Rota--Baxter algebras and left weak composition quasi-symmetric functions

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by  Li Guo

ROTA–BAXTER ALGEBRAS AND LEFT WEAK COMPOSITION
QUASI-SYMMETRIC FUNCTIONS
LIGUO,HOUYIYU,ANDJIANQIANGZHAO
6 Abstract. Motivated by a question of Rota, this paper studies the relationship between Rota–
1 Baxter algebras and symmetric related functions. The starting pointis the fact that the space of
0 quasi-symmetricfunctionsisspannedbymonomialquasi-symmetricfunctionswhichareindexed
2
by compositions. When composition is replaced by left weak composition (LWC), we obtain

n theconceptofLWCmonomialquasi-symmetricfunctionsandtheresultingspaceofLWCquasi-
a symmetricfunctions. InlinewiththequestionofRota,thelatterisshowntobeisomorphictothe
J
freecommutativenonunitaryRota–Baxteralgebraononegenerator.Thecombinatorialinterpreta-
2
tionofquasi-symmetricfunctionsbyP-partitionsfromcompositionsisextendedtothecontextof
2
left weak compositions, leading to the conceptof LWC fundamentalquasi-symmetricfunctions.

] The transformation formulas for LWC monomial and LWC fundamentalquasi-symmetric func-
O
tionsareobtained,generalizingthecorrespondingresultsforquasi-symmetricfunctions. Extend-
C ingthe close relationshipbetweenquasi-symmetricfunctionsandmultiplezeta values, weighted
. multiplezeta valuesanda q-analogof multiplezetavaluesareinvestigatedanda decomposition
h
t formulaisestablished.
a
m
[

1 Contents
v
0 1. Introduction 1
3
2. Background 3
0
6 2.1. Rota’sstandard Rota–Baxteralgebras and symmetricfunctions 4
0 2.2. Mixableshuffle,quasi-shuffleand quasi-symmetricfunctions 5
.
1 3. Free nonunitaryRota–Baxteralgebras and LWCquasi-symmetricfunctions 7
0
4. P-partitionsand LWCfundamentalquasi-symmetricfunctions 12
6
1 4.1. P-partitionsandLWCquasi-symmetricfunctions 12
:
v 4.2. LWCfundamentalquasi-symmetricfunctions 14
i 5. Applicationto multiplezetavaluesand theirq-analogs 17
X
5.1. Left weak compositionmultiplezeta values 17
r
a
5.2. A q-analogofmultiplezetavalues 20
References 22
1. Introduction
Symmetric functions have played an important role in mathematics for a long time [21, 29].
More recently, their generalizations, especially quasi-symmetric functions and noncommutative
Date:January25,2016.
2010MathematicsSubjectClassification. 05E05,16W99,11M32.
Key words and phrases. Rota-Baxter algebras, symmetric functions, quasi-symmetric functions, left weak
compositions,monomialquasi-symmetricfunctions,fundamentalquasi-symmetricfunctions,P-partitions,lmultiple
zetavalues,q-multiplezetavalues.
1
2 LIGUO,HOUYIYU,ANDJIANQIANGZHAO
symmetric functions, have played similar roles. Quasi-symmetric functions were defined ex-
plicitly by Gessel [9] in 1984 with motivation from the P-partitions of Stanley [30] in 1972.
Most of their study was carried out after the middle 1990s. The last decade witnessed a surge
in the research on quasi-symmetric functions with broad applications. Conceptually, the central
role played by quasi-symmetric functions was demonstrated by the result of Aguiar, Bergeron
and Sottile [2] that the Hopf algebra of quasi-symmetric functions is the terminal object in the
category of combinatorial Hopfalgebras. This Hopf algebra is the graded dual of the Hopfalge-
bra of noncommutativesymmetric functions [8], another important generalization of symmetric
functions. Generalizationsofquasi-symmetricfunctionshavealsobeenintroduced(seeforexam-
ple [18, 22]). Further details of quasi-symmetric functions can be found in the monograph [20]
and the references therein. In this paper we study the close relationship between Rota–Baxter
algebras and the algebras of quasi-symmetric functions, and investigate a new class of multiple
zetavaluesthat arisefromthisrelationship.
