Combinatorial Hopf Algebras

Alessandro Iraci

October 25, 2017

1

1 25/09/2017The goal of the first lecture is to define what an Hopf algebra is. In some sense, a Hopf algebra is to agroup what a vector space is to a set.First of all, let’s try to understand what extra structure there is in a group with respect to a set. We needto to this in category theory language.

Definition 1.1. A group G is a set with two maps m: G ù G ô G and e: {?} ô G such that m isassociative, there it is an unit element, and there it is an inverse map.

The associative property and the existence of the unity correspond to the following diagrams.

G ù G ù G G ù G

G ù G G

mùid

idùm m

m

G

{?} ù G G ù {?}

G ù G

Ì Ì

m

eùid idùe

Now let’s take the dual maps �: G ô G ù G (diagonal map) and ⇡ : G ô {?} (projection) and thecorresponding dual diagrams.

G ù G ù G G ù G

G ù G G

�ùid

idù� �

�

G

{?} ù G G ù {?}

G ù G

Ì Ì

�⇡ùid idù⇡

The existence of the inverse map i: G ô G corresponds to the following commutative diagram.

G ù G G ù G

G

{?}G

G ù G G ù G

idùi

⇡ e

iùid

�

�

m

m

We just translated the group properties in category theory language. Now, we want to go further with ouranalogy. If we have a set X and a commutative ring k, we can define kX as the free k-module spannedby X. Now, if we take kG instead of kX, all the extra structure we get is encoded in some commutativediagrams as above.Recall that we define kG := {u

g

› g À G} with the algebra structure given by u

g

� uh

= u

gh

.

2

We have a bilinear multiplication m: kG ‰ kG ô kG, and an inclusion map ⌘: k{?} ô kGThe associative property and the existence of the unity translate to the following diagrams.

kG ‰ kG ‰ kG kG ‰ kG

kG ‰ kG kG

m‰id

id‰m

m

m

G

k{?}‰ kG kG ‰ k{?}

kG ‰ kG

Ì Ì

m

⌘‰id id‰⌘

Now, as we’ve done before, let’s take the dual maps �: kG ô kG ‰ kG and ✏: kG ô k{?} and thecorresponding dual diagrams.

kG ‰ kG ‰ kG kG ‰ kG

kG ‰ kG kG

�‰id

id‰� �

�

kGk{?}‰ kG kG ‰ k{?}

kG ‰ kG

Ì Ì

�✏‰id id‰✏

And now the existence of the inverse translates into the existence of another map, called antipodal map(or antipode), S : kG ô kG, which corresponds to the following commutative diagram.

kG ‰ kG kG ‰ kG

kG k{?} kG

kG ‰ kG kG ‰ kG

id‰S

✏

⌘

S‰id

�

�

m

m

There are some di�erences. The map � is not the diagonal map, since

�(ug

+ u

h

) = �(ug

) + �(uh

) = u

g

‰ u

g

+ u

h

‰ u

h

ë (ug

+ u

h

)‰ (ug

+ u

h

)

so � acts as the diagonal map on the basis {ug

› g À G} extended by linearity. Same holds for theprojection map ⌘, which maps every element of the base to 1 À k Ù k{?}.Now, the first two diagrams encode the property of being an associative k-algebra. By just replacing kGwith A, and k{?} with k, we get the following.

A‰A‰A A‰A

A‰A A

m‰id

id‰m

m

m

Ak ‰A A‰ k

A‰A

Ì Ì

m

⌘‰id id‰⌘

3

The next two diagrams encode the property of being a coalgebra. Namely, a k-coalgebra is a k-module Cwith a comultiplication �: C ô C‰C and a counit ✏: C ô k such that the following diagrams commute.

C ‰ C ‰ C C ‰ C

C ‰ C C

�‰id

id‰� �

�

Ck ‰ C C ‰ k

C ‰ C

Ì Ì

�✏‰id id‰✏

For our convenience, we define a new notation as follows: �(c) = c(1) ‰ c(2), where c(1) ‰ c(2) representsa sum over certain indices of elements that depend on c. Since C is coassociative, we define

c(1) ‰ c(2) ‰ c(3) := �(c(1))‰ c(2) = c(1) ‰ �(c(2))

and so on.

