SURJECTIONS AND DOUBLE POSETS LOÏC FOISSY AND FRÉDÉRIC PATRAS

The theory and structure of the Hopf algebra of surjections (known as WQSym, the Hopf algebra of word quasi-symmetric functions) parallels largely the one of bijections (known as FQSym, the Hopf algebra of free quasi-symmetric functions or as the MalvenutoReutenauer Hopf algebra). Recall besides that bijections can be encoded by pictures in the sense of Zelevinsky and more generally by pictures between double posets. This encoding is, among others, a key ingredient of Zelevinsky's proof of the Littlewood-Richardson rule and of the Robinson-Schensted-Knuth correspondence. The study of surjections from a picture and double poset theoretic point of view, which is the subject of the present article, seems instead new. The article is organized as follows. We introduce rst a family of double posets, weak planar posets, that generalize the planar posets and are in bijection with surjections or, equivalently, packed words. The following sections investigate their Hopf algebraic properties, which are inherited from the Hopf algebra structure of double posets and their relations with WQSym. Abstract.

1. Introduction The algebraic and Hopf algebraic structures associated to permutations [18, 5, 8] have been intensively studied and applied in various contexts. A reason for their ubiquity is that they occur naturally in geometry, algebraic topology and classical and stochastic integral calculus because their (noncommutative) shue products encode cartesian products of simplices [21]. In combinatorics, they appear naturally in the theory of twisted algebras and twisted Hopf algebras (aka Hopf species) [22, Sect. 5] [2]. In the theory of operads, through the associative operad [1] and its Hopf operad structure [17, Sect. 2], and so on... In another direction, permutations can be encoded by pictures in the sense of Zelevinsky [24].

This encoding is a key ingredient of his proof of

the Littlewood-Richardson rule and of the Robinson-Schensted-Knuth correspondence. Recently, the rst of us introduced a new approach to Hopf algebras of permutations by means of the notion of double posets [19], closely related to the notion of picture [10, 9, 11]. Besides making a bridge between the combinatorial, picture-theoretic, and the Hopf algebraic approaches to permutations, this new construction is natural in that it originates in the encoding of the statistics of inversions in symmetric groups (recall that, given a permutation

σ

in the symmetric group

1

Sn , the pair of integers (i, j) in [n]2 ,

2

LOÏC FOISSY AND FRÉDÉRIC PATRAS

with

i < j,

σ(i) > σ(j).

denes an inversion if and only if

Inversions are a

key notion from the Coxeter group point of view on permutations; among others, their cardinal computes the length of the permutation). The present article is dedicated to constructing and proving analog objects and results for surjections. Although they are a relatively less familiar object in view of applications of algebra and combinatorics than permutations, there are several reasons to be interested in surjections. For example, through the usual bijection with ordered partitions of nite initial subsets of the integers, they appear naturally in the study of the geometry of Coxeter groups [3] and of twisted Hopf algebras (aka Hopf species) [23, 2]. They have also been involved recently in the modelling of quasi-shue products [20, 15, 14] and their applications in stochastics [7, 6]. The theory and structure of the Hopf algebra of surjections (known as WQSym, the Hopf algebra of word quasi-symmetric functions) [16, 4] parallels largely the one of bijections. The study of surjections from a picture and double poset theoretic point of view, which is the subject of the present article, seems instead new. The article is organized as follows. We introduce rst a family of double posets, weak planar posets, that generalize the planar posets of [10] and are in bijection with surjections or, equivalently, packed words. The following sections investigate their Hopf algebraic properties, which are inherited from the Hopf algebra structure of double posets and their relations with WQSym. 2. Weak plane posets Recall that a double poset is a set equipped with two orders. A quasi-order is a binary relation

≤

which is reexive and transitive but not necessarily

antisymmetric (so that one may have

x ≤ y

and

y ≤ x

x 6= y ). A x≤y write x ≡ y and

with

quasi-order is total when all elements are comparable (it holds that or

y≤x

say that

for arbitrary

x

and

y

x

and

y ).

