A Solomon descent theory for the wreath products G≀Sn

arXiv:math/0503011v1 [math.CO] 1 Mar 2005

A Solomon des ent theory for the wreath produ ts

G ≀ S

n

Pierre Baumannand ChristopheHohlweg

∗

Abstra t

WeproposeananalogueofSolomon'sdes enttheoryforthe aseofawreathprodu t

G≀S

n

,where

G

isaniteabeliangroup. Our onstru tionmixesanumberofingredients: Manta i-Reutenaueralgebras,Spe ht'stheoryfortherepresentationsofwreathprodu ts, Okada'sextensiontowreathprodu tsoftheRobinson-S hensted orresponden e,Poirier's quasisymmetri fun tions. Weinsistonthefun torialaspe tofourdenitionsandexplain therelationofourresultswithpreviouswork on erningthehypero taedralgroup.

Introdu tion

The problem studied in this arti le has its roots in a dis overy by Solomon in 1976. Let

(W, (s

i

)

i∈I

)

be a Coxeter system. For any subset

J ⊆ I

, all

W

J

the paraboli subgroup generated by the elements

s

j

with

j ∈ J

. In ea h left oset

wW

J

of

W

modulo

W

J

, there is a unique element of minimal length, alled the distinguished representative of that oset. We denote the set of these distinguished representatives by

X

J

, and we form thesum

x

J

=

P

w∈X

J

w

inthegroup ring

Z

W

. Finallywe denoteby

Σ

W

the

Z

-submoduleof

Z

W

spanned byall elements

x

J

.

Now let

R(W )

be the hara ter ring of

W

, and let

ϕ

J

∈ R(W )

be the hara ter of

W

indu ed from the trivial hara ter of

W

J

. Given two subsets

J

and

K

of

I

, ea h double oset

C ∈ W

J

\W/W

K

ontains a unique element

x

of minimal length, and a result of Tits, Kilmoyer[18 ℄and/orSolomon[33 ℄assertsthattheinterse tion

x

−1

_W

J

x ∩ W

K

istheparaboli subgroup

W

L(C)

, where

L(C) = {k ∈ K | ∃j ∈ J, x

−1

_s

j

x = s

k

}

. Joint to Ma key's tensor produ t theorem,this yields the multipli ation rule intherepresentation ring

R(W )

ϕ

J

ϕ

K

=

X

L⊆I

a

JKL

ϕ

L

,

where

a

JKL

=



{C ∈ W

J

\W/W

K

| L = L(C)}



.

With these notations, Solomon's dis overy [33℄ is the equality

x

J

x

K

=

P

L⊆I

a

JKL

x

L

in the ring

Z

W

. It impliesthat

Σ

W

isa subring of

Z

W

and itshows theexisten e amorphism of rings

θ

W

: Σ

W

→ R(W )

su h that

θ

W

(x

J

) = ϕ

J

. This result means that (a part of) the hara ter theory of

W

an be lifted to a subring of its group ring. Additional details (for instan e, a more pre ise des ription of the image of

θ

W

) an be found in the paper [7 ℄ by F. Bergeron, N.Bergeron, Howlettand Taylor.

It isnatural to lookfor a similartheoryfor groups otherthan Coxeter systems. The rst examplesthat ometomindarenitegroupsofLietypeandnite omplexree tiongroups.

∗

PartiallysupportedbyCanadaResear hChairs.

MSC:Primary16S99,Se ondary05E0505E1016S3416W3020B3020E22. Keywords: wreathprodu ts,Solomondes entalgebra,quasisymmetri fun tions.

Among the latter, the groups of type

G(r, 1, n)

are wreath produ ts

(Z/rZ) ≀ S

n

of a y li group

Z

/rZ

bythe symmetri group

S

n

. Oneis thenledto investigate the ase of ageneral wreathprodu t

G ≀ S

n

. Tobuildthetheory,itisne essarytohavesomeknowledgeaboutthe representation theory of

G

itself,and we assumeinthis paperthat

G

isabelian. One ofour mainresultsexplainshowto onstru tasubring

MR

n

(ZG)

insidethegroupring

Z



G≀S

n

]

and asurje tiveringhomomorphism

θ

G

from

MR

n

(ZG)

onto therepresentation ring

R(G ≀ S

n

)

of the wreath produ t. Here the notation

MR

refersto thenames of Manta i and Reutenauer; indeedit turnsout thattheremarkable subring inside

Z



G ≀ S

n

]

dis overed in1995 by these two authors [24℄ isadequateto our purpose.

Ausually e ient methodto ta kle problemswiththesymmetri group

S

n

istotreat all

n

at the same time. For instan e, Malvenuto and Reutenauer observed in 1995 [23 ℄ thatthe dire tsum

F

=

L

n≥0

Z

[S

n

]

anbeendowed withthestru tureofagradedbialgebrainsu h a way that the submodule

Σ =

L

n≥0

Σ

S

_n

is a gradedsubbialgebra. A similar phenomenon appears here: the dire tsum

F

(ZG) =

L

n≥0

Z



G ≀ S

n



an be endowed withthe stru ture ofa graded bialgebra, of whi h

MR

(ZG) =

L

n≥0

MR

n

(ZG)

isa subbialgebra. (Aparti ular aseofthis onstru tion waspreviously onsideredbyAguiar andMahajan; thepaper [2℄ by Aguiar,N.BergeronandNymanpresentsana ountoftheirresult. Aguiarandhis oauthors view the hypero taedral group of order

2

n

_n!

as the wreath produ t

{±1} ≀ S

n

, that is, as the group of signed permutations. Then they onstru t the graded bialgebra

F Z



{±1}



andits subbialgebra

MR Z



{±1}



. Using the morphism ofgroup`forgetting thesigns'from

{±1} ≀ S

n

onto

S

n

,they ompare these gradedbialgebras withMalvenuto andReutenauer's bialgebra

F

and itssubbialgebra

Σ

. Our onstru tion andits fun toriality generalize Aguiar and his oauthors'resultsto the aseofallwreathprodu ts

G ≀ S

n

.) Thisbialgebrastru ture on

F

(ZG)

will be the starting point of our story; indeed we dene a `free quasisymmetri algebra'

F

(V )

for any

Z

-module

V

and investigate itsproperties.

We nowpresent theplan andthemain results ofthis paper.

InSe tion1,wedene thefreequasisymmetri algebraonamodule

V

overa ommutative ground ring

K

: this is a graded module

F

(V ) =

L

n≥0

F

n

(V )

, whi h we endow with an `external produ t' and a oprodu t to turn it into a graded bialgebra (Theorem 1). In the ase where

V

is endowed with the stru ture of a oalgebra,

F

(V )

ontains a remarkable subbialgebra

MR

(V )

, the so- alled Manta i-Reutenauer bialgebra, whi h is a freeasso iative algebraassoon as

V

isa freemodule(Propositions 3 and 4).

In Se tion 2, we show that the fun tor

V F (V )

is ompatible with the duality of

K

-modules, in the sense that any pairing between two

K

-modules

V

and

W

gives rise to a pairing ofbialgebras between

F

(V )

and

F

(W )

(Proposition 5). Inparti ular, thebialgebra

F

(V )

isself-dual assoonasthe module

V

is endowed withaperfe t pairing.

