn−1+ fn−2)zn+z+1

=

∑

n≥2

fn−1zn+

∑

n≥2

fn−2zn+z+1

=z

∑

n≥1

fn−1zn−1+z2

∑

n≥2

fn−2zn−2+1

=zF(z) +z2F(z) +1,

which is equivalent to:

F(z) = 1

1−z−z2.

Though this “introduction” is not showing the strength of the generating functions we want to remark some things:

• When you have a generating function that encodes the information of a combinatorial object, such as the Fibonacci numbers, by studying it assymptotically you may get an explicit formula for the recurrence.

• There are combinatorial objects for which we don’t know any “nice” formula in order to compute its values such as pn, the number of integer partitions of a number n, it is with those numbers that the generating functions become of great use. Usually, the generating functions become a handy tool, in contrast to working with explicit numbers.

• Equality of generating functions implies that the two combinatorial objects that they represent are “equivalent”.

• Multiplication, sum and even evaluating certain values, allow us to understand new combinatorial identities and relations between different combinatorial objects.

For more information on generating functions we recommend [5].

1.2.2 Partially ordered sets

Definition 1.2.2. A poset (partially ordered set) P is a finite set, also denoted P, together with a binary relation denoted≤satisfying the following axioms:

• (reflexivity)x≤x for allx ∈P.

• (antisymmetry) Ifx ≤yandy≤x, thenx =y.

• (transitivity) If x≤yandy≤z, thenx≤z.

We say that y covers x in P if x < y and there is no other element in between, we denote it by xly. A chain Cin a poset is a totally ordered subset ofP, i.e., if x,y∈ Cthenx ≤yor y≤ xinP. The length of the chain is defined by the number of elements in the chain, and if it hasn+1 elements the chain is said to be of lengthn. Finally, we say that a finite poset isgraded of rank nif every maximal chain has lengthn.

Example.

1. One classical example is the boolean algebra: Let n ∈ N and E = {1, 2, ...,n} be a set. We call

B_n= (2E,⊂)theboolean algebraon nelements, where 2E denotes the power set of E.

You can depict graphically a poset by drawing itsHasse diagram, that is, put the elements of P as vertices and join them with edges when one of them is covered by the other. In Figure 1 we present the Hasse diagram ofB₃.

1.2.3 Graphs

Here we give the basic definitions and results concerning graphs.

Definition1.2.3. A graph is a pairG= (V,E)such that:

1. Vis a set.

2. Eis a multiset of two element subsets ofV.

The elements ofV are called vertices and the elements ofEare its edges.

When we have a given graphGwe will useV(G)to denote its vertices andE(G)to denote its edges. For two verticesu,v∈ V(G)we will denote the edge whose endpoints areuandvas uv.

Example.

1. The pair (∅,∅) is a graph.

2. For any setV, the pair(V,∅)is a graph where none of the vertices are connected by any edge.

3. The complete graph ofnverticesK_n is a graph which has as vertex set ann-element set, and as edges all the two-element subsets ofV(Kn).

Definition1.2.4. Let G be a graph.

1. A vertexvis incident with an edgee ifvis an endpoint ofe.

2. Two vertices are adjacent, or neighbours, ifuv ∈ E(G). Two edges e 6= f are adjacent if they share a vertex.

3. The degree ofv, denoted asdeg(v)=|{e ∈E(G)|e is incident tov}|.

4. Let x and y be two vertices of G. A walk from u to v is a sequence of vertices v₁,v₂, ...,v

k such that v1 =u,vk =vandvivi+1 ∈E(G)for alli∈[k]. The length of a walk is the number of edges associated

to the sequence, in the definition the path would have lengthk−1.

5. Letuandvbe two vertices ofG. A path fromutovis a walk such that all the elements of the sequence are different.

6. A cycle is a pathCwhose startpointc₁ and endpointc_k are the same vertex. There is a special kind of graphs calleddirected graphsthat will be our main interest:

Definition1.2.5. A directed graph is a pair G= (V,E)such that:

1. Vis a set.,

2. Eis a multiset satisfying that ife∈ E, thene∈ V×V.

Here all of our conventions of vertices and edges hold. The reader should note that the difference between an undirected graph and a directed graph is that for the undirected graph we havevu=uvfor allu,v∈V(G) but for a directed graph we may haveuv ∈ V(G) butvu not necessarily an edge. To show this we present Figure2.