Proposition 1.2. If C is a coalgebra, then C< := Homk(C,k) is an algebra.

Proof. We define (f < g)(c) := f (c(1)) � g(c(2)), and it gives an associative product.The converse, unfortunately, doesn’t work. The main problem is that (A ‰ A)< ‘ A<

‰ A<, and theequality doesn’t always hold. It obviously holds if A is finite dimensional over k; if it’s not, then we mightbe able to solve the problem by defining a “reduced” dual, which we’ll see later.Now we can go a little further, and give the definition of Hopf algebra.

Definition 1.3. A bialgebra is a k-module A which is both an algebra and a coalgebra, such that thetwo structures are compatible (i.e. � and ✏ are algebra morphisms, or equivalently m and e are coalgebramorphisms). If it also has an antipodal map S : A ô A, then it’s an Hopf algebra.

If A is a Hopf algebra, and M , N are A-modules, then M ‰k N is an A-module with the producta � (m‰ n) = (a(1) � m)‰ (a(2) � n).Example 1.4. Let V be a vector space, and T (V ) its tensor algebra.It also has a coalgebra structure given by �(1) := 1Ì 1, �(v) := 1Ì v + v Ì 1, ✏(1) = 1, ✏(v) = 0 (wedenote by Ì the “external” tensor product T (V ) Ì T (V ), in order to distinguish it from the “internal”tensor product of T (V )).At last, it also has an antipodal map S defined by S(v) = *v, and S(v‰w) = w‰v. S is an antialgebramap, which means that S(ab) = S(b) � S(a).

To prove stu� with Hopf algebra, there it is an useful trick. If A is an algebra and C is a coalgebra, thenHom(C,A) is an algebra with the convolution product defined in Proposition 1.2. Notice that f < g =m˝(f ‰ g)˝�. If A is a Hopf algebra, then we can choose C = A, and ⌘˝✏ is the unit element.Now, take Hom(A‰A,A‰A), and look at the maps a‰ b ≠ S(a‰ b) and a‰ b ≠ S(a)‰S(b). It’spretty easy to check that their inverses are equal, so they must be equal as well.

Definition 1.5. A bialgebra A is graded if it can be decomposed as a direct sum of k-modules Ai

suchthat A =

ªNA

i

, m: Ai

‰Aj

ô Ai+j , �: A

i

ôª

jfiAj

‰Ai*j .

Definition 1.6. A graded bialgebra A is connected if A0 = k.

4

Proposition 1.7. A connected bialgebra is a Hopf algebra.

Proof. By hypothesis, ✏: A0 ô k, and ✏An

= 0 for n > 0 (it’s a graded morphism). We want to findS : A ô A such that ✏(a) � 1 = S(a(1)) � a(2).By definition, �(a) = a(1)‰a(2) = 1‰a+a®(1)‰a

®(2)+a‰1 for some a®(1), a

®(2), so from ⌘˝✏ = m˝(S‰id)˝�

we get ✏(a) �1 = S(1) �a+S(a®(1)) �a®(2) +S(a) �1 and now one can determine S by induction on the grade

of the element.Namely, one can prove that S =

≥k>0(*1)kmk*1(id*⌘˝✏)‰k�k*1.

By definition, an algebra A is commutative if, for any a, b À A, we have ab = ba. Now, if we define⌧ : A ‰ A ô A ‰ A as ⌧(a ‰ b) = b ‰ a extended by linearity (the twist map), we can translatecommutativity to the following diagram.

A‰A A‰A

A

⌧

m m

And of course we can take the dual.

Definition 1.8. A coalgebra C is cocommutative if the following diagram commutes.

C ‰ C C ‰ C

A

⌧

��

Theorem 1.9 (Milnor-Moore). If A is a cocommutative Hopf algebra over k algebraically closed fieldwith char(k) = 0, then A = U (L) # O(G).Here L is a Lie algebra, G is a group, and the # sign denotes some sort of combination we won’t discuss.

Many other concepts regarding algebras have their dual in coalgebras.

Definition 1.10. Let C be a coalgebra, J ” C. We say that J is a coideal, and we denote it by J / C, if�(J ) ” J ‰ C + C ‰ J , and ✏(J ) = 0.