Rota–Baxter algebra is an abstraction of the algebra of continuous functions equipped with
the integration operator, characterized by the integration by parts formula. Shortly after its
introduction by G. Baxter [3] in 1960 from his probability study, Rota–Baxter algebra (called
Baxter algebra in the early literature)attracted theinterest of well-knowncombinatoristssuch as
Cartierand Rota. Rotagavethefirst constructionoffree commutativeRota–Baxteralgebras [25,
27] and thereby showed that the Spitzer’s identity that he established for Rota–Baxter algebras
hasoneofitsincarnationsastheWaringformulaofsymmetricfunctionsrelatingelementaryand
powersum symmetricfunctions (see Eq. (2)). He also proposed to studysymmetricfunctions in
the framework of Rota–Baxter algebras. His interest in Rota–Baxter algebras continued into the
mid1990swhen hewrote, in hisinspiringsurveyarticle[26],
By analyzing the work of Baxter and Atkinson, I was led to conjecture that a
very close relationship exists between the Baxter identity (1) and the algebra of
symmetricfunctions.
and concluded
The theory of symmetric functions of vector arguments (or Gessel functions) fits
nicely with Baxter operators; in fact, identities for such functions easily translate
into identities for Baxter operators. ··· In short: Baxter algebras represent the
ultimateand mostnatural generalizationofthealgebraofsymmetricfunctions.
Partly motivated by Rota’s enthusiasm on Rota–Baxter algebras1, their study has experienced
a remarkable renaissance this century with broad applications in mathematics and mathematical
physics,in areas includingcombinatorics,Hopfalgebras, operads, numbertheory,quantumfield
theoryandclassicalYang–Baxterequations[1, 5, 7, 11,24]. See[10],aswellas[19], foramore
detailedintroductionto thissubject.
Inparticular,theconnectionofRota–Baxteralgebrawithquasi-symmetricfunctionenvisioned
by Rota was partly established by the equivalence [6] of the mixable shuffle product introduced
in [11] (also knownas overlappingshuffles [14]) in afree commutativeRota–Baxter algebra and
thequasi-shuffleproduct[16]whichisknowntobeageneralizationofquasi-symmetricfunctions.
This realizes the algebra of quasi-symmetric functions as a large part of a free commutative
Rota–Baxter algebra (on one generator). To further develop Rota’s insight, it would be useful
1Forexample,L.GuowasintroducedtothesubjectoftheRota–BaxteralgebrabyRota’ssuggestiontoextenda
jointworkofL.GuoandW.KeighersubmittedtohimtothecontextofRota–Baxteralgebras,resultingin[11,12]
andtheirsequels.
ROTA–BAXTERALGEBRASANDLEFTWEAKCOMPOSITIONQUASI-SYMMETRICFUNCTIONS 3
to interpret the full free commutative Rota–Baxter algebra as a suitable generalization of quasi-
symmetric functions. This is the purpose of our study. In this paper, we study free commutative
nonunitaryRota–Baxteralgebras. Theunitarycasewillbeconsidered inaseparatework.
The benefit of this connection between quasi-symmetric functions and Rota–Baxter algebras
is bidirectional. On the one hand, this interaction allows us to realize the abstractly defined free
Rota–Baxter algebras as subalgebras of concretely defined polynomial or power series algebras.
On the other hand, this interaction allows us to find generalizations of these symmetric related
functions, in the spirit of Rota’s aforementioned quotes. As a result, we obtain new classes of
functions that share properties similar to quasi-symmetric functions, including their canonical
bases consistingofmonomialandfundamentalquasi-symmetricfunctions.