InSe tion 3,weinvestigate the asewherethemodule

V

isa

K

-algebra. Then

F

(V )

an beendowedwithan`internalprodu t',whi hturnsea hofthegraded omponents

F

n

(V )

into analgebra. Theinterestingpointhereistheexisten eofasplittingformulathatdes ribesthe ompatibility between this internal produ t, theexternal produ tand the oprodu t (Theo-rem10). Thisformulais a generalization ofthesplitting formulaof Gelfand, Krob, Las oux, Le ler , Retakh and Thibon [13 ℄; it entails that the Manta i-Reutenauer bialgebra

MR

(V )

isa subalgebra of

F

(V )

for the internal produ twhenever

V

is endowed withthe stru ture of a o ommutative bialgebra (Corollaries 11 and 12). In Se tion 3.5, we onsider for

V

the ase of the groupalgebra

K

Γ

of a nite group

Γ

and justify that thegraded omponent

F

_n

_(KΓ)

is anoni ally isomorphi to thegroup algebra

K



om-ponent

MR

n

(KΓ) = MR(KΓ) ∩ F

n

(KΓ)

oin ideswiththesubalgebradenedbyManta iand Reutenauerin [24℄.

InSe tion4,weatlastprovidethelinkbetweenthese onstru tionsandaSolomondes ent theoryfor wreathprodu ts. We rst re all Spe ht's lassi ation of the irredu ible omplex hara tersofawreathprodu t

G ≀ S

n

and Zelevinsky'sstru tureofagradedbialgebraonthe dire tsum

Rep(G) =

L

n≥0

R(G ≀ S

n

)

fortheindu tionprodu tandtherestri tion oprodu t (Se tion4.2). We thenfo usonthe ase where

G

is abelian. We denotethedualgroupof

G

by

Γ

,we observe thatthegroup ring

Z

Γ

is a o ommutative bialgebra, sothat the Manta i-Reutenauerbialgebra

MR

(ZΓ)

isdenedandisasubalgebraof

F

(ZΓ)

fortheinternalprodu t, andwedeneamap

θ

G

: MR(ZΓ) → Rep(G)

. Thenweshowthat

θ

G

isasurje tivemorphism of graded bialgebras, and that in ea h degree,

θ

G

: MR

n

(ZΓ) → R(G ≀ S

n

)

is a surje tive morphism of rings whose kernel is the Ja obson radi al of

MR

n

(ZΓ)

(Theorem 16). We also showthat

θ

G

enjoys aremarkablesymmetrypropertyanalogousto thesymmetrypropertyof Solomon'shomomorphisms

θ

W

proved byJöllenbe kand Reutenauer [17℄and byBlessenohl, Hohlweg and S ho ker [8℄ (Theorem 19). Finally we ompare our results with the work of Bonnafé and Hohlweg, who treated in [10 ℄ the ase of the hypero taedral group

{±1} ≀ S

n

usingmethods fromthetheoryof Coxetergroups (Se tion4.5).

Thequestionsabout thebialgebras

F

(V )

investigated inSe tions1 to 3arefun torialin the

K

-module

V

. Asusual,themostinteresting pointinthisassertion isthe ompatibilityof the onstru tionswiththehomomorphisms,namelyherethe

K

-linearmaps. Onthe ontrary the questions studied in Se tion 5 require that

V

be a free

K

-module and depend on the hoi eofa basis

B

of

V

. Su habasis

B

anbe viewedasthedataofastru ture ofapointed oalgebraon

V

,whi hyieldsinturnaManta i-Reutenauersubbialgebra

MR

(V )

inside

F

(V )

. The hoi e of

B

also givesrise to ase ondsubbialgebra

Q

(B)

,biggerthan

MR

(V )

,whi hwe all the opla ti bialgebra. Thedenitionof

Q

(B)

involvesa ombinatorial onstru tion due toOkada[28℄,whi hextendsthewell-knownRobinson-S hensted orresponden eto` oloured' situations; at this point, we take the opportunity to provide an analogue of Knuth relations for Okada's orresponden e (Proposition 24). In the ase where

B

is a singleton set, the bialgebra

Q

(B)

is one of the`algèbres de Hopf de tableaux' of Poirier and Reutenauer [30 ℄. Extending the work of these authors, we dene a surje tive homomorphism

Θ

B

of graded bialgebrasfrom

Q

(B)

ontoabialgebra

Λ(B)

of` oloured' symmetri fun tions(Theorem 31). Wethengo ba ktothesituation investigated inSe tion4 andtake thegroup algebra

Z

Γ

for

V

andthegroup

Γ

for

B

;here

Θ

Γ

an beviewed asa liftof

θ

G

: MR(ZΓ) → Rep(G)

to

Q

(Γ)

thatyieldsani e des riptionofthesimplerepresentationsofall wreathprodu ts

G ≀ S

n

. We re over Jöllenbe k's onstru tion of the Spe ht modules [16 ℄ as the parti ular ase where

G

isthegroupwithone element; werefer thereaderto Blessenohland S ho ker's survey[9 ℄ for additional details aboutJöllenbe k's onstru tion.

Finally we present in Se tion 6 a realization of thebialgebra

F

(V )

in termsof free qua-sisymmetri fun tions. Asin Se tion 5, the

K

-module

V

is assumed to be free; we hoose a basis

B

of

V

andendow

B

withalinearorder. When

V

hasrankone,ourfreequasisymmetri fun tions oin idewith theusual ones[14 ℄. Inhigher rank however, our freequasisymmetri fun tions are dierent from those dened by Novelli and Thibon in [27℄. This disagreement has its rootsin thefa tthat Novelli andThibon's onstru tion andours were designed with dierent aims: roughly speaking, Novelli and Thibon's goal was to nd a non ommutative version of Poirier's quasisymmetri fun tions [29℄; on the other side, we view thedual alge-bra

MR

(V )

∨

asa quotient of

F

(V )

and des ribe itinterms of ommutative quasisymmetri fun tions.

Atthis point, we shouldmentionthat theassignment

(V, B) MR(V )

∨

enjoys a ertain fun toriality property; this property and theisomorphism between

MR

(K)

∨

and the graded bialgebra

QSym

of usual quasisymmetri fun tions yield in turn homomorphisms of graded bialgebrasfrom

F

(V )

and

MR

(V )

∨

to

QSym

,whi hamountstosaythat

F

(V )

and

MR

(V )

∨

are` ombinatorial Hopf algebras'inthesense of Aguiar,N.Bergeronand Sottile [3℄.

TheauthorswishtothankJean-ChristopheNovelliandJean-YvesThibonforfruitfuland instru tive onversations,whi htookpla eon Mar h30,2004inOttrott andonMay3,2004 at theInstitutGaspardMonge(UniversityofMarne-la-Vallée). Theirpreprint[27℄ inuen ed our writing of Se tions1 and 3. The mainpart ofthis workwas arriedout when C. H.was at theInstitut deRe her he Mathématique Avan éeinStrasbourg.

Wexa ommutativegroundring

K

. Conne ted

N

-graded

K

-bialgebrasappeareverywhere in the paper. Su h bialgebras areindeed automati ally Hopf algebras, at least when

K

is a eld. However we will neithermake use of this property nor attempt to work out expli itly anyantipode.

1 Free quasisymmetri bialgebras

In this se tion, we present our main obje ts of study, namely the free quasisymmetri bial-gebras and the generalized des ent algebras, among whi htheNovelli-Thibon bialgebras and the Manta i-Reutenauer bialgebras. Before that, we introdu e some notations pertaining permutations.