Proposition 1.11. If A is a bialgebra, and I /A is both an ideal and a coideal, then A_I is a bialgebra.If A is also a Hopf algebra, and S(I) ” I , then A_I is also a Hopf algebra.

Now we can give the definition of “reduced” dual we anticipated above.

Definition 1.12. Let A be a bialgebra (resp. Hopf algebra). We define its reduced dual A0 as

A0 := {f À A< › « I / A, I ” ker f : dimA_I < ÿ}.

Notice that the definition depends on the structure of A.

Proposition 1.13. If A is a graded bialgebra, then A0 =ª

NAn

<” A< =

±NA

n

<.

5

At last, we have these two interesting facts.

Fact 1.14. If G is an algebraic group and L is its associated Lie algebra, then O(G)0 = U (L).

Fact 1.15. If A is a Hopf algebra, then

G := {x À A › �(x) = x ‰ x, ✏(x) = 1}

is a group, andL := {x À A › �(x) = 1‰ x + x ‰ 1, ✏(x) = 0}

is a Lie algebra.

2 09/10/2017We now want to introduce two specific Hopf algebras, namely the Hopf algebra of the symmetric functionsand the Hopf algebra of the quasisymmetric functions.Let we start by defining the space of the symmetric functions.

Definition 2.1. Let k be a field, char k = 0. Let X be a countable, totally ordered set of variables. Letk[[X]] be the algebra of formal power series in those variables. We say that f À k[[X]] is a symmetricfunction if for all n À N, for all ↵ À Nn, for all x, y À X

n, the coe�cients of x↵ and y

↵ in f are equal (wedefine x

↵ = x

↵11 5 x

↵

n

n

). We denote with Sym(X) the space of those symmetric functions.

Proposition 2.2. Sym is a graded algebra.

To understand what we’re doing, let’s think about polynomials first. On k[x1,… , x

n

] we have an actionof the symmetric group S

n

obtained by permuting the variables. We call ⇤n

= k[x1,… , x

n

]Sn the fixedpoints set, which is indeed a subalgebra.Now, if we take X = {x1, x2,…}, we have an algebra morphism Sym(X) ô ⇤

n

defined by x

i

≠ x

i

for i f n, and x

i

≠ 0 if i > n. Our algebra Sym(X) is the inverse limit of those algebras. Another wayto think at it is as the fixed group of S ¬ k[[X]], where S = lim

},,,,,,,,,,S

n

, and each S

n

permutes the first nvariables.It’s probably best to think at Sym(X) as polynomials in a fixed and large enough but finite number ofvariables (notice that no element of Sym(X) is actually a polynomial), since basically all the theory worksin the same way.Let’s now see some of those symmetric functions.Example 2.3. e

n

=≥

x1<5<x

n

x1x25 x

n

and h

n

=≥

x1f5fxn

x1x25 x

n

are both symmetric functions.If the number of variables is finite (say N g n), then the sums are finite as well and those functions arejust polynomials in x1,… , x

N

. If it’s not, then those are formal power series which collapse to the samepolynomials if you set x

i

= 0 for i > N .For example, for n = 2 and N = 3, we have e2 = x1x2 + x1x3 + x2x3 and h2 = x1

2 + x22 + x3

2 + x1x2 +x1x3 + x2x3.

In order to fully understand symmetric functions, we need to introduce partitions.

Definition 2.4. Given n À N, a partition of n, denoted by � Ô n is an element � À Nk for some k suchthat �1 g �2 g 5 g �

k

and �1 + �2 +5 + �

k

= n. We say that the length of the partition is l(�) = k.

6

Definition 2.5. Given n À N, a composition of n, denoted by ↵ ı n is an element ↵ À Nk for some k suchthat ↵1 + ↵2 +5 + ↵

k

= n. We say that the length of the composition is l(↵) = k.

If ↵ ı n, we define �(↵) Ô n as the unique partition of n obtained by rearranging the ↵

i

’s in weaklydecreasing order.

Proposition 2.6. I : {↵ ı n} ô 2[n*1] defined by I(↵) = {↵1, ↵1 + ↵2,… , ↵1 +5+ ↵

k*1} is a bijection.

Definition 2.7. If ↵, � ı n, we say that ↵ refines �, denoted by ↵ … �, if I(�) ” I(↵).