Another benefit of this connection is in the study of multiple zeta values (MZVs for short),
started withEulerand Goldbacherinthetwo variablecase, and systematicallyinvestigatedsince
the1990s with thework ofHoffman [15] and Zagier [31]. Quasi-symmetricfunctions specialize
toMZVsbysuitableevaluationsthroughwhichthemultiplicationoftwoMZVs(stuffleproduct)
comesfromthequasi-shuffleproductoftwomonomialquasi-symmetricfunctions[16,17]. Inthe
contextofRota–Baxteralgebras,arelatedconstructioncanbefoundintheworkofCartier[4]on
free commutativeRota–Baxter algebras. Extending this well-knownconnection, we obtain from
thesegeneralized quasi-symmetricfunctionsa classofMZVswithweights(coefficients).
The paper is organized as follows. Section 2 is devoted to a quick review of well-known
results and definitions on Rota–Baxter algebras and quasi-symmetric functions. In particular, as
motivation, we recall the connection of symmetric functions with Rota’s standard Rota–Baxter
algebrasand,aspreparation,werecalltheconnectionofmixableshuffleproduct,theprimarypart
ofthefreecommutativeRota-Baxteralgebra, withmonomialquasi-symmetricfunctionsindexed
bycompositions. ToputthefullfreecommutativenonunitaryRota–Baxteralgebrainthecontext
of quasi-symmetricfunctions, we introducein Section 3 a generalization of thequasi-symmetric
functions, called LWC quasi-symmetric functions, by generalizing monomial quasi-symmetric
functions to be indexed by left weak compositions. These functions, called monomial LWC
quasi-symmetricfunctionsformabasisoftheLWCquasi-symmetricfunctions. Arelationamong
monomial LWC quasi-symmetric functions of the same degree is established by applying this
construction. In Section 4, we apply the concept of a P-partition of Stanley to define left weak
fundamental quasi-symmetric functions that form the second basis for LWC quasi-symmetric
functions. Transformation formulas between these two bases are established. The LWC quasi-
symmetricfunctionspermitustodefinealeftweakMZVanditsq-analoginSection5. Particular
relationsareestablishedforthesetypesofMZVs. Especially,ageneralizationofthefinitedouble
shufflerelationstotheproductoftwo leftweak MZVsis provided.
Convention. Unless otherwise specified, an algebra in this paper is assumed to be commuta-
tive, defined over a unitary commutative ring k. Let N and P denote the set of nonnegative and
positiveintegersrespectively.
2. Background
To providebackgroundand motivationforourstudy,we recall in thissection Rota’s construc-
tion of standard Rota–Baxter algebras and their relationship with symmetric functions. This is
followed by the concepts of mixable shuffle product, quasi-shuffle product and their relationship
withquasi-symmetricfunctionsand MZVs.
4 LIGUO,HOUYIYU,ANDJIANQIANGZHAO
2.1. Rota’s standard Rota–Baxter algebras and symmetric functions. For a fixed λ ∈ k, a
Rota–Baxter k-algebra of weight λ is a pair (R,P) consisting of an algebra R and a k-linear
operator P : R → Rthat satisfiesthe Rota–Baxterequation
(1) P(x)P(y) = P(xP(y))+P(P(x)y)+λP(xy) for all x,y ∈ R.
Then P is called a Rota–Baxter operator (RBO) of weight λ. If R is only assumed to be a
nonunitaryk-algebra, wecall R anonunitaryRota–Baxterk-algebra ofweightλ.
Throughouttherestofthepaperwewillassumethat λ = 1 and dropλ fromthenotations.
A Rota–Baxter algebra homomorphism is an algebra homomorphism that is compatible with
the Rota–Baxter operators. Given a commutative k-algebra A that is not necessarily unitary,
the free commutative Rota–Baxter k-algebra on A is defined to be a Rota–Baxter k-algebra
(F(A),P ) together with a k-algebra homomorphism j : A → F(A) with the property that, for
A A
anyRota–Baxterk-algebra(R,P)andanyk-algebrahomomorphism f : A → R,thereisaunique
Rota–Baxterk-algebrahomomorphism f˜: (F(A),P ) → (R,P)suchthat f = f˜◦ j ask-algebra
A A
homomorphisms.