1.1 Notations related to permutations

For ea h positive integer

n

, we denote the symmetri group of all permutations of the set

{1, 2, . . . , n}

by

S

n

. By onvention,

S

0

is thegroup with one element. The unit element of

S

_n

is denoted by

e

n

. The group algebra over

K

of

S

n

is denoted by

KS

n

. In pra ti e, a permutation

σ ∈ S

n

iswritten astheword

σ(1)σ(2) · · · σ(n)

withlettersin

Z

>0

= {1, 2, . . .}

. Let

A

betotallyorderedset(analphabet). Thestandardizationofaword

w = a

1

a

2

· · · a

n

oflength

n

with letters in

A

is the permutation

σ ∈ S

n

with smallest number of inversions su hthatthe sequen e

a

_σ

−1

₍₁₎

, a

_σ

−1

₍₂₎

, . . . , a

_σ

−1

_(n)

is non-de reasing. In other words, the word

σ(1)σ(2) · · · σ(n)

that represents

σ

is obtained by putting the numbers

1

,

2

, ...,

n

in the pla e of the letters

a

i

of

w

; in this pro ess of substitution, the diverse o urren es of the smallest letter of

A

get repla ed rst by the numbers

1

,

2

, et . from left to right; thenwe repla e theo urren es of these ond-smallest elementof

A

bythefollowing numbers;andsoon,uptotheexhaustionofalllettersof

w

. An example laries this explanation: given the alphabet

A

= {a, b, c, . . .}

with theusualorder, the standardizationof theword

w = bcbaba

is

σ = 364152

.

A ompositionof a positive integer

n

isa sequen e

c

= (c

1

, c

2

, . . . , c

k

)

of positiveintegers whi h sum up to

n

. The usual notation for thatis to write

c

|= n

. Given two ompositions

c

= (c

1

, c

2

, . . . , c

k

)

and

d

= (d

1

, d

2

, . . . , d

l

)

ofthesameinteger

n

,wesaythat

c

isarenement of

d

and wewrite

c < d

ifthere holds

Therelation

4

is a partial orderon the setof ompositions of

n

. For instan e, the following hain ofinequalities holdamong ompositions of

5

:

(5) ≺ (4, 1) ≺ (1, 3, 1) ≺ (1, 2, 1, 1) ≺ (1, 1, 1, 1, 1).

Let

c

= (c

1

, c

2

, . . . , c

k

)

bea ompositionof

n

andset

t

i

= c

1

+ c

2

+ · · · + c

i

forea h

i

. Given a

k

-uple

(σ

1

, σ

2

, . . . , σ

k

) ∈ S

c

1

× S

c

2

× · · · × S

c

k

ofpermutations,wedene

σ

1

× σ

2

× · · · × σ

k

∈

S

_n

as the permutation that maps an element

a

belonging to the interval

[t

i−1

+ 1, t

i

]

onto

t

i−1

+ σ

i

(a − t

i−1

)

. Thisassignment denes an embedding

S

c

1

× S

c

2

× · · · × S

c

k

֒→ S

n

;we denote its image by

S

c

. Su ha

S

c

is alleda Young subgroup of

S

n

. We obtainfor free an embedding for the groupalgebras

KS

_c

1

⊗ KS

c

2

⊗ · · · ⊗ KS

c

k

≃

−→ KS

c

⊆ KS

n

.

Themap

c

7→ S

c

is an order reversingbije tion from theset of ompositions of

n

, endowed withtherenementorder, onto thesetofYoungsubgroups of

S

n

,endowedwiththein lusion order.

Let again

c

= (c

1

, c

2

, . . . , c

k

)

be a omposition of

n

and set

t

i

= c

1

+ c

2

+ · · · + c

i

. The subset

X

c

=



σ ∈ S

n





∀i, σ

isin reasing on theinterval

[t

i−1

+ 1, t

i

]

isasystemof representatives oftheleft osetsof

S

c

in

S

n

. Herearesome examples:

X

(2,2)

= {1234, 1324, 1423, 2314, 2413, 3412},

X

(n)

= {id}

and

X

(

1, 1, . . . , 1

|

{z

}

n

times

)

= S

n

.

We dene anelement of thegroupring

KS

n

bysetting

x

c

=

P

σ∈X

c

σ

.

Let

d

= (d

1

, d

2

, . . . , d

l

)

be a omposition ofan integer

n

. Then a omposition

c

of

n

is a renementof

d

ifandonlyif

c

anbeobtainedasthe on atenation

f

1

f

2

· · · f

l

ofa omposition

f

1

of

d

1

,a omposition

f

2

of

d

2

,...,and a omposition

f

l

of

d

l

. Ifthis holds,thenthemap

(ρ, σ

1

, σ

2

, . . . , σ

l

) 7→ ρ ◦ (σ

1

× σ

2

× · · · × σ

l

)

isabije tionfrom

X

d

× X

f

1

× X

f

2

× · · · × X

f

l

onto

X

c

,for

X

f

1

× · · · × X

f

l

is asetofminimal oset representativesof

S

c

in

S

d

. Therefore theequality

x

c

= x

d

(x

f

1

⊗ x

f

2

⊗ · · · ⊗ x

f

l

)

(1)

holdsinthegroupring

KS

n

. Asa parti ular aseof (1), wesee that

x

_(n,n

′

_,n

′′

₎

= x

_(n,n

′

_+n

′′

₎

x

_(n)

⊗ x

_(n

′

_,n

′′

₎

= x

_(n+n

′

_,n

′′

₎

x

_(n,n

′

₎

⊗ x

_(n

′′

₎

(2)

holdstruefor anythree positive integers

n

,

n

′

and

n

′′

.

Let

σ ∈ S

n

. One may partition the word

σ(1)σ(2) · · · σ(n)

that represents

σ

into its longestin reasing subwords; the omposition of

n

formed by thesu essive lengthsof these subwords is alled the des ent omposition of

σ

and is denoted by

D(σ)

. For instan e, the des ent omposition of

σ = 51243

is

D(σ) = (1, 3, 1)

. Then for any omposition

c

of

n

,the assertions

σ ∈ X

c

and

D(σ) 4 c

areequivalent.

1.2 Denition of the free quasisymmetri bialgebra

F

(V )

Let

V

be a

K

-module. The group

S

n

a ts on the

n

-th tensor power

V

⊗n

;the submodule of invariants, that is, thespa e of symmetri tensors, is denotedby

TS

n

_{(V )}

. We may formthe tensor produ tof

V

⊗n

by

kS

n

. To distinguish this tensor produ t from those usedto build thetensorpower

V

⊗n

,wedenoteitwithasharpsymbol. Wedenotetheresult

(V

⊗n

_)#(KS

n

)

by

F

n

(V )

. Thea tions dened by

π ·



(v

1

⊗ v

2

⊗ · · · ⊗ v

n

)#σ



=



(v

π

−1

₍₁₎

⊗ v

_π

−1

₍₂₎

⊗ · · · ⊗ v

_π

−1

_(n)

)#(πσ)



and



(v

1

⊗ v

2

⊗ · · · ⊗ v

n

)#σ



· π =



(v

1

⊗ v

2

⊗ · · · ⊗ v

n

)#(σπ)



endow

F

n

(V )

withthestru tureofa

KS

n

-bimodule,where

(v

1

, v

2

, . . . , v

n

) ∈ V

n

and

π ∈ S

n

. For instan e,

F

n

(K)

isthe(left andright)regular

KS

n

-module.