Figure2: On the left we have a graph, and in the right a directed graph.

Then, despite having the same vertices and apparently having the same edges, none of the graphs are equal.

1.2.3.1 Dual graded graphs

The study of combinatorial Hopf algebras in this thesis was motivated by a particular family of graphs that are calleddual graded graphs. These graphs are due to Fomin [6] and they are of huge importance since they generalize a combinatorial structure called differential posets. Before describing this graphs in full we begin with some basics:

Definition1.2.6. Agraded graphis a tupleG= (V,E,ρ)where (V,E)is a directed graph such that:

1. Vis a discrete set,

2. _ρ:V →Nis a rank function,

3. For alluv∈ Ewe have thatρ(v) =ρ(u) +1.

From now on, the set Pn = {u|ρ(u) = n}is called thenth-level of Gandm(u,v)is the multiplicity of an edgeuv∈ E, that is, the number of edges joining uandv. Before defining duality let’s discuss a little about an algebraic structure on G. Let kbe a field of characteristic 0 and define kG to be the vector space which consists of formal linear combinatoin of vertices ofG. Since the directed graphs give an orientation onGwe can define endomorphismsUandDsuch that:

Uv=

∑

vu∈E

m(v,u)u

and

Du=

∑

vu∈E

m(v,u)v

where u,v ∈ V. These operations are of immense importance since you can study the number of certain kinds of paths by just taking words consisting of letters{U,D}and applying the word to G. In this thesis we are interested in the case where we have a pair of graded graphs G1 andG2, in this case we are going to

deviate a little from the definition so that

Uv=

∑

vu∈E1

m1(v,u)u

and

Du=

∑

vu∈E2

m2(v,u)v

By restricting this operations to the levels ofG1andG2we can win more combinatorial information from

the relationship of the graphs, we denoteDn=D|Pn andUn=U|Pn the respective restrictions.

Definition1.2.7. Let G1 andG2 be graded graphs with common set of vertices and common rank function,

the tuple(G1,G2)is said to ber−dualif the following relations hold:

Dn+1Un−Un−1Dn =rnIn.

wherer= {rn}n∈Nand In is the identity operator.

The vector r is called the differential coefficient, and we call the pair P = (G1,G1)a differential poset if r = {r}_n∈N. In the case of differential posets [9], those two operators are the ones which determine the existence of the Robinson-Schensted correspondence (the reason for which differential posets were studied in the first place).

1.2.4 Partitions

Here we define the partitions of numbers, which appear in many branches of mathematics. More acknowl-edgedly, in the representation theory of the symmetric group.

Definition 1.2.8. We call an l-tuple λ = (λ1,λ2, ...,λl) a partition if λ1 ≥ ... ≥ λl ≥ 0. Additionally if l

∑

i=1

λi =nwe say thatλis a partition ofnand denote it byλ`n.

There are many ways of thinking of a partition, and maybe the most useful way is by its geometric representation, or what is best called as its Young diagram. To depict the partition as itsYoung diagram we think each number in the tuple as a given number of blocks of squares placed from left to right, and the order of the tuple will give us the way to place this “set” of blocks vertically. InFigure3aandFigure3bwe show two different examples of Young diagrams.

(a)λ= (5, 4, 3, 2, 1). (b)µ= (1, 1, 1).

Figure3: Two partitions in their respective geometric representation.

Now, let’s discuss some operations on the partitions:

Definition 1.2.9. Let m,n ∈ _N. If λ ` n and µ ` m, we define the product of two partitions λ∪µ˙ as the partition obtained by:

1. taking the multiset union_λ∪µ˙ of the parts of_λand_µ.

2. Reordering the elements of_λ∪µ˙ so that is weakly decreasing. Let’s see how this product works with an example:

Example.