Definition 2.8. If ↵ ı m and � ı n are compositions, we define their product ↵� ı m + n as theconcatenation (↵1,… , ↵l(↵), �1,… , �l(�)).

Now we’re ready to introduce some of the bases of the algebra Sym(X).

Theorem 2.9 (Fundamental theorem of the symmetric functions). Let � Ô n, e�

:=±

e

�

i

. Then the set{e

�

› � Ô n, n À N} is a basis of Sym(X). Equivalently, Sym(X) = k[e1, e2,…].

In fact, we have several possible bases.

• e

�

:=±

e

�

i

is the basis of elementary symmetric functions.

• h

�

:=±

h

�

i

is the basis of (complete) homogeneous symmetric functions.

• p

�

:=±

p

�

i

, where p

k

=≥

X

x

k, is the basis of the power symmetric functions.

• m

�

:=≥

�(↵)=�≥

x1<5<xl(�)x

↵ is the basis of the monomial symmetric functions.

• s

�

:=≥

TÀSSYT(�) xT is the basis of the Schur symmetric functions.

Here, SSYT(�) denotes the set of semi-standard Young tableaux of shape �, of which I’ll skip the definition(there’s plenty of literature everywhere). Those are all bases of Sym(X).Remark 2.10. From now on, I’ll write Sym(X) as Sym, since it’s structure does not depend on X.

Definition 2.11. We define !: Sym ô Sym as !(en

) = h

n

.

Proposition 2.12. We have the following.

• !

2 = id

• !(pn

) = (*1)n*1pn

• !(s�

) = s

�

® , where �

® is the transpose of �.

Now, we want to endow Symwith a Hopf algebra structure. We need to define a comultiplication, a counit,and an antipode.

Definition 2.13. We define the counit of Sym as ✏: Sym ô k, ✏(f ) = f (0) (all xi

≠ 0).

Definition 2.14. We define the comultiplication of Sym as �: Sym ô Sym‰Sym as follows.Suppose you have two sets of variables X, Y . Since X, Y , and X·Y are all countable, totally ordered set(we define x < y in X·Y for all x À X, y À Y ), we have f À Sym(X) ˆ Sym(Y ) ˆ Sym(X·Y ), and wecan expand f (x, y) À Sym(X · Y ) as f (x, y) =

≥a(x)b(y), where a(x) À Sym(X) and b(y) À Sym(Y ).

We can now define �(f ) =≥

a ‰ b.

7

It takes some time to convince yourself that this is a well-defined comultiplication. Let’s look at someexamples.Example 2.15. We have the following expansions.

• e

n

(x, y) =≥

ifn ei(x)en*i(y), so �(en

) =≥

ifn ei ‰ e

n*i.

• p

n

(x, y) = p

n

(x) + p

n

(y), so �(pn

) = p

n

‰ 1 + 1‰ p

n

.

• �(m�

) =≥

�(↵�)=� m↵

‰ m

�

, where ↵, � are partitions.

Finally, we need an antipode. Since Sym is a connected bialgebra, by Proposition 1.7, it must have one.It can be constructed from the identity 0 = ✏(p

n

) � 1 = S(pn

) � 1 + S(1) � pn

= S(pn

) + p

n

, which impliesS(p

n

) = *pn

= (*1)n!(pn

).It follows that S(f ) = (*1)deg f!(f ), for f homogeneous, is an antipode map. So, we have the following.

Theorem 2.16. Sym is a Hopf algebra.

We actually have a bit more structure. In particular, we can endow Sym with a scalar product.

Definition 2.17. We define the Hall scalar product on Sym as Íh�

,m

�

Î := �

��

.

Proposition 2.18. We haveÍs�

, s

�

Î = �

��

® (so, s�

is an orthonormal basis), and also ! is an isometry.

It also follows that the map f ≠ Íf , �Î is an isomorphism between Sym and Sym0 (the reduced dual wediscussed in the previous lecture). This implies that Sym is cocommutative.We can now move to quasisymmetric functions.

Definition 2.19. We say that f À k[[X]] is a quasisymmetric function if for all n À N, for all ↵ À Nn,for all x, y À X

n such that x1 < 5 < x

n

, y1 < 5 < y

n

, the coe�cients of x↵ and y

↵ in f are equal. Wedenote with QSym(X) the space of those quasisymmetric functions.