Ouraim nowisto endowthespa e

F

(V ) =

L

n≥0

F

n

(V )

withthestru ture of agraded bialgebra. Wedene the produ tof two elements

α ∈ F

n

(V )

and

α

′

_{∈ F}

n

′

(V )

of theform

α =



(v

1

⊗ v

2

⊗ · · · ⊗ v

n

)#σ



and

α

′

₌



_(v

′

1

⊗ v

2

′

⊗ · · · ⊗ v

n

′

′

)#σ

′



bytheformula

α ∗ α

′

= x

_(n,n

′

₎

·



(v

₁

⊗ v

₂

⊗ · · · ⊗ v

_n

⊗ v

₁

′

⊗ v

₂

′

⊗ · · · ⊗ v

′

_n

′

)#(σ × σ

′

)



.

(This formula an be made more on rete by noting that

x

(n,n

′

)

(σ × σ

′

₎

is the sum in the groupalgebra

KS

n+n

′

of all permutations

π

su h that

σ

is the standardization of the word

π(1)π(2) · · · π(n)

and

σ

′

isthestandardizationoftheword

π(n + 1)π(n + 2) · · · π(n + n

′

₎

.) We extendthisdenitionbymultilinearity toanoperation denedon thewholespa e

F

(V )

and allthis latterthe external produ t.

We dene the oprodu tof an element

α =



(v

1

⊗ v

2

⊗ · · · ⊗ v

n

)#σ



of

F

n

(V )

as

∆ (v

1

⊗v

2

⊗· · · ⊗v

n

)#σ

=

n

X

n

′

₌₀



(v

1

⊗v

2

⊗· · · ⊗v

n

′

)#π

_n

′



⊗



(v

_n

′

₊₁

⊗v

_n

′

₊₂

⊗· · · ⊗v

_n

)#π

′

_n−n

′



,

where

π

n

′

∈ S

n

′

isthe inverse of thestandardization of theword

σ

−1

_{(1) σ}

−1

_{(2) · · · σ}

−1

_(n

′

₎

and

π

′

n−n

′

∈ S

n−n

′

is the inverse of the standardization of the word

σ

−1

_(n

′

_{+ 1) σ}

−1

_(n

′

₊

2) · · · σ

−1

_(n)

. In other words,

π

n

′

and

π

′

n−n

′

are su h that the two sequen es of letters

(π

n

′

(1), π

_n

′

(2), . . . , π

_n

′

(n

′

))

and

(n

′

_{+ π}

′

n−n

′

(1), n

′

+ π

_n−n

′

′

(2), . . . , n

′

+ π

′

_n−n

′

(n − n

′

))

appear inthis order inthe word

σ(1)σ(2) · · · σ(n)

. We all themap

∆ : F (V ) → F (V ) ⊗ F (V )

the oprodu tof

F

(V )

.

We dene the unit of

F

(V )

as the inje tion of the graded omponent

F

0

(V ) = K

into

F

_{(V )}

;we dene the ounit of

F

(V )

astheproje tion of

F

(V )

onto

F

0

(V ) = K

.

We now give an example to illustrate these denitions. Given sixelements

v

1

,

v

2

,

v

3

,

v

4

,

v

₁

′

,

v

′

2

in

V

,the produ tof

α =



(v

1

⊗ v

2

)#e

2



and

α

′

₌



_(v

′

2

⊗ v

1

′

)#21



= (21) ·



(v

₁

′

⊗ v

₂

′

)#e

2



is

α ∗ α

′

= (1243 + 1342 + 1432 + 2341 + 2431 + 3421) ·



(v

1

⊗ v

2

⊗ v

′

1

⊗ v

′

2

)#e

4



=



(v

1

⊗ v

2

⊗ v

′

2

⊗ v

′

1

)#1243



+



(v

1

⊗ v

′

2

⊗ v

2

⊗ v

′

1

)#1342



+



(v

1

⊗ v

′

2

⊗ v

′

1

⊗ v

2

)#1432



+



(v

2

′

⊗ v

1

⊗ v

2

⊗ v

′

1

)#2341



+



(v

₂

′

⊗ v

1

⊗ v

′

1

⊗ v

2

)#2431



+



(v

2

′

⊗ v

′

1

⊗ v

1

⊗ v

2

)#3421



,

andthe oprodu t of

α =



(v

3

⊗ v

1

⊗ v

2

⊗ v

4

)#2314



= (2314) ·



(v

1

⊗ v

2

⊗ v

3

⊗ v

4

)#e

4



is

∆(α) =



()#e

0



⊗ α +



(v

3

)#1



⊗



(v

1

⊗ v

2

⊗ v

4

)#123



+



(v

3

⊗ v

1

)#21



⊗



(v

2

⊗ v

4

)#12



+



(v

3

⊗ v

1

⊗ v

2

)#231



⊗



(v

4

)#1



+ α ⊗



()#e

0



=



()#e

0



⊗ α +



(v

3

)#e

1



⊗



(v

1

⊗ v

2

⊗ v

4

)#e

3



+

+ (21) ·



(v

1

⊗ v

3

)#e

2



⊗



(v

2

⊗ v

4

)#e

2



+ (231) ·



(v

1

⊗ v

2

⊗ v

3

)#e

3



⊗



(v

4

)#e

1



+ α ⊗



()#e

0



.

Theorem 1 The unit,the ounit,andtheoperations

∗

and

∆

endow

F

(G)

withthestru ture of a graded bialgebra.

Proof. Itis learthatthefouroperationsrespe tthegraduation. Theasso iativityof

∗

follows immediately fromEquation (2). A moment's thought su es to he kthe oasso iativity of

∆

and theaxiomsfor theunitandthe ounit. Itremainsto showthepentagonaxiom, whi h asksthat

∆

be multipli ative withrespe tto theprodu t

∗

.

FollowingMalvenutoandReutenauer's method[23 ℄,we rstre alla lassi al onstru tion inthe theoryof Hopf algebras. Let

A

be a set, let

hA i

denote the set of words on

A

, and let

K

hA i

be the free

K

-modulewith basis

hA i

. The shue produ tof two words

w

and

w

′

oflength

n

and

n

′

respe tively isthe sum

w

x

w

′

₌

X

ρ∈X

_{(n,n′ )}

b

_ρ

−1

₍₁₎

b

_ρ

−1

₍₂₎

· · · b

_ρ

−1

_(n+n

′

₎

,

wheretheword

b

1

b

2

· · · b

n+n

′

isthe on atenationofthe words

w

and

w

′

. Thisoperation xis thenextended bilinearly to a produ t on

K

hA i

. The de on atenation is the oprodu t

δ

on

K

_{hA i}

su h that

δ(w) =

n

X

n

′

₌₀

a

1

a

2

· · · a

n

′

⊗ a

_n

′

₊₁

a

_n

′

₊₂

· · · a

_n

foranyword

w = a

1

a

2

· · · a

n

. Itis knownthattheoperationsxand

δ

endow

K

hA i

withthe stru tureof abialgebra (see Proposition 1.9in[31 ℄ for aproof).