When

x≤y

and

are equivalent (the relation

≡

y ≤ x,

we

is an equivalence relation).

A quasi-poset is a set equipped with a quasi-order. All the posets, double posets, quasi-posets...

we will consider are assumed to be nite (we omit

therefore nite in our denitions and statements).

Denition 1. A weak plane poset is a double poset (P, ≤1 , ≤2 ) such that: (1) For all x, y ∈ P , (x ≤1 y and x ≤2 y ) =⇒ (x = y). (2) The relation  dened on P by (x ≤1 y or x ≤2 y ) is a total quasiorder. In particular, the relation dened by (x relation (we denote it as above by

Example. •

For all

x = y.

y

A plane poset is a double poset

x, y ∈ P ,

if

x

and

y

and

y  x)

is an equivalence

≡). (P, ≤1 , ≤2 )

such that:

are comparable for both

≤1

and

≤2 ,

then

SURJECTIONS AND DOUBLE POSETS

•

For all

x, y ∈ P , x

and

By proposition 11 of [10], if

y

are comparable for

(P, ≤1 , ≤2 )

P

is a weak plane poset, then

plane: by (2), two distinct elements that



x, y

or for

≤2 .

is a plane poset, then

order, and obviously (1) is also satised. Moreover, if

≤1

3



is a total

So plane posets are weak plane.



is an order if, and only if,

P

is

are always comparable and the fact

is an order implies that they cannot be comparable for both

≤1

and

≤2 .

Lemma 2. Let (P, ≤1 , ≤2 ) be a weak plane poset. Then: (3) The relation  dened on P by (y ≤1 x or x ≤2 y ) is a total order. Proof.

Let

x, y, z ∈ P , such that x  y

and

y  z.

Three cases are possible:

• (y ≤1 x and z ≤1 y ) or (x ≤2 y and y ≤2 z ). Then (z ≤1 x or x ≤2 z ), so x  z . • x ≤2 y and z ≤1 y . As  is a total quasi-order, two subcases are possible.

 x  z , then x ≤1 z

or x ≤2 z . If x ≤2 z , then x  z . x ≤1 y and x ≤2 y . By (1), x = y , so x  z .  z  x, then z ≤1 x or z ≤2 x. If z ≤1 x, then x  z . then z ≤2 y and z ≤1 y . By (1), y = z , so x  z . • y ≤1 x and y ≤2 z . Similar proof.

If

x ≤1 z ,

If

z ≤2 x,

then

 is transitive. x, y ∈ P , such that x  y

Therefore, Let

and

y  x.

Two cases are possible:

≤1 x and x ≤1 y ) or (x ≤2 y and y ≤2 x): then x = y . y ≤1 x and y ≤2 x, or x ≤1 y and x ≤2 y : by (1), x = y .

(1) (y (2) So



is an order.

x, y ∈ P , x  y x  y or y  x:  is

For all so

Remark.

or

y  x,

so

y ≤1 x

or

x ≤2 y

or

total.

x ≤1 y

or

y ≤2 x, 

(3) implies (1), but not (2).

3. Surjections and packed words

f from [n] to [k] is represented by the sequence of wf := f (1) . . . f (n), the word wf is packed: its set of letters identies with an initial subset of the integers (in that case, [k]). Conversely, an arbitrary packed word of length n can always be obtained in that way: packed words (on length n) are in bijection with surjections (with domain [n] and codomain an initial subset of the integers). Recall also that total quasi-orders  on [n] are in bijection with packed words. An example gives the general rule: assume that n = 6 and that the When a surjection

its values,

quasi-order is dened by

2 ≡ 5  1 ≡ 3 ≡ 6  4.

4

LOÏC FOISSY AND FRÉDÉRIC PATRAS

Then, the corresponding packed word is

{2, 5}

212312:

gives the position of letter 1, the second,

the rst equivalence class

{1, 3, 6},

the positions of

letter 2, and so on.