Of course every symmetric function is quasisymmetric, but the converse does not hold in general. QSymis a graded vector space, with a basis {M

↵

:=≥

x

↵ › ↵ ı n, n À N}.QSym is actually a Hopf algebra, but before giving all the structure we need a couple more definition.

Definition 2.20. Let ↵, � be compositions. We define the shu�e ↵—� as the set of compositions of lengthl(↵) + l(�) such that all the ↵

i

and the �

i

appear, preserving their relative order.We define the quasi-shu�e ↵ É— � as the same set, but including the compositions obtained by collapsingtogether two adjacent parts belonging to di�erent compositions.

Hopefully, the two following examples will clarify the definition. Let ↵ = (2, 1) and � = (1). Then

↵ — � = {(2, 1, 1), (2, 1, 1), (1, 2, 1)}

and↵ É— � = {(2, 1, 1), (2, 1, 1), (1, 2, 1), (3, 1), (2, 2)}.

Since QSym is closed under product, it’s a Hopf subalgebra of k[[X]], where �(M�

) =≥

↵�=� M↵

‰

M

�

(it’s the same as before), and the same counit. The antipod is defined as S(M↵

) =≥

�…rev(↵)M�

,where rev(↵) is just ↵ reversed. We can also consider the dual Hopf algebra QSym0, which is the socalled noncommutative symmetric functions Hopf algebra; since it’s not commutative, then QSym is notcocommutative.

8

3 23/10/2017We’re now going to see a couple examples of combinatorial Hopf algebras.

3.1 Posets Hopf algebraDefinition 3.1. A (finite) poset (partially ordered set) is a pair (P ,f) where P is a (finite) set, and f is apartial order relation on P .We will sometimes identify a poset (P ,f) with the set of its elements P .

We say that a bijection �: P ô P

® is a poset isomorphism between (P ,f) and (P ®,f®) if, for all p, q À P ,

we have p f q ⌥ �(p) f®�(q).

Our base space will be the vector space P spanned by all the isomorphism classes of finite posets. Wewant to endow it with a Hopf algebra structure.

Definition 3.2. We define the multiplication map m: P ‰ P ô P on the elements of the form P ‰ P

®

as m(P ‰ P

®) := P · P

®, and then extended by linearity.Here, by disjoint union of P and P

® we mean the poset (P · P

®,û) where p f q if and only if p, q À P

and p f q, or p, q À P

® and p f®q (so, û := f · f®).

Definition 3.3. We define the unit as ⌘: k ô P as ⌘(1) = (…,…) (the empty poset).

It’s clear that the definition we just gave are enough to endowP with a graded, connected algebra structure,where the degree of a poset (P ,f) is just the cardinality of P . It’s connected because the only poset withzero elements is the empty poset.Before going any further, we need one more definition.

Definition 3.4. Let (P ,f) be a poset. A subset I ” P is an order ideal of P (denoted by I f P ) if for allx À P , y À I , we have x f y ⌃ x À I .

Now we can describe the coalgebra structure.

Definition 3.5. We define the comultiplication map �: P ô P ‰ P on the basis as

�(P ) =…IfP

I ‰ (P ‰ I),

extended by linearity. Here I and P ‰ I are posets with the order given by the restriction of f on P .

Definition 3.6. We define the counit as ✏: P ô k as the projection on the subspace spanned by the emptyposet (identifying k with k(…,…)).

It’s easy to check that the comultiplication is coassociative (because if I f P , then J f I ⌃ J f P ,and J f P ‰ I ⌃ J ‰ I f P ), and that the counit is such that the relevant diagram is commutative.This implies that P is also a coalgebra. If we prove that � and ✏ are both algebra morphisms, then, sinceP is graded connected, it must be a Hopf algebra.By using the formula S =

≥k>0(*1)kmk*1(id*⌘˝✏)‰k�k*1 from Proposition 1.7, we get

S(P ) =…

I0<5<I

k

(*1)kI0 · (I1 ‰ I0) ·5 · P ‰ Ik

which is just a sum of copies of the set P with a weaker order relation.

9

3.2 Graded posets Hopf algebra (Rota)There’s another Hopf algebra that can be defined starting from posets, which is the graded posets Hopfalgebra. We need some more definitions, though.