We are now ready to show the pentagon axiom in the ase where the

K

-module

V

is free. We take a basis

B

of

V

and we set

A

= Z

>0

× B

. We observe that the elements

(b

1

⊗ b

2

⊗ · · · ⊗ b

n

)#σ

form a basisof

F

n

(V )

, where

(b

1

, b

2

, . . . , b

n

) ∈ B

n

and

σ ∈ S

n

. We maythus dene linear maps

j

k

: F (G) → KhA i

(depending on the hoi e of a non-negative integer

k

) by mapping anelement

α =



(b

1

⊗ b

2

⊗ · · · ⊗ b

n

)#σ



to

j

k

(α) = a

1

a

2

· · · a

n

,where

a

i

= (k + σ

−1

(i), b

i

)

. In the other dire tion, we dene a linear map

s : KhA i → F (V )

as follows: given a word

w = a

1

a

2

· · · a

n

with letters in

A

, we write

a

i

= (p

i

, b

i

)

and set

s(w) = (b

1

⊗ b

2

⊗ · · · ⊗ b

n

)#σ

, where

σ

is the inverse of the standardization of the word

p

1

p

2

· · · p

n

.

One easily he ks that

s ◦ j

k

= id

F

(G)

and that

(s ⊗ s) ◦ δ = ∆ ◦ s

. Moreover, let

w = a

1

a

2

· · · a

n

and

w

′

_{= a}

′

1

a

′

2

· · · a

′

n

′

be twowords withletters in

A

. Ifwewrite

a

i

= (p

i

, b

i

)

and

a

′

i

= (p

′

i

, b

′

i

)

,then

s(w

x

w

′

_{) = s(w) ∗ s(w}

′

₎

assoonas every integer

p

i

isstri tly smaller than everyinteger

p

′

i

.

We nowtake

α ∈ F

n

(G)

and

α

′

_{∈ F}

n

′

(G)

. We ompute:

∆(α ∗ α

′

) = ∆

h

s j

0

(α)

∗ s j

n

(α

′

)

i

= ∆ ◦ s

j

0

(α)

x

j

n

(α

′

₎

= (s ⊗ s)

h

δ j

0

(α)

x

j

n

(α

′

₎

i

= (s ⊗ s)

h

δ j

0

(α)

x

δ j

n

(α

′

₎

i

=



(s ⊗ s) ◦ δ ◦ j

0

(α)



∗



(s ⊗ s) ◦ δ ◦ j

n

(α

′

)



=



∆ ◦ s ◦ j

0

(α)



∗



∆ ◦ s ◦ j

n

(α

′

)



= ∆(α) ∗ ∆(α

′

).

Thisrelationprovesthepentagonaxiomfor

F

(V )

inthe asewhere

V

isafree

K

-module. Inthegeneral ase,wemaynd afree

K

-module

V

˜

and asurje tive morphism of

K

-modules

f : ˜

V → V

. Then

f

indu es asurje tivemapfrom

F

( ˜

V )

onto

F

(V )

whi h isa morphism of algebras and of oalgebras. Sin e the operations

∗

and

∆

on

V

˜

satisfy thepentagonaxiom, theiranalogueson

V

satisfyalsothepentagonaxiom. This ompletestheproofofthetheorem.



We note that the assignment

V F (V )

is a ovariant fun tor from the ategory of

K

-modulesto the ategory of

N

-graded bialgebras over

K

.

Thealgebras

F

(V )

werealsoindire tlydenedbyNovelliandThibon;in[27℄,theydenote our

F

(K

l

₎

by

FQSym

(l)

andstatethatitisafreeasso iativealgebra,when e thename`free quasisymmetri bialgebras.'

Remark 2. Given a

K

-module

V

, one an endow the dire t sum

L

n≥0

V

⊗n

with two stru -tures of a graded bialgebra: the tensor algebra, denoted by

T

(V )

,and the otensor algebra, sometimesdenotedby

T

c

_{(V )}

. (The bialgebra

K

hA i

usedintheproofofTheorem 1isindeed the otensor algebra on the free

K

-module

KA

with basis

A

.) One he ks easily that the maps

ι : T(V ) → F (V ), v

1

⊗ v

2

⊗ · · · ⊗ v

n

7→

X

σ∈S

n

σ · (v

1

⊗ v

2

⊗ · · · ⊗ v

n

#e

n

)

and

p : F (V ) → T

c

(V ), (v

1

⊗ v

2

⊗ · · · ⊗ v

n

#σ) 7→ v

1

⊗ v

2

⊗ · · · ⊗ v

n

are morphisms of graded bialgebras. Moreover the omposition

p ◦ ι

is the symmetrization map

T

_{(V ) → T}

c

_{(V ), v}

₁

_{⊗ v}

₂

_{⊗ · · · ⊗ v}

_n

_7→

X

σ∈S

n

v

_σ(1)

⊗ v

_σ(2)

⊗ · · · ⊗ v

_σ(n)

.

For detailsand appli ationsof this onstru tion,we refer thereader to[26 ℄ and [32℄. 1.3 The des ent subbialgebras

Σ(W )

In this se tion, we investigate a lass of graded subalgebras of

F

(V )

, alled the des ent algebras. We nd a riterion for a des ent algebra to be a subbialgebra of

F

(V )

and give a oupleof examples.

Wexherea

K

-module

V

. Toanygradedsubmodule

W =

L

n≥0

W

n

ofthetensoralgebra

T

_{(V ) =}

L

n≥0

V

⊗n

,weasso iate thesubalgebra

Σ(W )

of

F

(V )

generated byallelements of the form

(t#e

n

)

with

t ∈ W

n

. We all su h asubalgebra

Σ(W )

ades ent algebra. A des ent algebrais ne essarilygraded, for itisgenerated byhomogeneous elements.

Proposition 3 Assume that

V

is at and that ea h module

W

n

is free of nite rank. For ea h

n ≥ 1

, pi ka basis

B

n

of

W

n

. Then

Σ(W )

isthefree asso iativealgebra on theelements

(b#e

n

)

, where

n ≥ 1

and

b ∈ B

n

.

Proof. By the way of ontradi tion, we assume that there existsa nite family

(u

i

)

i∈I

on-sisting of distin t nite sequen es

u

i

=

c

(i)

1

, b

(i)

1

, c

(i)

₂

, b

(i)

₂

, . . . , c

(i)

_k

i

, b

(i)

k

i

of elements in

S

n≥1

{n} × B

n

and anite family

(λ

i

)

i∈I

ofelements of

K

\ {0}

su h that

X

i∈I

λ

i

h

b

(i)

₁

#e

c

(i)

₁



∗



b

(i)

₂

#e

c

(i)

₂



∗ · · · ∗



b

(i)

_k

i

#e

c

(i)

_ki

i

= 0.

(3)

Using thegraduation, we may supposewithout loss ofgenerality thatall the sequen es

c

i

=

(c

(i)

₁

, c

(i)

₂

, . . . , c

(i)

_k

_i

)

are ompositions ofthe same integer

n

. Then(3)yields

X

i∈I

λ

i

x

c

i

·



(b

(i)

₁

⊗ b

(i)

₂

⊗ · · · ⊗ b

(i)

_k

i

)#e

n



= 0.

(4)

We hoose a maximalelement

c

= (c

1

, c

2

, . . . , c

k

)

among theset

{c

i

| i ∈ I}

withrespe t to the renement order, we set

J = {i ∈ I | c

i

= c}

, and we hoose a permutation

σ ∈ S

n

whose des ent omposition is

c

. Then for any

i ∈ I

,

σ ∈ X

c

i

⇐⇒ c 4 c

i

⇐⇒ i ∈ J.