Proposition 3. Let w be a packed word of length n. The double poset Pw (also written Dp(w)) is dened by Pw = ([n], ≤1 , ≤2 ), with: ∀i, j ∈ [n], i ≤1 j ⇐⇒ (i ≥ j and w(i) ≤ w(j)), i ≤2 j ⇐⇒ (i ≤ j and w(i) ≤ w(j))

It is a weak plane poset. The total quasi-order  is the one associated bijectively to w. Proof.

The fact that

i ≤1 j

and

i ≤2 j ,

Pw is a double poset is obvious. i ≥ j and i ≤ j , so i = j : (1) is

then

For all

i, j ∈ [n],

if

veried. Moreover:

i  j ⇐⇒ w(i) ≤ w(j), 

so

is indeed a total quasi-order. Finally, remark that the total order



agrees with the natural order:

i  j ⇐⇒ i ≤ j. 

Theorem 4. The set of packed words of length n and of isomorphism classes of weak plane posets are in bijection through Dp. The inverse map pack is given as follows. Let P a weak double poset. By Lemma ?? we can assume that P = [n] with  the natural order. Then, Dp−1 (P ) =: pack(P ) is the packed word associated to the total quasi-order . Proof.

Let us show rst that for any packed words

w, w0 ,

the double posets

Pw and Pw0 are isomorphic if, and only if, w = w0 . Let f : Pw → Pw0 be an 0 0 0 isomorphism. Then f is increasing from ([n], ) to ([n ],  ). As  and  [n] and [n0 ], n = n0 that i ≤ j :

are the usual total orders of for all

i, j ∈ [n],

assuming

and

f = Id[n] .

Consequently,

w(i) ≤ w(j) ⇐⇒ i ≤2 j ⇐⇒ i ≤02 j ⇐⇒ w0 (i) ≤ w0 (j). As

w

and

are packed words,

w = w0 .

Dp(pack(P )) = P . We can assume that (P, ) = ([n], ≤). The packed word w = pack(P ) is 0 0 such that for all i, j ∈ [n], i  j ⇐⇒ w(i) ≤ w(j). We denote by ≤1 and ≤2 Let

P

w0

now be a weak plane poset and let us show that

SURJECTIONS AND DOUBLE POSETS the orders of

Pw .

i, j ∈ [n]:

Then, for all

i ≤01 j ⇐⇒ (j ≤ i)

So

≤01 =≤1 .

(w(i) ≤ w(j))

and

⇐⇒ (j  i)

and

⇐⇒ (i ≤1 j

or

(i  j)

j ≤2 i)

(i ≤1 j

and

⇐⇒ (i ≤1 j)

or

(j ≤2 i

⇐⇒ (i ≤1 j) ⇐⇒ i ≤1 j.

or

(i = j)

Similarly,

5

or

i ≤2 j)

i ≤2 j)

and

≤02 =≤2 .



4. The self-dual Hopf algebra structure We denote by

HW P P

the vector space generated by isomorphism classes

of weak plane posets and show below how denitions and results in [19, 13] apply in this context (denitions and results relative to double posets are taken from [19]). Let

P, Q

be two double posets. Two preorders are dened on

∀i, j ∈ P t Q, i ≤1 j

(i, j ∈ P

if

or

i ≤2 j

HDP

(i, j ∈ Q

(i, j ∈ P

if

(i, j ∈ Q

or

(i ∈ P P Q.

P Q: HW P P

i ≤1 j);

i ≤2 j)

and

and

i ≤2 j)

j ∈ Q).

Extending this product by bi-

of double posets an associative algebra,

whose unit is the empty double poset denoted posets, then so is

by:

i ≤1 j)

and

and

or

This denes a double poset denoted by linearity makes the linear span

and

P tQ

1.

is a subalgebra of

If P and Q HDP .

are weak plane

Denition 5. Let P be a double poset (resp. weak double poset) and let X ⊆ P. • X is also a double poset (resp. weak) by restriction of ≤1 and ≤2 : we denote this double poset (resp. weak) by P|X . • We shall say that X is an open set of P if: ∀i, j ∈ P, i ≤1 j and i ∈ X =⇒ j ∈ X.