1. Let_λ= (4, 2, 2)and_µ= (3, 3, 3, 2). Then,_λ∪µ˙ = (4, 3, 3, 3, 2, 2, 2).

2. Let_λ= (5, 4, 3, 2, 1)and_µ= (1, 1, 1). Then,_λ∪µ˙ = (5, 4, 3, 2, 1, 1, 1, 1).

3. This operation is associative since the union is associative and reordering does not affect it: Let λ = (5, 4, 3, 2, 1),µ = (1, 1, 1) and e = (4, 2, 2). We know that λ∪µ˙ = (5, 4, 3, 2, 1, 1, 1, 1) and µ∪e˙ = (4, 2, 2, 1, 1, 1). It is easy to check that(λ∪µ)˙ ∪e˙ = (5, 4, 4, 3, 2, 2, 2, 1, 1, 1, 1) =λ∪(µ˙ ∪e)˙ .

Remark.

1. _λ∪µ˙ =_µ∪λ˙ for all _λ,_µpartitions.

2. The empty partition_λøis the identity of this product.

Additionally, we can define many poset structures on the partitions. We will be interested in particular in the ordering that gives rise to the so calledYoung’s lattice.

Definition 1.2.10. Let Y= {λ|λ ` nfor somen ∈ N}the set of all partitions. We can define an order in Y

by saying that for anyλ= (λ1, ...λk),µ= (µ1, ...,µl)∈Ywe have:

λ≤ µif and only ifk≤landλi ≤µi

where i is a number less than or equal to l. Note that this is well defined, in the case that the size of µis strictly larger than that ofλyou can think ofλas the concatenation ofλand the number of zeros needed to match the size. It is easy to check that this is exactly the same as ordering the partitions by inclusion of their Young diagrams.

As we stated before, this posetY is what we call theYoung lattice. Clearly, Y is a graded poset and each level is given by the partitions of a positive integer, if we considerY = (Y,Y)we will get a 1−dualgraded graph (1-differential poset) as it is shown in the next proposition:

Proposition 1.2.11. The Young lattice is a dual graded graph with differential coefficient of1, that is, the following equation holds:

DU−UD= I.

Proof. Here the operators are simpler than those described in Subsection1.2.2.1, since

Uλ=

∑

λlµ

µ

and

Dµ=

∑

λlµ

λ.

Before proving the proposition we should note two things:

1. In order to create a partition _λ∗ that covers _λ we have to either add a new 1 at the end of the tuple, λ∗ = (λ1, ...,λk, 1), or add a 1 to one of the terms of the partition,i.e. λ∗= (λ1, ...,λi+1, ...λk).

2. If a _λ∗ from the second case exists then there exists a _λ∗ that is covered by _λ. Namely, _λ∗ = (λ₁, ...,λi+1−1, ...λ_k whereiis the term where you added the 1 in order to getλ∗.

From this it’s clear that for a given partition λ = (λ1, ...,λk) there are at most k+1 partitions that cover λ and at mostk partitions that are covered byλ. Moreover, we know that the number of partitions that cover λand the number of partitions covered byλdiffer by one. From this it follows that:

(DU−UD)λ= DUλ−UDλ

=

∑

λlµe

∑

lµ

e−

∑

µlλµ

∑

le

e

=λ+A.

We need to show that A = 0 and we are done with the proof. For this suppose that A 6= 0. Then, there existse6=λthat is in the sum described above. We know that eis a partition of the same number asλand that they differ in two terms, becauseewas not anhilitated in the equation. That means, that two different partitions that cover λ cover e and that only one partition covered by λ is covered by e. By 2. this can’t happen, so A=0. We conclude thatYis a dual graded graph.

The order allows us to define “restricted” partitions, such as the skew partitionλ/µ= (λ1−µ1, ...,λ_l−µ_l), where µ≤ λ. Finally, we can study different combinatorial properties of the partitions byfillingthe blocks of the Young diagrams with numbers. We will be interested particularly in one way to fill blocks, called the

Littlewood-Richardson coefficients. First, we describe some of these “fillings” in the next definition:

Definition1.2.12. Let λ= (λ1, ...,λl)be a partition.