Taking the image of(4) bythe linearmap

p : F

n

(V ) → V

⊗n

dened by

p (v

1

⊗ v

2

⊗ · · · ⊗ v

n

)#ρ

=

(

v

_ρ(1)

⊗ v

_ρ(2)

⊗ · · · ⊗ v

_ρ(n)

if

ρ = σ

,

0

otherwise, weobtain

X

i∈J

λ

i

b

(i)

1

⊗ b

(i)

2

⊗ · · · ⊗ b

(i)

k

= 0.

(5)

By assumption however, the sequen es

(b

(i)

1

, b

(i)

2

, . . . , b

(i)

k

)

are distin t when

i

runs over

J

. Thereforetheelements

b

(i)

1

⊗ b

(i)

2

⊗ · · · ⊗ b

(i)

k

arelinearlyindependent in

W

c

1

⊗ W

c

2

⊗ · · · ⊗ W

c

k

, for

B

c

1

⊗ B

c

2

⊗ · · · ⊗ B

c

k

isabasisofthis module. Sin e

V

andthe

W

c

i

areat modules,the imagesof the elements

b

(i)

1

⊗ b

(i)

2

⊗ · · · ⊗ b

(i)

k

in

V

⊗n

arelinearlyindependent. Wethen rea h a ontradi tion withEquation(5).



Before we look for a ondition on

W

that would ensures that

Σ(W )

is a subbialgebra of

F

_{(V )}

, we introdu e a pie e of notation that will be needed later, espe ially in Se tion 3.3. Let

c

= (c

1

, c

2

, . . . , c

k

)

be a omposition (possibly with parts equal to zero)

1

of

n

. Sin e 1

Itis onvenientinthis ontexttoallow ompositions tohavepartsequaltozero. We oulduseaspe ial terminology,followingforexampleReutenauerwho oinedin[31 ℄thewordpseudo ompositionforthatpurpose. Tolimittheadventofnewwords,wewillhoweversimplysay` omposition(possiblywithpartsequaltozero).'

V

⊗n

_{= V}

⊗c

1

_{⊗ V}

⊗c

2

_{⊗ · · · ⊗ V}

⊗c

k

,ea htensor

t ∈ V

⊗n

an bewritten asalinear ombination ofprodu ts

t

1

⊗ t

2

⊗ · · · ⊗ t

k

,where

t

i

∈ V

⊗c

i

for ea h

i

. We denotesu ha de omposition by

t =

P

_(t)

t

(c)

₁

⊗ t

(c)

₂

⊗ · · · ⊗ t

(c)

_k

. Inthisequation, the symbol

t

(c)

i

ismeant asapla e-holder for the a tualelements

t

i

. Withthisnotation, the oprodu t of an element oftheform

t#e

n

is

∆(t#e

n

) =

n

X

n

′

₌₀

h

t

((n

₁

′

,n−n

′

))

#e

n

′

i

⊗

h

t

((n

₂

′

,n−n

′

))

#e

n−n

′

i

.

(6)

Let us now return to our study of the des ent algebras. We introdu e the following ondition ona gradedsubmodule

W =

L

n≥0

W

n

of

T

(V )

: (A) Thereholds

W

n

⊆ W

c

1

⊗ W

c

2

⊗ · · · ⊗ W

c

k

for any omposition(possiblywithparts equal tozero)

c

= (c

1

, c

2

, . . . , c

k

)

of apositive integer

n

.

2

Inotherwords,forany omposition

c

= (c

1

, c

2

, . . . , c

k

)

ofapositiveinteger

n

andany

t ∈ W

n

, we may assume that in thewriting

t =

P

(t)

t

(c)

1

⊗ t

(c)

2

⊗ · · · ⊗ t

(c)

k

, all the elements of

V

⊗c

i

representedbythepla e-holder

t

(c)

i

anbepi kedin

W

c

i

. We annowndasu ient ondition for

Σ(W )

to bea subbialgebra of

F

(V )

.

Proposition 4 If

W

satises Condition (A), then

Σ(W )

isa graded subbialgebra of

F

(V )

. Proof. Wehavealreadyseenthat

Σ(W )

is agradedsubalgebraof

F

(V )

. It remainsto prove the in lusion

Σ(W ) ⊆ {x ∈ F (V ) | ∆(x) ∈ Σ(W ) ⊗ Σ(W )}.

The set

E

on the right of the symbol

⊆

above is a subalgebra of

F

(V )

, be ause

∆

is a morphism ofalgebrasand

Σ(W ) ⊗ Σ(W )

isasubalgebra. Moreover, Equation(6)showsthat if

W

satisesCondition (B),then

E

ontains all theelements

t#e

n

with

t ∈ W

n

. Sin e these elementsgenerate

Σ(W )

asanalgebra, itfollows that

E

ontains

Σ(W )

.



Besides the trivial hoi e

W = T(V )

,there are two main examples. The rst one o urs with

W = TS(V )

, the spa e of all symmetri tensors on

V

.

3

We all the orresponding subbialgebra

Σ(W )

the Novelli-Thibonbialgebraandwedenoteitby

NT

(V )

. Onemaynoti e that theassignment

V NT(V )

is fun torial.

The se ond interesting example on erns the ase where

V

is the underlying spa e of a oalgebra. We rst x two rather standard notations that are onvenient for dealing with oalgebras; we will usethem not only inthepresentation below, but also later inSe tion 3.3 with the omultipli ative stru ture of

F

(V )

. Let

C

be a oalgebra with its oasso iative oprodu t

δ

and its ounit

ε

. We dene the iterated oprodu ts

δ

n

: C → C

⊗n

by setting

δ

0

= ε

,

δ

1

= id

C

,

δ

2

= δ

,and

δ

n

= δ ⊗ (id

C

)

⊗n−2

◦ δ ⊗ (id

C

)

⊗n−3

◦ · · · ◦ δ

for all

n ≥ 3

. The Sweedler notation proposes to write theimage of an element

v ∈ C

by

δ

n

as

δ

n

(v) =

X

(v)

v

(1)

⊗ v

(2)

⊗ · · · ⊗ v

(n)

;

2 We abusively onfuse

W

c

1

⊗ W

c

2

⊗ · · · ⊗ W

c

k

withitsimage in

V

⊗c

1

⊗ V

⊗c

2

⊗ · · · ⊗ V

⊗c

k

_{= V}

⊗n

. Of oursenoambiguityariseswhen

K

isaeldor

V

istorsion-freemoduleoverap.i.d.

3

Condition(A)holdsfor

W

= TS(V )

assoonas

V

isproje tiveor

K

isaeldoraDedekindring. Wedo notknowiftheserestri tions anbelifted.

inthis writing, thesymbol

v

(i)

is apla e-holderfor ana tual element of

C

whi hvariesfrom oneterm to the other.

Now we assume that the module

V

on whi h the free quasisymmetri algebra

F

(V )

is onstru tedisendowedwithastru tureofa oalgebra, witha oprodu t

δ

and a ounit

ε

. In this ase,wemay onsidertheimage

W

n

oftheiterated oprodu t

δ

n

: V → V

⊗n

andwemay set

W =

L

n≥0

W

n

. Forany omposition(possiblywithpartsequaltozero)

c

= (c

1

, c

2

, . . . , c

k

)

of

n

and any element

v ∈ V

,the oasso iativityof

δ

implies

δ

n

(v) =

X

(v)

δ

c

1

(v

(1)

)

|

{z

}

(δ

n

(v))

(c)

₁

⊗ δ

c

2

(v

(2)

)

|

{z

}

(δ

n

(v))

(c)

₂

⊗ · · · ⊗ δ

c

k

(v

(k)

)

|

{z

}

(δ

n

(v))

(c)

_k

,

(7)

whi h shows that Condition (B) holds. Therefore

Σ(W )

is a subbialgebra of

F

(V )

. We all it the Manta i-Reutenauer bialgebra of the oalgebra

V

and we denote it by

MR

(V )

. Theassignment

V MR(V )

isa ovariant fun tor from the ategory of

K

- oalgebras to the ategory of

N

-graded bialgebras over

K

. As we will see in Se tion 3.3, this onstru tion is mainly useful when

V

isaproje tive

K

-moduleand the oprodu tof

V

is o ommutative; in this ase,

MR

(V )

is asubbialgebra of

NT

(V )

.