The set of open sets of P is denoted by T op(P ). • A coproduct is dened on HDP (resp. HW P P ) by ∆(P ) =

X

P|P \O ⊗ P|O .

O∈T op(P )

Theorem 6. The product and the coproduct equip HDP and therefore its subspace HW P P with the structure of a graded, connected Hopf algebra.

6

LOÏC FOISSY AND FRÉDÉRIC PATRAS See [19] for a proof of the compatibility properties between the product

and the coproduct characterizing a Hopf algebra.

P = (P, ≤1 , ≤2 ), ι(P ) := (P, ≤2 , ≤1 ). If P is a weak plane poset, then so is ι(P ). Recall also that there exists a pairing on HDP dened, for two double posets P, Q by Recall that, for

hP, Qi := ]P ic(P, Q), where

P ic(P, Q) stands for the number P and Q is a bijection f

picture between

of pictures between

P

and

Q

(a

such that

i ≤1 j ⇒ f (i) ≤2 f (j), f (i) ≤1 f (j) ⇒ i ≤2 j.) The Hopf algebra of double poset

HDP

is self-dual for this pairing.

By

Proposition 24 of [13]:

Proposition 7. The Hopf algebra HW P P is a self-dual Hopf subalgebra of the Hopf algebra of double poset HDP . 5. Linear extensions and

WQSym.

Denition 8. Let P = (P, ≤1 , ≤2 ) be a weak plane poset. We assume that (P, ) = ([n], ≤). A linear extension of P is a surjective map f : [n] −→ [k] such that: (1) For all i, j ∈ [n], i ≤1 j =⇒ f (i) ≤ f (j). (2) For all i, j ∈ [n], f (i) = f (j) =⇒ i ≡ j . The set of linear extensions of P is denoted by Lin(P ). f is a linear extension word f (1) . . . f (n). If

Let us denote by

WQSym

P,

we see it as a packed

the set of packed words of length

n.

Recall that

is given two products, its usual one, denoted by ., and the shifted

shue product both

P W (n)

of a weak plane poset

(WQSym,

[12].

, ∆)

∆ the (WQSym, ., ∆)

Denoting by and

usual coproduct of

WQSym,

are Hopf algebras.

For example:

(1)

(1) = (12) + (21),

(1).(2) = (12) + (21) + (11).

Proposition 9. The following map is a Hopf algebra morphism:   HW P P φ: P 

Proof.

−→ (WQSym, X −→ f.

, ∆)

f ∈Lin(P )

We omit it since it is similar to the proof of Theorem 18 of [10].

Proposition 10. For all f, g ∈ P W (n), we shall say that f ≤ g if: (1) For all i, j ∈ [n], i ≥ j and f (i) ≤ f (j) =⇒ g(i) ≤ g(j). (2) For all i, j ∈ [n], g(i) = g(j) =⇒ f (i) = f (j).



SURJECTIONS AND DOUBLE POSETS

7

Then ≤ is an order on P W (n). Moreover, for all f ∈ P W (n): φ(Pf ) =

X

g.

f ≤g

Proof. The relation ≤ is clearly transitive and reexive. Let us assume that f ≤ g and g ≤ f . By (2), for all i, j ∈ [n], f (i) = f (j) if, and only if, g(i) = g(j). Hence, putting k = max(f ) = max(g), there exists a unique −1 (i) = g −1 (σ(i)). By (1), if permutation σ ∈ Sk such that for all i ∈ [k], f i ≥ j , then g(i) ≤ g(j) ⇐⇒ f (i) ≤ f (j). Hence: max f −1 (1) = max{i ∈ [n] | ∀j ≤ i, f (i) ≤ f (j)} = max{i ∈ [n] | ∀j ≤ i, g(i) ≤ g(j)} = max g −1 (1) = max(g −1 (σ(1)), so

σ(1) = 1.