As a motivation, we recall the first construction of free commutative Rota–Baxter algebras
given by Rota [25, 27], called the standard Rota–Baxter algebra, and their relationship with
symmetricfunctions.
Let X be a given set. For each x ∈ X, let t(x) denote a sequence (t(x),··· ,t(x),···) of distinct
1 n
symbolst(x),n > 1, suchthatthesets{t(x1)} and {t(x2)} are disjointfor x , x in X. Denote
n n n n n 1 2
X = t(x) | n > 1
n
[x∈X n o
and let A(X) = k[X]P denote the algebra of sequences with entries in the polynomial algebra
k[X]. The addition, multiplicationand scalar multiplicationby k[X] in A(X) are defined compo-
nentwise.
Define
Pr : A(X) → A(X), (a ,a ,a ,···) 7→ (0,a ,a +a ,a +a +a ,···).
X 1 2 3 1 1 2 1 2 3
So each entry of Pr(a) for a = (a ,a ,···) is the sum of the previous entries of a. Then Pr
X 1 2 X
defines a Rota–Baxter operatoron A(X). The standard Rota–Baxter algebra on X is theRota–
Baxter subalgebra S(X) of A(X) generated by the sequences t(x) := (t(x),··· ,t(x),···), x ∈ X. An
1 n
importantresult ofRotais
Theorem 2.1. ([25, 27])(S(X),Pr) isthefreecommutativeRota–Baxteralgebraon X.
X
Consider the special case of X = {x}. Let the sequence (Pr x)[n],n > 1, be defined by the
X
recursion (Pr x)[1] := Pr(x) and(Pr x)[n+1] := Pr(x(Pr x)[n]),n > 1. Then
X X X X X
(Pr x)[n] = (0,e (x ),e (x ,x ),e (x ,x ,x ),···)
X n 1 n 1 2 n 1 2 3
where e (x ,··· ,x ) = x x ···x is the elementary symmetric function of degree
n 1 m i1 i2 in
ninthevariables x ,··· 1,6xi1<iw2P<i·t··h<int6hmeconventionthate (x ,··· ,x ) = 1ande (x ,··· ,x ) = 0if
1 m 0 1 m n 1 m
m < n. Alsoby definition,
Pr(xk) = (0,p (x ),p (x ,x ),p (x ,x ,x ),···),
X k 1 k 1 2 k 1 2 3
ROTA–BAXTERALGEBRASANDLEFTWEAKCOMPOSITIONQUASI-SYMMETRICFUNCTIONS 5
where p (x ,··· ,x ) = xk +xk +···+xk isthepowersumsymmetricfunctionofdegreek inthe
k 1 m 1 2 m
variables x ,··· ,x . Thesetwoclasses ofsymmetricfunctionsarerelated by Waring’sformula
1 m
∞ ∞
(2) exp − (−1)ktkp (x ,··· ,x )/k = e (x ,··· ,x )tn forall m > 1.
 k 1 m  n 1 m
As mentioned inthXek=1introduction, Rota derived thXni=s0formula as a special case of his algebraic
formulation of Spitzer’s identity. Cartier [4] gave the second construction of free commutative
Rota–Baxter algebras.
2.2. Mixable shuffle, quasi-shuffle and quasi-symmetric functions. Another construction of
free commutative Rota–Baxter k-algebras was given by using the mixable shuffle algebra. The
mixable shuffle algebra generated by a commutative (unitary or nonunitary) algebra A, denoted
by MS(A), hasitsunderlyingmoduleas thatofthetensoralgebra
T(A) = A⊗k = k⊕A⊕A⊗2 ⊕··· , where A⊗k = A ⊗ A ⊗· · · ⊗ A,
Mk>0 k−factors
equippedwiththemixableshuffleproduct∗ defined as follows. | {z }
For pure tensors a = a ⊗ ··· ⊗ a and b = b ⊗ ··· ⊗ b , a shuffle of a and b is a tensor list
1 m 1 n
from the factors of a and b in which the natural orders of the a’s and the b ’s are preserved. The
i j
shuffle product of a and b, denoted aXb, is the sum of all shuffles of a and b. For example, we
have
a1X(b1 ⊗b2) = a1 ⊗b1 ⊗b2 +b1 ⊗a1 ⊗b2 +b1 ⊗b2 ⊗a1.