Definition 3.7. A poset P is ranked if it has a minimum element 0, a maximum element 1, and all themaximal chains 0 = p0 < p1 < 5 < p

k

= 1 between the two have the same length.If that’s the case, we say that the poset P has rank k.

If we have two posets P , P ®, then P ù P

® is also a poset, where (x, x®) f (y, y®) ⌥ x f y and x

® f y

®.It’s easy to check that if P and P

® are graded, then P ù P

® is also ranked, and its rank is the sum of theranks of P and P

®.Let R be the vector space spanned by the isomorphism classes of ranked posets. As usual, we want toendow it with a Hopf algebra structure.

Definition 3.8. We define the multiplication map m: R‰R ô R on the elements of the form P ‰ P

®

as m(P ‰ P

®) := P ù P

®, and then extended by linearity.

Definition 3.9. We define the unit as ⌘: k ô P as ⌘(1) = ({?}) (the poset with one element).

With these two operations, R is an algebra. Now we describe the coalgebra structure.

Definition 3.10. We define the comultiplication map �: R ô R‰R on the basis as

�(P ) =…xÀP

[0, x]‰ [x, 1],

extended by linearity. Here, and [x, y] := {z À P › x f z f y}.

Definition 3.11. We define the counit as ✏: R ô k as the projection on the subspace spanned by theposet of cardinality 1.

Once again, it’s fairly easy to check that � and ✏ are algebra morphisms, so R is a graded connectedbialgebra, hence it’s a Hopf algebra.By using the formula S =

≥k>0(*1)kmk*1(id*⌘˝✏)‰k�k*1 from Proposition 1.7, we get

S(P ) =…

x0<5<x

k

(*1)k[0, x0] ù [x0, x1] ù5 ù [xk

, 1].

Given any poset P , let Int(P ) := {[x, y] › x f y À P } be the set of its intervals.

Definition 3.12. We define the incidence algebra of P as the set I(P ) := {f : Int(P ) ô k} with theusual sum and a convolution product defined as

f < g([x, y]) :=…

zÀ[x,y]f ([x, z]) � g([z, y]).

We usually write f (x, y) instead of f ([x, y]). The unit element is the function �(x, y) = �(x = y). Anotherimportant function is ⇣ (x, y) = 1. It’s invertible because of this general result.

Proposition 3.13. f À I(P ) < ⌥ f (x, x) ë 0 for all x À P .

10

Proof. We need to prove that there exists a function g À I(P ) such that f (x, x)g(x, x) = 1 for all x À P ,and …

zÀ[x,y]f ([x, z]) � g([z, y]) = 0

for all x < y À P . The first condition implies that f (x, x) ë 0, and also fixes the value of g on all theintervals of rank 0. The second one allows us to define g(x, y) by induction on the rank of [x, y], using theformula

g(x, y) = *f (x, x)*1…

x<zfyf (x, z) � g(z, y).

An extremely important function, while talking about posets, is the Möbius function � = ⇣

*1, because ofthe following theorem.

Theorem 3.14 (Möbius inversion formula). Let P be a locally finite poset (i.e. every interval is finite),let F : P ô k any function and suppose that there exists an element p À P such that F (x) = 0 for allx f p. Then the function G(x) :=

≥yfx F (y) is well defined, and F (x) =

≥yfx �(y, x)G(y).

Notice that�(x, y) =

…CÀC(x,y)

(*1)l(C)

where C(x, y) denotes all the chains in [x, y]. We can extend the Möbius function to any ranked posetjust by replacing [x, y] with P , and now the formula for the antipode in the Rota’s Hopf algebra clearlygeneralizes the Möbius function.

3.3 Permutations Hopf algebra (Malvenuto-Reutenauer)One of the most important combinatorial Hopf algebras is the Malvenuto-Reutenauer Hopf algebra ofpermutations. Our base space will be the vector space SSym spanned by

¿nÀN S

n

, graded by n. Toavoid confusion, we’ll denote by F

u

the basis element corresponding to the permutation u, which will bewritten in the one-line notation. We’ll denote by 1 the basis element of degree zero (the only permutationon …). Notice that F1 ë 1, since F1 is the basis element corresponding to the only permutation of {1}.We need to define the operations.