1. A semistandardad Young tableau (SSYT) of shape_λ is an array T = (T_ij)of positive integers such that 1 ≤ i≤ l, 1 ≤ j ≤ λi, that is weakly increasing in every row and strictly increasing in every column. The size of an SSYT is its number of entries.

2. Let _µbe a partition such that_µ ≤ _λ. Asemistandard tableau of skew shape_λ/_µis an array T = (T

ij) of positive integers such that 1 ≤ i ≤ l, µi ≤ j ≤ λi, that is weakly increasing in every row and strictly increasing in very column.

3. Astandardad Young tableau(SYT) of shape_λis an arrayT= (T_ij)of positive integers such that 1≤i≤ l, 1 ≤ j≤ λi, that is strictly increasing in every row and strictly increasing in every column. The size of an SSYT is its number of entries.

4. Let _µ be a partition such that _µ ≤ _λ. A standard tableau of skew shape_λ/_µ is an array T = (T

ij) of positive integers such that 1≤i≤l,µi ≤ j≤ λi, that is strictly increasing in every row and column.

Remark. Note that these tableaux can be thought of as a filling of the (skew) Young diagram by an array satifying the restrictions originally inherent to the tableau (Figure4).

(a) A SSYT of shape(5, 4, 3, 2, 1).

(b) A semistandard skew tableau of shape(5, 4, 3, 2, 1)/(3, 2, 1).

Figure4: Examples of both a semistandard skew tableau and SSYT.

Now, we will describe a “game” which will help us understand what are the Littlewood-Richardson coefficients. Let λ/µ be a skew shape. Consider the boxes b that can be added to λ/µ, so that b shares at least one edge with λ/µ, and {b} ∪λ/µ is a valid skew shape. Two types of such boxes may occur depending on which side of λ/µ that they are on. We mark by a bullet • the dashed boxes that share a lower or a right edge withλ/µ, while those that share an upper or a left edge are marked by a circle◦, as is shown inFigure5.

Figure5

Suppose we are given an SYT T of shape λ/µ. To each box b marked • or ◦, we will associate a new tableau jdtb(T)of Tgiven by:

1. Ifbis marked by•renamebasb₀. We know, becauseb₀ is marked by a bullet, that there exists at least one box b1 in λ/µthat is adjacent to b0; if there are two such boxes, letb1 be the one with a smaller

entry. Move the entry occupyingb1intob0. Then look at the tableau entries to the right and below b1,

and repeat the procedure.

2. Itbis marked by◦renamebasb₀. We know, because b₀ is marked by a circle, that there exists at least one boxb1 inλ/µthat is adjacent tob0; if there are two such boxes, letb1 be the one with the greatest

entry. Move the entry occupying b1 intob0. Then look at the tableau entries to the left and above b1,

and repeat the procedure.

Definition1.2.13. The standard Young tableau jdtb(T)ofT is thejeu de taquin slideofT intob.

Remark. The reader should note that ∪ here means the union of the diagrams, as adjoining one box to the skew diagram. Additionaly, we say that two SYT arejeu de taquinequivalent if they can be constructed from the other by a sequence of jeu de taquin transformations.

For an example on how to construct a jeu de taquin slide, look atFigure6.

Figure6

At last, we state the relationship between the jeu de taquin with the Littlewood-Richardson coeffficients without a proof. For more information the reader should look at [11].

Theorem 1.2.14. Let ν,µ,λ be partitions such that µ ≤ λ. Fix an SYT P of shapeν. The Littlewood-Richardson

coefficient cλ

µ,νis equal to the number of SYT of shapeλ/µthat are jeu de taquin equivalent to P.

Example. Takeλ= (4, 3, 2),µ= (2, 1), andν= (3, 2, 1). With a little work you can see that cλ_µ,ν =2.

(a) (b) (c)

Figure7: The tableau of_νand the possible_λ/_µskew tableau that are jeu de taquin equivalent.

1.2.5 Compositions

Definition1.2.15. We call an l-tuplec= (c1,c2, ...,cl)sucht thatci >0 for alli, a composition. Additionally

if l

∑

i=1

ci =n we say thatcis anl-composition ofnand denote it by c|=l n. We omit the subscript when we

are talking of nol-composition in particular.