≤

is an order. Let

σ = Idk , so f = g : the relation f, g ∈ P W (n). Then: ( ∀i, j ∈ [n], i ≥ j and f (i) ≤ f (j) =⇒ g(i) ≤ g(j), g ∈ Lin(f ) ⇐⇒ ∀i, j ∈ [n], g(i) = g(j) =⇒ f (i) = f (j) Iterating this, one shows that

⇐⇒ f ≤ g. So

Lin(Pf ) = {g ∈ P W (n), f ≤ g}.



Corollary 11. φ is a Hopf algebra isomorphism. Here are the Hasse graphs of

(P W (2), ≤)

and

(P W (3), ≤).

21 B

12

| || || | ||

BB BB BB B

11

hhh 321EEVVVVVVVV hhhh yyy EE VVVV h h h EE y VVVV hhh h y h VVVV h E y h h E h y VVVV h h y h h V hh 221 VVVVV 231E 312 hh 211 h h E y VVVV h h EE VVVV yyy hhhh VVVV hhhEEh h h V h y V h E VVVV hhh yy VV hhhh 212 213 111 132 R EE RRR 121 ll RRR EE yy lll y l RRR l EE y ll y RRR l E y l l E y RRR l y lll

112

Remark.

123

If

f

and

g

122

are two permutations, then:

f ≤ g ⇐⇒ ∀i, j ∈ [n], i > j

and

f (i) < f (j) =⇒ g(i) < g(j)

⇐⇒ Desc(f ) ⊆ Desc(g). So the restriction of

≤

to

Sn

is the right weak Bruhat order.

8

LOÏC FOISSY AND FRÉDÉRIC PATRAS

Denition 12. Let P = (P, ≤1 , ≤2 ) be a weak plane poset. We assume that (P, ) = ([n], ≤). A weak linear extension of P is a surjective map f : [n] −→ [k] such that: (1) For all i, j ∈ [n], i ≤1 j =⇒ f (i) ≤ f (j). (2) For all i, j ∈ [n], if i ≤1 j and f (i) = f (j) =⇒ i ≡ j . The set of weak linear extensions of P is denoted by W Lin(P ). In [12], an order is dened on (1) For all (2) For all

P W (n):

for all

f, g ∈ P W (n), f ≺ g

if

i, j ∈ [n], g(i) ≤ g(j) =⇒ f (i) ≤ f (j). i, j ∈ [n], i < j and g(i) > g(j) =⇒ f (i) > f (j).

It is proved that the following map is a Hopf algebra isomorphism:

ψ:

  (WQSym,

, ∆) −→ (WQSym, ., ∆) X f −→ g.



gf

Lemma 13. Let P be a weak plane poset. Then: W Lin(P ) =

G

{g ∈ P W (n), g  f }.

f ∈Lin(P )

Proof. ⊆.

g ∈ W Lin(P ).

p ∈ [max(g)],

Pp = P|g−1 (p) . Then Pp is a weak plane poset, so there exists a unique packed word fp such 0 0 that Pp is isomorphic to Pfp . Let us dene g by g (i) = fp (i) + max(f1 ) + 0 . . . + max(fp−1 ) for any i ∈ Pp ; g is a packed word. 0 Let us show rst that g ∈ Lin(p). Assume that i ≤1 j . Then g(i) ≤ g(j). 0 0 Let us show that we also have g (i) ≤ g (j): •

Let

For any

we put

i ≡ j , then, as g is a weak linear extension of P , g(i) < g(j), which implies g 0 (i) < g 0 (j). • If g(i) = g(j) = p, then i ≡ j in Pp , so fp (i) = fp (j) and nally g 0 (i) = g 0 (j). If we don't have

g 0 (i) = g 0 (j), then g(i) = g(j) = p and fp (i) = fp (j), so i ≡ j in Pp 0 and nally i ≡ j : g ∈ Lin(P ). 0 0 0 Let us show nally that g  g . If g (i) ≤ g (j), then necessarily g(i) ≤ g(j). Let us assume i < j and g 0 (i) > g 0 (j). Then g(i) ≥ g(j). If g(i) = g(j) = p, then fp (i) > fp (j), so j ≤i 1 and we do not have i ≡ j : this contradicts the fact that g is a weak linear extension. So g(i) > g(j), and 0 nally g  g .