Definition 3.15. Let u À S

p

, v À S

q

. We define the multiplication map m: SSym‰SSym ô SSymon the elements of the form F

u

‰ Fv

as m(Fu

‰ Fv

) :=≥

wÀu—v

Fw

, and then extended by linearity.Here, by u — v we mean the following. Think at u À S

p

= S([1, p]), and v À S

q

= S([p + 1, p + q]).Now, if you write u and v in the one line notation, then u — v is a set of strings that are permutations ofthe integers in [1, p + q], which we can identify with the corresponding elements of S

p+q .

Equivalently, we can give the following definition. Let

S

(p,q) := {� À S

p+q › �1 < �2 < 5 < �

p

, �

p+1 < �

p+2 < 5 < �

p+q}

which is a collection of minimal (in length) representatives of left cosets of Sp

ù S

q

< S

p+q , that we willcall Grassmannian permutations. Now, for (u, v) À S

p

ù S

q

< S

p+q , we define

m(Fu

‰ Fv

) :=…

wÀS(p,q)

F(u,v)�w*1

which is equivalent to the previous one.

11

Definition 3.16. We define the unit as ⌘: k ô SSym as ⌘(1) = 1 (the permutation on …).

As usual, this is enough to endow SSym with a graded, connected algebra structure; and as usual, weneed another little definition before going any further.

Definition 3.17. For any sequence (a1,… , a

n

) of distinct integers, we define its standard permutation(or flattening) as the permutation u = st(a1,… , a

n

) À S

n

such that ui

< u

j

⌥ a

i

< a

j

(we’re justrenormalizing the a

i

’s so that they range from 1 to n).

For example, st(625) = 312. Now we can define the coalgebra structure.

Definition 3.18. We define the comultiplication map �: SSym ô SSym‰SSym on the basis as

�(u) =n…

p=0Fst(u1,…,u

p

) ‰ Fst(up+1,…,u

n

)

for u À S

n

, and then extended by linearity.

Definition 3.19. We define the counit as ✏: R ô k as the projection on the subspace spanned by 1.

Since� and ✏ are algebra morphisms, soSSym is a graded connected bialgebra, hence it’s a Hopf algebra.The explicit definition of the antipode is quite complicated, so we’ll skip it.For the Hopf algebra QSym, we defined a basis M

↵

indexed by the compositions of n. Now we needanother basis, namely

F

↵

:=…↵…�

M

�

where the ordering is by refinement. The basis {F↵

› ↵ ı n, n À N} is called the basis of the Gessel’sfundamental quasisymmetric functions (or fundamental basis) of QSym. The fact that {F

↵

} is a basisfollows from the Möbius inversion formula

M

↵

=…↵…�

(*1)l(�)*l(↵)F�

.

We need another definition.

Definition 3.20. For u À S

n

, we define the descent set of u as Des(u) := {i À [n * 1] › ui

> u

i+1}, andthe number of descents as des(u) := #Des(u).

Theorem 3.21. We have a graded Hopf algebra morphism SSym ß QSym defined by Fu

≠ F

I

*1(Des(u)),where I is the map that identifies subsets of [n * 1] with compositions of n (and the degree is taken intoaccount).

We want to use this map to define another basis of SSym, which should be the analogue of the monomialbasis on QSym. First of all, we need a partial order on S

n

.

Definition 3.22. We define the (left) weak order on S

n

as u f v ⌥ Inv(u) ” Inv(v), where

Inv(u) := {(i, j) À [n]2 › i < j, u

i

> u

j

}

is the set of inversions of u.

12

This order can be defined in a di�erent way. S

n

is generated, as a group, by the adjacent transpositionss

i

= (i, i + 1) for i À [n * 1], so every element can be written as a product of the s

i

’s. We say that u f v

if there exist two minimal decompositions u =±

s

i

j

, v =±

s

i

k

, such that ij

’s are a sublist of the i

k

’s(maintaining the order). This means that u is smaller than v if you can obtain it from v by deleting someadjacent transpositions.Now, we can define another basis of SSym as

Mu

:=…ufv

�(u, v) � Fv

where � is the Möbius function on S

n

equipped with the weak order.The basis M

u

has the property that the antipode map S has a completely explicit combinatorial definitionon this basis, in terms of the weak order on S

n

.