Note that a composition is only the sum decomposition of a given number:

Example.

1. The partition(5, 4, 3, 2, 1)is a composition too, but the composition(5, 3, 4, 1, 1, 1)is not a partition.

2. In the last examples(5, 4, 3, 2, 1)|=₅15 and(5, 3, 4, 1, 1, 1)|=₆15. Proposition1.2.16. For a fixed n∈Nthere are2n−1compositions of n.

Proof. Consider an n-tuple of 1’s. We can think of a tuple of any size as a sum decomposition of a given number, the n-tuple of 1’s is just a decomposition of n in a sum whose parts are all 1’s. If we change a comma in the tuple for a sum symbol we get a new compostion ofn. This suggests that by choosing wether a comma stays the same or changes into a sum symbol we can find every composition ofn. Since there are

n−1 commas, there are 2n−1compositions ofn.

Proposition1.2.17. For fixed n,l∈Nsuch that l ≤n there are(n_l−−₁1)l-compositions of n.

Proof. Consider the same n-tuple of 1’s from the last proof. We want to count the number of l-composition of n, that is, we want to assure that in the tuple there arel−1 commas. There are(n_l−−₁1)to do so, hence the proposition follows.

Definition 1.2.18. We call an l-tuple c = (c1,c2, ...,cl)sucht that ci ≥ 0 for alli, a weak composition.

Addi-tionally if l

∑

i=1

ci = nwe say thatcis a weakl-composition ofnand denote it byc|=0l n. We omit the subscript

when we are talking of nol- weak composition in particular.

Proposition1.2.19. For fixed n,l∈_Nsuch that l ≤n there are(n+_nl−1)weak l-compositions of n.

Proof. Let c= (c1,c2, ...,cl)a weakl-composition of n. Then the following holds:

n=c1+c2+...+cl

n+l= (c1+1) + (c2+1) +...+ (cl+1).

Note that this shows that there is a 1−1 correspondence between weakl−compositions ofnandl-compositions ofn+l. Therefore counting one will be equivalent to counting the other. ByProposition1.2.11the theorem follows.

There are two really important operations on compositions that we will use throughout the text:

Definition1.2.20.

1. Ifc= (c₁, ...,c

l)is a composition, we call the compositioncrev= (cl, ...,c1)the reverse composition ofc.

2. Letn,m∈ Nandc,dtwo compositions ofn andm, respectively. We call theconcatenation cdof cand

d, the composition obtained by concatenatingdonc.

Remark.

1. c_øis the empty composition and is the identity of the concatenation operation. 2. Concatenation is not commutative.

Example.

1. c= (1, 2, 0, 2)is a weak composition but it is not a composition.

2. Ifc= (5, 4, 3, 2, 1)is a composition, then crev corresponds to the composition(1, 2, 3, 4, 5).

3. Letc= (5, 4, 3, 2, 1)andd= (1, 1, 1, 2)be two compositions of 15 and 5, respectively. The composition

cdis defined by(5, 4, 3, 2, 1, 1, 1, 1, 2).

4. Letc= (5, 4, 3, 2, 1)andd= (1, 1, 1, 2)be two compositions of 15 and 5, respectively. The composition

dcis defined by(1, 1, 1, 2, 5, 4, 3, 2, 1).

And, lastly, we define a partial order for the compositions:

Definition1.2.21. Letcanddtwo compositions. We say thatc≤dif the terms ofcare obtained by merging adjacent terms ofd.

Example.

1. Consider c = (2, 1, 1), d = (1, 1, 1, 1) and e = (1, 1, 2). We have that c ≤ d and e ≤ d but we can’t comparecande. This shows that the order is not a total order.

2

C O M B I N AT O R I A L H O P F A L G E B R A S

In this chapter we will give the definition of a combinatorial Hopf Algebra and some classical examples that will be used throughout the thesis. At the end of the chapter we will present some results on the theory of combinatorial Hopf algebras. We start with the basics:

Definition2.0.22.