Amixableshuffleofaandbisashuffleofaandbwheresome(ornone)ofthepairsa ⊗b are
i j
replaced byab . Themixableshuffleproduct a∗bofaandbisthesumofallmixableshuffles.
i j
Forexample
a ∗(b ⊗b ) = a ⊗b ⊗b +b ⊗a ⊗b +b ⊗b ⊗a +a b ⊗b +b ⊗a b ,
1 1 2 1 1 2 1 1 2 1 2 1 1 1 2 1 1 2
where a b ⊗ b comes from a ⊗b ⊗b by “mixing” or merging a ⊗b and b ⊗a b comes
1 1 2 1 1 2 1 1 1 1 2
fromb ⊗a ⊗b by“mixing”ormerginga ⊗b . Thelastshuffleb ⊗b ⊗a doesnotyieldany
1 1 2 1 2 1 2 1
mixedterm sincea isnotbefore anyb , j = 1,2.
1 j
Similartotherecursiveformulaoftheshuffleproduct
aXb := a1 ⊗((a2 ⊗···⊗am)Xb)+b1 ⊗(aX(b2 ⊗···⊗bn)), 1Xa = aX1 = a,
themixableshuffleproduct can alsobedefined by therecursion
(3) a∗b = a ⊗((a ⊗···⊗a )∗b)+b ⊗(a∗(b ⊗···⊗b ))+a b ⊗((a ⊗···⊗a )∗(b ⊗···⊗b ))
1 2 m 1 2 n 1 1 2 m 2 n
with the convention that 1 ∗ a = a ∗ 1 = a. It is known as the quasi-shuffle [16] of Hoffman.
Its equivalence to the mixable shuffle product was proved in [6]. The mixable shuffle product
equipped MS(A)withacommutativealgebrastructure.
Acompositionisafiniteorderedlistofpositiveintegers. Givenacompositionα = (α ,··· ,α ),
1 k
we define its weight or size to be |α| = α + ··· + α and its length to be ℓ(α) := k. We call
1 k
α,1 6 i 6 k, the components of α. If |α| = n, then we say α is a composition of n and write
i
α |= n. For convenience we denote by ∅ the unique composition whose weight and length are 0,
called the empty composition. There is a natural bijection between compositions of size n and
subsets of [n−1] = {1,··· ,n−1} which maps a composition α = (α ,··· ,α ) |= n to the set of
1 k
itspartial sums,notincludingnitself,that is,
set(α) := {α ,α +α ,··· ,α +α +···+α }.
1 1 2 1 2 k−1
6 LIGUO,HOUYIYU,ANDJIANQIANGZHAO
Given compositions α and β of n, we say that α is a refinement of β (or β is a coarsening
of α), denoted α (cid:22) β, if summing some consecutive components of α gives β. For example,
(1,2,1,4,1)(cid:22) (3,1,5).
Let X = {x ,x ,···} be a countably infinite totally ordered set of commuting variables. Recall
1 2
thataquasi-symmetricfunctionisaformalpowerseriesoffinitedegree f ∈ k[[X]]suchthatfor
each compositionα = (α ,··· ,α ), all monomials xα1 ···xαk in f with indices 1 6 n < ··· < n
1 k n1 nk 1 k
have the same coefficient. The set of all quasi-symmetric functions, denoted QSym, is a graded
algebrawithfunctionmultiplication,and withgrading
QSym := QSym ,
n
Mn>0
whereQSym is spannedby M = 1and all otherQSym are spannedby {M } where
0 ∅ n α α|=n
M := xα1 ···xαk
α n1 nk
16nX1<···<nk
for a composition α = (α ,··· ,α ) of n. This basis is called the basis of monomial quasi-
1 k
symmetric functions. A second basis for QSym consists of fundamental quasi-symmetric
functions, alsoindexedbycompositions. Theyare defined by F = 1and
∅
F = M .