Definition 3.23. For u À S

n

, we define the global descents set of u as

GDes(u) := {i À [n * 1] › j f i < i + 1 f k ⌃ u

j

> u

k

}

and the number of global descents as gdes(u) := #GDes(u).

Definition 3.24. For v À S

n

and A = {a1 < 5 < a

k

} ” [n * 1], we define

v

A

:= st(u1,… , u

a1) ù st(u

a1+1,… , a

u2) ù5 ù st(u

a

k

+1,… , u

n

).

Definition 3.25. Let v,w À S

n

. We define (v,w) as the number of u À S

A := {u À S

n

› Des(u) ” A},with A = GDes(v), such that

1. v

A

u

*1 f w,

2. v is a maximal element (with respect to the weak order) among those that satisfy 1,

3. A is a minimal set (with respect to inclusion) among those that contain Des(u) and satisfy 1.

We have the following result.

Theorem 3.26.S(M

v

) = (*1)gdes(v)+1 �…wÀS

n

(v,w) �Mw

.

This gives a completely explicit and combinatorial descriptions of the coe�cients of the antipode map,with respect to the basis M

u

.For further reference, see [1]

3.4 Chromatic Hopf algebraDefinition 3.27. A (finite, simple, unoriented) graph is a pair G = (E,V ) such that

• V is a finite set, whose elements are the vertices of the graph,

• E ”

0V

2

1is the set of the edges of the graph.

13

We use the standard notation0X

k

1to denote the set of the subsets of X of cardinality k.

We say that v,w À V are adjacent if {v,w} À E. Two graphs G = (E,V ),G® = (E®,V

®) are isomorphicif there exists a bijection �: V ô V

® such that {v,w} À E ⌥ {�(v),�(w)} À E

®.Our base space will be the vector space G spanned by all the isomorphism classes of graphs. We want toendow it with a Hopf algebra structure.

Definition 3.28. We define the multiplication map m: G‰ G ô G on the elements of the form G ‰ G

®

as m(G ‰G

®) := G · G

®, and then extended by linearity.Here, by disjoint union of G and G

® we mean the graph G · G

® whose vertices set is V · V

® and whoseedges set is E · E

®.

Definition 3.29. We define the unit as ⌘: k ô G as ⌘(1) = (…,…) (the empty graph).

It’s clear that the definition we just gave are enough to endow G with a graded, connected algebra structure,where the degree of a graph G is just cardinality of its vertices set V . It’s connected because the only graphwith zero vertices is the empty graph.Before going any further, we need one more definition.

Definition 3.30. Let G be a graph, S ” V a subset of the vertices. We define the subgraph of G inducedby S as

G ›S

:= (S,E „0S

2

1).

A graph G is connected if it’s not disjoint union of two nonempty subgraphs, and a connected componentof G is a maximal connected subgraph.

Definition 3.31. We define the comultiplication map �: G ô G‰ G on the basis as

�(G) =…S”V

G ›S

‰ G ›V ‰S

extended by linearity.

Notice that the comupltiplication is cocommutative, since applying the twist map ⌧ simply switches S

with V ‰ S, which also appears in the sum.

Definition 3.32. We define the counit as ✏: G ô k as the projection on the subspace spanned by theempty graph (identifying k with k(…,…)).

It’s easy to check that the comultiplication is coassociative, and that the counit is such that the relevantdiagram is commutative. This implies that G is also a coalgebra. If we prove that � and ✏ are both algebramorphisms, then, since G is graded connected, it must be a Hopf algebra.Those are actually two pretty easy computations.

�(G � G®) =…

T”V ·V

®(G · G

®) ›T

‰ (G · G

®) ›(V ·V

®)‰T

=

H…S”V

G ›S

‰ G ›V ‰S

I�

H …S

®”V

®G

® ›S

® ‰ G

® ›V

®‰S®

I= �(G) � �(G®)

14

where S = T „ V and S

® = T „ V

®. The check for ✏ is even easier, since ✏(G � G®) is 1 if and only if Gand G

® are both empty, which is true if and only if G · G

® is empty.This proves that G is a Hopf algebra.

References[1] Marcelo Aguiar and Frank Sottile, Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, Adv. Math. 191 (2005),

no. 2, 225–275. MR2103213

15