1. A combinatorial Hopf algebra (H,α) is a graded connected Hopf algebra H =

M

n≥0

Hn over k with

dim(Hn) < ∞ for all n ∈ N, together with an algebra map α : H → k. We call such α a character of H.

2. LetHandAtwo combinatorial Hopf algebras with respective characters_αand_β. We say that f : H→

Ais a morphism of combinatorial Hopf algebras if it is a graded Hopf alegbra morphism such that the following diagram commutes:

H

α

f _// A

β





k

or equivalently, β(f(h)) =α(h)for allh∈ H.

2.1 t h e c o m b i nat o r i a l h o p f a l g e b r a o f s y m m e t r i c f u n c t i o n s

First let us introduce some notation: let ¯x = (x1,x2, ..)be an enumerable list of distinct commuting

indeter-minates, and ¯α= (α1,α2, ...)be an enumerable list of natural numbers where only a finite number ofαi’s are nonzero. We denote by ¯xα¯ _{= x}α1

1 x

α2

2 ..., where ¯αcan be thought of as a weak composition of∑αi, suggesting that we define the degreedeg(x¯α¯_{)to be}∑

αi. Now we are ready to define what a symmetric function is:

Definition2.1.1. Asymmetric functionoverkis a function

f(x¯) =

∑

¯

α

cα¯x¯α¯

such that f is of bounded degree and f(x_σ(1),xσ(2), ...) = f(x1,x2, ...) for every permutation σ of Z

+_{. We}

denote the set of symmetric functions asSym.

Remark. We say that f(x) =

∑

¯

α

cα¯x¯α¯ is of bounded degree if it exists a bounddfor whichdeg(x¯α¯)>dimplies

cα¯ = 0. Additionaly, we can define the symmetric functions as the G-invariant subalgebra R(x¯)G of the

ring of formal power series with bounded degree R(x¯), where Gis the group that permutes finitely many variables while leaving the other ones fixed. This definition has many interesting interpretations ofSymbut will not be used that much on the thesis.

Example.

1. The function x2

1+x22+x32+... is symmetric since all the indeterminates have the same exponent and

coefficient.

2. The function

∑

i6=j6=k6=l

x3_ix1_jx_k1x1_l is symmetric because for every permutationσ we have that ifi6= jthen

3. The function

∑

i<j

x2_ix1_j is not symmetric because there are permutations for which σ(i) > σ(j) when

i<j.

We denote Symn the set of all symmetric functions of degreen. Notice thatSym has an algebra structure with the sum and multiplication of indeterminates, moreover, this algebra is graded by the degree function,

i.e. Sym= M

n≥0

Symn. We want to giveSyma Hopf algebra structure that is compatible with the algebra

struc-ture defined before, for this we have to show that Sym has a compatible coalgebra structure, an antipode and thatdim(Symn)<∞.

Let us begin by showing that dim(Symn) < ∞. In the preliminaries we defined a partition as a tuple

λ = (λ1, ...,λl) where the λi ≥ λj for any i ≥ j. From now on we think of the partition λ to be exactly as in the preliminaries but with the distinction that here λ_l is followed by a sequence of 0’s, that is λ = (λ₁, ...,λl, 0, 0, ...). Throughout the thesis we won’t bother about explicitly writing the 0’s, but they are thought of being there. Moreover, when we talk about a permutation ofλwe are just refering to a new order of the tuple(λ₁, ...,λ_l, 0, 0, ...). We show this in the following example:

Example.

1. Let (12435) be a permutation and _λ the partition (3, 3, 1, 1). The weak composition (12435)λ = (0, 3, 1, 3, 1)is a permutation of λ. Notice that a permutation ofλis not necesarilly a partition.

2. Let_λas in the last example and(145)(2)(3)(678)be a permutation. The weak composition

(145)(2)(3)(678)λ= (0, 3, 1, 3, 1)is a permutation ofλbut is the same as the permutation(12435)λ= (0, 3, 1, 3, 1).

Definition2.1.2. For any partitionλdefine themonomial symmetric function

mλ =

∑

α∈I

¯

xα_,

where I is the set of distinct permutations ofλ.