α β
Xβ(cid:22)α
Forinstance, F = M + M . Moreover,QSym = span{M |α |= n} = span{F |α |= n}.
(2,1) (1,1,1) (2,1) n α α
Themultiplicationoftwomonomialquasi-symmetricfunctionsisdictatedbythequasi-shuffle
product, or mixable shuffle product, on the indexing compositions. This is a special case of the
quasi-shuffleproduct defined in Eq. (3). In the special case when the algebra A is the nonunitary
algebra xk[x]ofpolynomialsinonevariablewithouttheconstantterm,abasisof MS(A)consists
of pure tensors xα := xα1 ⊗···⊗ xαk where α = (α ,··· ,α ) is a composition, together with the
1 k
identity1. Throughthenaturallinearbijection
ρ : kC → MS(A), α 7→ xα,
onthespacespannedbythesetCofcompositions,thequasi-shuffleproduct∗ on MS(A)defines
a product, also denoted ∗, on kC by transport of structures. More precisely, if α = (α ,··· ,α ),
1 m
β = (β ,··· ,β ), thenwehavetherecursion
1 n
(4) α∗β = (α ,(α ,··· ,α )∗β)+(β ,α∗(β ,··· ,β ))+(α +β ,(α ,··· ,α )∗(β ,··· ,β ))
1 2 m 1 2 n 1 1 2 m 2 n
withtheconventionthat1∗α = α∗1 = α.
Convention 2.2. To simplify notations in the rest of the paper, we will adapt the following
convention. LetkX bethefreemoduleonaset X andlet · : X ×X → kX beamultiplicationthat
extendstooneon kX. Then for x,y ∈ X, wehave x·y = c z which wedenoteby z. Thus
z
z∈X z∈(x·y)
P P
V isak-algebrawhosemultiplicationissuppressed. Thenalinearmap f : kX → V preservesthe
multiplications: f(x)f(y) = f(x·y)ifand onlyif f(x)f(y) = f(u). Notethat hereu ∈ (x·y)
u∈(x·y)
P
does notmeanthat u isabasiselementin X.
Inparticular,weletα∗β = γdenotethelinearcombinationresultingfromα∗β. By[16]
γ∈(α∗β)
we obtain an algebra isomorphisPmof MS(A) with the algebra QSym by sending xα to M . Here
α
ROTA–BAXTERALGEBRASANDLEFTWEAKCOMPOSITIONQUASI-SYMMETRICFUNCTIONS 7
the multiplication on QSym is the natural multiplication of power series. Therefore, given two
compositionsαand β,we have
M M = M ,
α β γ
γ∈X(α∗β)
as inEq. (4). In otherwords, let
(5) φ : kC → QSym, α 7→ M ,α ∈ C.
α
Then
M M = φ(α∗β).
α β
Evaluating x at n−1, M givestheMZV
n α
1
(6) ζ(α) := ζ(α ,··· ,α ) := ,
1 k nα1 ···nαk
16nX1<···<nk 1 k
whichconverges forα > 1,1 6 j 6 k and α > 2. Thequasi-shufflerelation orstufflerelationof
j k
MZVsisa directconsequenceofthequasi-shuffleproducton QSym.
3. Free nonunitary Rota–Baxter algebras and LWCquasi-symmetric functions
In this section, we first outline a construction for free commutative nonunitary Rota–Baxter
algebras on one generator by making use of the mixable shuffle algebras recalled in the last
section. Wethenintroduceaclassofpowerseriesthatontheonehandcontainsquasi-symmetric
functionswithsimilarpropertiesandontheotherhandisisomorphictothefreenonunitaryRota–
Baxter algebraononegenerator.