Now, if

⊇. Let f ∈ Lin(P ) and g  f . If i ≤1 j , then f (i) ≤ f (j), so g(i) ≤ g(j). If moreover g(i) = g(j), as i ≥ j (because i ≤1 j ), we can not have f (i) < f (j) as g  f , so f (i) = f (j) and i ≡ j . So g ∈ W Lin(P ).

Disjoint union. g

Let

f, f 0 ∈ Lin(P ), such that there exists g ∈ P W (n), i < j . If f (i) > f (j), then g(i) > g(j). If f 0 (i) ≤

f, f 0 . Let us consider

SURJECTIONS AND DOUBLE POSETS

f 0 (j),

we would have

∀i, j ∈ [n]

g(i) ≤ g(j),

such that

9

contradiction. Hence, by symmetry:

i < j, f (i) > f (j) ⇐⇒ f 0 (i) > f 0 (j), f (i) ≤ f (j) ⇐⇒ f 0 (i) ≤ f 0 (j).

i < j and f (i) = f (j). Then i ≡ j , and f 0 (i) ≤ f 0 (j). As P is isomorphic to Ph for a certain packed word h, i < j and h(i) = h(j), 0 0 0 0 so j ≤1 i in P ; consequently, f (j) ≤ f (i) and nally f (i) = f (j). As a

Let us assume that

conclusion:

∀i, j ∈ [n]

such that

i < j, f (i) > f (j) ⇐⇒ f 0 (i) > f 0 (j), f (i) = f (j) ⇐⇒ f 0 (i) = f 0 (j), f (i) < f (j) ⇐⇒ f 0 (i) < f 0 (j).

So

Pf = Pf 0 ,

which implies

f = f 0.



Proposition 14. The following map is a Hopf algebra isomorphism:   HW P P φ0 = ψ ◦ φ : Pf 

Proof.

Indeed, for any packed word

X

ψ ◦ φ(Pf ) =

−→ (WQSym, ., ∆) X −→ f. f ∈W Lin(P )

f,

by the preceding lemma:

X

f ∈Lin(P ) gf By composition,

Examples.

φ0

X

g=

f.

f ∈W Lin(P )



is an isomorphism.

We order the packed words of degree

2

in the following way:

(11,12,21). (1) The matrices of

P W (2)

φ

and

φ0

from the basis

(Pf )f ∈P W (2)

to the basis

are respectively given by:

  1 0 0 0 1 0 , 1 1 1



 1 1 0 0 1 0  . 1 1 1

(2) The matrix of the pairing of

HW P P

in the basis

(Pf )f ∈P W (2)

is given

by:



 1 1 0 1 2 1  . 0 1 0 (3) Via

φ

and

φ0 , (WQSym,

, ∆)

and

(WQSym, ., ∆)

inherit nonde-

generate Hopf pairings. The matrices of these pairings in the basis

P W (2)

are respectively given by:



 1 0 0 0 0 1  , 0 1 0



 1 −1 0 −1 1 1 . 0 1 0

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LOÏC FOISSY AND FRÉDÉRIC PATRAS References

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LMPA Joseph Liouville, Université du Littoral Côte d'opale, Centre Universitaire de la Mi-Voix, 50, rue Ferdinand Buisson, CS 80699, 62228 Calais Cedex, France, email [email protected] Université Côte d'Azur, UMR 7351 CNRS, Parc Valrose, 06108 Nice Cedex 02 France, email [email protected]