To see that for anyλwe have thatmλ is in fact a symmetric function the reader should note that because

the indeterminates are commutative we can rewritemλas

∑

σ

(σx¯)λ _where

σ is a permutation ofZ+.

Proposition2.1.3. S={mλ|λ` n}is a basis for Symn.

Proof. That S is linearly independent follows from the fact that if λ 6= µ then mλ 6= mµ. To see that S

spans Symn let f(x¯) =

∑

¯

α

cα¯x¯α¯ be a symmetric function of degreen. If we fix one term ¯xα¯ of f and reorder

the variables so that the exponents are weakly decreasing we will have a term of a monomial symmetric function, by grouping all the terms of f that come from permuting the variables of that fixed ¯x¯α _{we have}

that f = f0+mλ whereλ is the reordering of our given ¯α. We can keep applying this procedure for f 0 _and

so on. Therefore,Sis a basis.

Corollary2.1.4. dim(Symn)<∞for all n∈N.

Though we have a basis forSym, there may be calculations that are easier to carry out with other bases.

Definition2.1.5. The elementary symmetric functions are

en=

∑

i1<...<in

xi1....xin

eλ = eλ1...eλl, whereλ= (λ1, ...,λl).

One can show that the transition matrix to express the eλ’s in terms of the mλ’s is lower triangular, in

order to be concise and “well focused” throughout the thesis we are only going to state this as a result. If the reader wants to see a proof we recommend that he looks at [11].

Proposition 2.1.6. The transition matrix T to express {eλ} in terms of {mλ} is lower triangular with 1’s on the

diagonal.

From this result it easily follows that theeλ’s form a basis:

Corollary2.1.7. {eλ|λ`n}is a basis for Symn.

Proof. By the proposition we have that the linear transformation Thas determinant different from 0 so it is in fact an automorphism ofSymn. Since linear transformations preserve bases, the corollary follows.

Proposition2.1.8. Sym'k[e1,e2, ...], that is, Sym is free.

Proof. One can think of Symas an algebra of polynomials where the indeterminates are the ei’s. From the examples ofSection1.1.3we know that this makesSyma free algebra.

Now we can start defining the additional structures on Sym:

Proposition2.1.9. Define for every partitionλ

∆(mλ) =

∑

(ν,µ):

ν∪˙µ=λ

mν⊗mµ.

And, define e in the usual manner for graded connected coalgebras, i.e. the identity on Sym0 and the annihilator

for every homogeneous element of degree greater than0. For any other symmetric function extend by linearity. This induces a coalgebra structure on Sym.

Proof. We have to prove that the counit is compatible with the comultiplication and that the comultiplication is coassociative.

• The counit is compatible: notice that the only element that survives in the sum is

∑

(ν,µ): ν∪˙µ=λ

mνe(mµ) and

∑

(ν,µ): ν∪˙µ=λ

e(mν)mµismλ.

• Coassociativity: it is sufficient to show that the comultiplication is coassociative on the basis. This ammounts to showing that(∆⊗Id)(∆(mλ) = (Id⊗∆)(∆(mλ) for allmλ. The right hand side of the

equation ends up being:

(∆⊗Id)(∆(mλ)) = (∆⊗Id)(

∑

(ν,µ):

ν∪˙µ=λ

mν⊗mµ)

=

∑

(ν,µ): ν∪˙µ=λ

∑

(ν1,ν2):

ν1∪˙ν2=ν

mν1⊗mν2⊗mµ

=

∑

(ρ,δ,γ):

(ρ∪˙δ)∪˙γ=λ

mρ⊗mδ⊗mγ.

For the left hand side we have:

(Id⊗_∆)(∆(mλ)) = (Id⊗∆)(

∑

(ν,µ): ν∪˙µ=λ

mν⊗mµ)

=

∑

(ν,µ): ν∪˙µ=λ

∑

(µ1,µ2):

µ1∪˙µ2=µ

mν⊗mµ1⊗mµ2

=

∑

(ρ,δ,γ):

ρ∪˙(δ∪˙γ)=λ

mρ⊗mδ⊗mγ.