Let A beacommutativeunitaryk-algebra. Define
X (A) := A⊗ MS(A) = A⊗k
k
Mk>1
to be the tensor product of the algebras A and MS(A). More precisely, for two pure tensors
a ⊗a = a ⊗a ⊗···⊗a andb ⊗b = b ⊗b ⊗···⊗b ,theaugmentedmixableshuffleproduct
0 0 1 m 0 0 1 n
⋄onX (A) isdefined by
k
a b , m = n = 0,
0 0
(a b )⊗a, m > 0,n = 0,
(7) (a0 ⊗a)⋄(b0 ⊗b) := (a00b00)⊗b, m = 0,n > 0,
Theorem 3.1. [11] The algebra (X (A),⋄)(,a0wbi0t)h⊗th(ae∗libn)e,armop>er0a,tonr>P0. : X (A) → X (A)
k A k k
sendingato1⊗a, isthefreecommutativeRota–Baxteralgebrageneratedby A.
Nowlet A be a commutativenonunitary k-algebra and let A˜ = k⊕A betheunitarization of A.
Define
X (A)0 := (A˜⊗k ⊗A)
k
Mk>0
with the convention that A˜⊗0 = k and hence A˜⊗0 ⊗ A = A. Then X (A)0 is a k-submodule of
k
X (A˜), additivelyspannedby puretensorsoftheform
k
a ⊗···⊗a , a ∈ A˜, 0 6 i 6 n−1, a ∈ A.
0 n i n
8 LIGUO,HOUYIYU,ANDJIANQIANGZHAO
With the restriction of P , denoted by P , X (A)0 is a subobject of X (A˜) in the category of
A˜ A k k
commutativenonunitaryRota–Baxteralgebras. By[12,Proposition2.6],(X (A)0,P )isthefree
k A
commutativenonunitaryRota–Baxteralgebrageneratedby Acharacterizedbyitsusualuniversal
property.
We now consider the special case when A = k[x] or A = xk[x]. Then Theorem 3.1 can be
restated as
Theorem 3.2. The k-module
(8) X(x) := X (k[x]) = k[x]⊗(k+1) = kxα0 ⊗···⊗ xαk,
k
Mk>0 αi>M0,06i6k
with the product in Eq. (7) and the operator P : a 7→ 1 ⊗ a, is the free unitary Rota–Baxter
x
algebraon x. The restrictionoftheproductand operatoronthesubmodule
(9) X(x)0 := X (xk[x])0 = k[x]⊗k ⊗ xk[x] = kxα0 ⊗···⊗ xαk
k
Mk>0 (cid:16) (cid:17) αi>0,06Mi6k−1,αk>1
gives thefreenonunitaryRota–Baxteralgebraon x.
Asrecalled in Section 2,themixableshufflealgebra
MS(xk[x]) = k1 k{xi1 ⊗···⊗ xik |(i ,··· ,i ) ∈ Pk,k > 1}
1 k
M
is isomorphic to the algebra of quasi-symmetric functions QSym in the formal power series
k[[X]]. Let
MS(xk[x])0 := k{xi1 ⊗···⊗ xik |(i ,··· ,i ) ∈ Nk−1 ×P,k > 1}.
1 k
Then, fromtheabovetheorem,wehaveX(x)0 = MS(xk[x])0 whichis largerthan MS(xk[x]).
InordertorealizeX(x)0 asformalpowerseries,wefirstneedtodosofor MS(xk[x])0. Tothis
end, we first generalize the definition of quasi-symmetric functions, by generalizing the concept
ofcompositions.
A weak composition [28] α = (α ,··· ,α ) of a nonnegative integer n, denoted α(cid:25)n, is a
1 k
sequence of nonnegative integers whose sum is n. We call the α the components of α and
i
ℓ(α) := k thelength ofα. Ifα(cid:25)n,then wecall nthesizeofα, denoted|α|.
Definition 3.3. A left weak composition2 is a weak composition that ends with a positivecom-
ponent, that is, if α = (α ,··· ,α ) with α being a positive integer. Let LWC denote the set of
1 k k
left weak compositions.