For onvenien e, we introdu e the following spe ial notation for the generators of the Manta i-Reutenauer bialgebra

MR

(V )

: given any positive integer

n

and any element

v ∈ V

, weset

y

n,v

=



δ

n

(v)#e

n



. Equations (6)and (7)entail thatthe oprodu tof

y

n,v

isgiven by

∆(y

n,v

) =

X

(v)

n

X

n

′

₌₀

y

n

′

_,v

(1)

⊗ y

n−n

′

,v

(2)

.

(8)

Moreover, Proposition 3 implies that if

V

is a free

K

-module, then the asso iative algebra

MR

_{(V )}

isfreelygenerated bythe elements

y

n,v

,where

n ≥ 1

and

v

is hosenin abasisof

V

.

2 Duality

Themainresultofthisse tionsaysthatthedualbialgebra

F

(V )

∨

ofthefreequasisymmetri bialgebra on

V

is the free quasisymmetri bialgebra

F

(V

∨

₎

on the dual module

V

∨

. This result is neither deep nor di ult, but has many interesting onsequen es, as we will see in Se tions 4 and 5. We begin by a general and easy dis ussion of duality for

K

-modules and

K

-bialgebras.

2.1 Perfe t pairings Wedenethedualityfun tor

?

∨

asthe ontravariantendofun tor

Hom

K

(?, K)

ofthe ategory of

K

-modules. In parti ular, this fun tor maps a morphism

f : M → N

to its transpose

f

∨

: N

∨

→ M

∨

. Restri tedto the full sub ategory onsistingof nitely generated proje tive

K

-modules,the dualityfun tor isan anti-equivalen e of ategories.

Giventwo

K

-modules

M

and

N

,thereisa anoni alisomorphism

(M ⊕ N )

∨

∼

_{= M}

∨

_{⊕ N}

∨

anda anoni almap

N

∨

_{⊗ M}

∨

_{→ (M ⊗ N )}

∨

;the latterisanisomorphism assoonas

M

or

N

isnitelygeneratedandproje tive. Givena

K

-module

M

,thereisa anoni alhomomorphism

M → M

∨∨

,whi his anisomorphism if

M

isnitely generated andproje tive.

Let

H

bea

K

-bialgebrawhose underlyingspa eisnitelygenerated andproje tive. Then the dual

H

∨

ounit of

H

∨

are the transposeof the oprodu t, themultipli ation, the ounit and theunit of

H

,respe tively.

A pairingbetween two

K

-modules

M

and

N

is a bilinear form

̟ : M × N → K

. It gives rive to two linearmaps

̟

♭

_{: M → N}

∨

_{, x 7→ ̟(x, ?)}

and

̟

#

_{: N → M}

∨

_{, y 7→ ̟(?, y)}

. The pairing

̟

is alled perfe t if the maps

̟

♭

and

̟

#

are isomorphisms. A pairing on a

K

-module

M

is a pairing between

M

and itself; su h a pairing

̟

is alled symmetri if

̟

♭

= ̟

#

.

Inthe ase where the

K

-modules

M

and

N

arenitelygenerated andproje tive,we may identify

M

and

N

with their respe tivebiduals, andfor anypairing

̟

between

M

and

N

,it holds

̟

#

_{= (̟}

♭

₎

∨

. Ifmoreover

M

and

N

arebialgebras,then

M

∨

and

N

∨

arealsobialgebras; in this situation, a pairing

̟

between

M

and

N

su h that

̟

♭

and

̟

#

are morphisms of bialgebrasis alledapairing ofbialgebras.

The above onstru tions on erning biduality or bialgebras are only valid with nitely generated proje tive modules. We an however relax therequirement of nite generation by workingwith

N

-graded modules. In this situation,we must adapt thedenition for thedual module: thedual of

M =

L

n≥0

M

n

is the gradedmodule

M

∨

₌

L

n≥0

(M

n

)

∨

,whose graded omponentsarethe dualmodules inthe previous senseof thegraded omponents of

M

. We must also make the further assumptions that the morphisms preserve the graduation and that pairings make graded omponents of dierent degrees orthogonal to ea h other. Then everything works as before, and biduality and duality of bialgebras go smoothly as soon as the modulesareproje tive withnitely generated homogeneous omponents.

2.2 Duality and the fun tor

F

Thefollowing propositionexaminestherelationship between thefun tor

F

andduality. Proposition 5 There is a natural transformation from the ontravariant fun tor

F

(?

∨

₎

to the ontravariant fun tor

F

(?)

∨

, whi hisan isomorphismwhenthe domainof these fun tors is restri ted tothe full sub ategory of nitely generated proje tive

K

-modules.

In other words, for any

K

-module

V

, we an dene a morphism of graded algebras

c

V

:

F

_(V

∨

₎

_{−→ F (V )}

≃

∨

, the onstru tion being su h that theassignment

V c

V

is natural in

V

,andthat

c

V

isan isomorphism of bialgebras if

V

isnitely generated and proje tive. Proof. Let

V

bea

K

-module. Withthehelpofthe anoni aldualitybra ket

h?, ?i : V × V

∨

_→

K

between

V

and

V

∨

,wedene for ea h

n ≥ 0

apairing

h?, ?i

n

between

F

n

(V )

and

F

n

(V

∨

₎

bythefollowing formula:



(v

1

⊗ v

2

⊗ · · · ⊗ v

n

)#σ



,



(f

1

⊗ f

2

⊗ · · · ⊗ f

n

)#π



_n

=

(Q

n

i=1

hv

σ(i)

, f

i

i,

if

σ = π

−1

,

0

otherwise, (9) where

(v

1

, v

2

, . . . , v

n

) ∈ V

n

,

(f

1

, f

2

, . . . , f

n

) ∈ (V

∨

₎

n

,and

σ

and

π

are elements of

S

n

. If

V

isassumed to be nitely generated and proje tive, the anoni al duality between

V

and

V

∨

isperfe tand extends to a perfe t pairingbetween

V

⊗n

and

(V

∨

₎

⊗n

,whi h impliesthatthe pairing

h?, ?i

n

isperfe t.

We ombinethesepie estodene apairing

h?, ?i

tot

between

F

(V )

and

F

(V

∨

₎

bysetting

hα, ξi

tot

=

X

n≥0

hα

n

, ξ

n

i

n

forall

α =

P

n≥0

α

n

and

ξ =

P

n≥0

ξ

n

,where

α

n

∈ F

n

(V )

and

ξ

n

∈ F

n

(V

∨

₎

. Themap

c

V

: F (V

∨

) → F (V )

∨

, x 7→ h?, xi

tot

isamorphismof

K

-modules;itisevenanisomorphismif

V

isnitelygeneratedandproje tive. Astraightforward veri ationshows thattheprodu t

∗

andthe oprodu t

∆

of

F

(V )

are adjointtothe oprodu t

∆

andtotheprodu t

∗

of

F

(V

∨

₎

withrespe ttothepairing

h?, ?i

tot

. Together with a similar statement about the unit and the ounits, this implies that

c

V

is a morphism of algebras, and even of bialgebras if

F

(V )

is proje tive with nitely generated homogeneous omponents. One he ks alsoeasily the ommutativityof thediagram

F

_(V

∨

₎

c

V

F

_(W

∨

₎

c

W

F

(f

∨

₎

F

_{(V )}

∨

F

(f )

F

_{(W )}

∨

∨

for any

K

-linear map

f : V → W

of

K

-modules. This means thatthe assignment

V c

V

is anatural transformation from

F

(?

∨

₎

to

F

(?)

∨

,whi h ompletes theproof.



Using the pre ise denitionof the maps

c

V

given in theproof of Proposition 5, one may he kthefollowing additional property: the two ompositions

F

_{(V ) −→ F (V}

∨∨

₎

_{−−−→ F (V}

c

(V ∨)

∨

₎

∨

and

F

(V ) −→ F (V )

∨∨ (c

V

)

∨

−−−→ F (V

∨

)

∨

areequal. Abusing thenotations, we willwrite the above equalityas

c

(V

∨

₎

= (c

_V

)

∨

.

Nowsupposethat

̟

isapairingbetween two

K

-modules

V

and

W

. We an thendene a pairing

̟

tot

between

F

(V )

and

F

(W )

bytheequality

̟

tot

♭

_{= c}

W

◦ F (̟

♭

)

;inotherwords, weset

̟

tot

(x, y) = c

W

◦ F (̟

♭

)

(x)(y),

where

x ∈ F (V )

and

y ∈ F (W )

. Then

̟

tot

#

= (̟

tot

♭

)

∨

= F (̟

♭

)

∨

◦ (c

W

)

∨

= F (̟

♭

)

∨

◦ c

(W

∨

₎

= c

_V

◦ F (̟

♭

)

∨

= c

_V

◦ F (̟

#

).

Theequalities

̟

tot

♭

_{= c}

W

◦ F (̟

♭

)

and

̟

tot

#

_{= c}

V

◦ F (̟

#

)

show that

̟

tot

isa pairing of bialgebras. Moreover if

̟

is perfe t, then so is

̟

tot

. In the ase

V = W

, one an also see that thesymmetryof

̟

entails thatof

̟

tot

.

2.3 Orthogonals and polars

Let

M

be a nitely generated proje tive

K

-module. We view it as an `ambient' spa e and identify it with its bidual

M

∨∨

. We dene the orthogonal of a submodule

S

of

M

as the submodule

S

⊥

_{= {f ∈ M}

∨

_{| f}





S

= 0}

of

M

∨

. Then

S

⊥

is anoni allyisomorphi to

(M/S)

∨

. Likewise,the orthogonal ofa submodule

T

of

M

∨

isa submodule

T

⊥

of

M

.

Let

S

bethesetofallsubmodules

S

of

M

su hthat

M/S

isproje tive,or inotherwords, that are dire t summands of

M

. If

S ∈ S

, then both

S

and

M/S

are nitely generated proje tive

K

-modules. Likewise, let

T

be theset of all submodules

T

of

M

∨

that aredire t summandsof

M

∨

. We endowboth

S

and

T

withthepartial ordergivenbythein lusion of submodules. The following results arewell-known inthis ontext:

•

The maps

S

→ T , S 7→ S

⊥

and

T

→ S , T 7→ T

⊥

are mutually inverse, order de reasingbije tions.

•

For any

S ∈ S

, there is a anoni al isomorphism

S

∨

∼

_{= M}

∨

_/S

⊥

. Moreover for ea h submodule

S

′

_{⊆ S}

,there isa anoni alisomorphism

(S/S

′

₎

∨

∼

_{= S}

′⊥

_/S

⊥

.

•

Let

S

and

S

′

be two elements in

S

. We always have

(S + S

′

₎

⊥

_{= S}

⊥

_{∩ S}

′⊥

and

S

⊥

₊

S

′⊥

⊆ (S ∩ S

′

)

⊥

. If moreover

S + S

′

belongs to

S

, then so does

S ∩ S

′

, and the equality

(S ∩ S

′

)

⊥

= S

⊥

+ S

′⊥

holds.

•

Assume that

M

is endowed with the stru ture of a bialgebra. Then a submodule

S ∈ S

is a subbialgebra of

M

if and only if

S

⊥

is a biideal of

M

∨

, and a submodule

T ∈ T

is a subbialgebra of

M

∨

ifand only if

T

⊥

is abiideal of

M

.

Given twosubmodules

S ∈ S

and

T ∈ T

,we have then sequen es of anoni almaps

T /(S

⊥

∩ T ) ∼

= (S

⊥

+ T )/S

⊥

= (S

⊥

+ T

⊥⊥

)/S

⊥

֒→ (S ∩ T

⊥

)

⊥

/S

⊥

∼

= S/(S ∩ T

⊥

)

∨

,

S/(S ∩ T

⊥

) ∼

= (S + T

⊥

)/T

⊥

= (S

⊥⊥

+ T

⊥

)/T

⊥

֒→ (S

⊥

∩ T )

⊥

/T

⊥

∼

= T /(S

⊥

∩ T )

∨

.

(10)

In other words, there is a anoni al pairing between

S/(S ∩ T

⊥

₎

and

T /(S

⊥

_{∩ T )}

, whi h is perfe t assoonas

(S + T

⊥

_{) ∈ S}

and

(S

⊥

_{+ T ) ∈ T}

.

We assumenowthat themodule

M

isendowed witha symmetri and perfe t pairing

̟

. Thento anysubmodule

S

of

M

we an asso iate its polar

P

◦

_{= ̟}

♭

−1

_S

⊥

withrespe t to

̟

. Using

̟

♭

,one an dedu epropertiesfor polarsubmodulesanalogous to theproperties for orthogonalsre alled above.

One analsoadapttheseresultsto the asewheretheproje tivemodule

M

isnot nitely generated,provided itis gradedwithnitely generated homogeneous omponents.

This material will prove useful in Se tions 4.3 and 5, where we will meet instan es of the following situation. Here

V

is a nitely generated proje tive

K

-module, endowed with a symmetri andperfe t pairing

̟

. Then

F

(V )

isa proje tive

K

-module,graded withnitely generated homogeneous omponents, and endowed with the perfe t and symmetri pairing

̟

tot

. Let moreover

S

be a graded subbialgebra of

F

(V )

, assumed to be a dire t summand ofthe graded

K

-module

F

(V )

. We have then thefollowing ommutative diagramof graded bialgebras,

F

_{(V )}

≃

F

_{(V )}

∨

S

F

_{(V )/S}

◦

≃

_S

∨

_.

S/(S ∩ S

◦

₎

_{S/(S ∩ S}

◦

₎

∨

(11)

Herethehorizontalarrows areindu ed by

̟

tot

♭

;theoneat thebottom lineis thepairingon

S/(S ∩ S

◦

)

dened bythe sequen es (10)withthe hoi e

T = ̟

tot

♭

_(S)

.

To on lude thisse tion, weshowthatthe framework aboveisgeneral enough to a omo-datethe ase ofa Manta i-Reutenauerbialgebra, viewed asa submoduleina free quasisym-metri bialgebra.