Fora leftweak compositionα, itisconvenienttogroup blocksofzeros togetherand denote
α = (0i1,s ,0i2,s ,··· ,0ij,s ), s > 1,i > 0,p = 1,2,··· , j,
1 2 j p p
where 0i means a string of i zero components. We also write α = 0i1s 0i2s ···0ijs for conve-
1 2 j
nience.
Recall that QSym can beequivalentlydefined tobe thesubspaceofformal powerseries in the
countable variables X := {x |i > 1} spanned by the monomial quasi-symmetricfunctions M , as
i α
αrun throughallthecompositions. Weextendthisnotionforleftweak compositionsas follows.
2Relating weak compositions to Rota-Baxter algebras needs a different strategy and will be considered in a
separatework.
ROTA–BAXTERALGEBRASANDLEFTWEAKCOMPOSITIONQUASI-SYMMETRICFUNCTIONS 9
Definition 3.4. we call a formal power series of finite degree f ∈ k[[X]] a left weak composi-
tion quasi-symmetric function, or simply an LWC quasi-symmetric function, if f is a linear
combinationofthefollowingformal powerseries
(10) M := xα1 ···xαk,
α n1 nk
16nX1<···<nk
indexedbyleftweakcompositionsα = (α ,··· ,α ). Wecallthe M theLWCmonomialquasi-
1 k α
symmetricfunction and denoteLWCQSym thespaceofLWCquasi-symmetricfunctions.
With the convention that m = 0 if m < n, we have the following formula for an LWC
n
monomialquasi-symmetricfu(cid:16)n(cid:17)ction M .
α
Lemma 3.5. Let α = (0i1,s ,0i2,s ,··· ,0ik,s ) be a left weak composition, where s > 1,i >
1 2 k p p
0,p = 1,2,··· ,k. Then
n −1 n −n −1 n −n −1
(11) M = 1 2 1 ··· k k−1 xs1xs2 ···xsk.
α i ! i ! i ! n1 n2 nk
16n1<Xn2<···<nk 1 2 k
Proof. By Eq. (10), weknow
M = x0 ···x0 xs1x0 ···x0 xs2 ···x0 ···x0 xsk.
α 16n11<···<n1i1<nX1<···<nk1<···<nkik<nk n11 n1i1 n1 n21 n2i2 n2 nk1 nkik nk
Forfixedn and n , p = 1,··· ,k,withtheconventionthatn = 0, onecan choosen ,··· ,n
p−1 p 0 p1 pip
satisfying n < n < ··· < n < n in np−np−1−1 ways. Notethat m = 0 if m < n and x0 = 1
p−1 p1 pip p ip n
forall indeterminants x. Then Eq.(11)foll(cid:16)ows. (cid:17) (cid:16) (cid:17) (cid:3)
Forexample,
n−1 n−1
M = x0 x0 x3 = x3 = x3.
(02,3) n1 n2 n3 2 ! n 2 ! n
16nX1<n2<n3 Xn>1 Xn>3
Clearly, M isa quasi-symmetricfunctionifand onlyifi = 0forall 1 6 p 6 k.
α p
Tomaketheparallelto monomialquasi-symmetricfunctionscomplete,we have
Lemma 3.6. The set{M |α ∈ Nk−1 ×P,k > 1}isa k-basisforLWCQSym.
α
Proof. It suffices to show that M for any finitely many distinct α are linear independent. Let
α
αj = (0ij1,sj1,0ij2,sj2,··· ,0ijkj,sjkj) be distinct left weak compositions, j = 1,··· ,n. Assume
thatc M +···+c M = 0 isanon-zero combinationwithc , 0, j = 1,··· ,n. Letr = ℓ(α ),
1 α1 n αn j j j
j = 1,··· ,n. By permuting r ,··· ,r if necessary, we may assume r = min{r ,··· ,r }. By
1 n 1 1 n
Lemma 3.5, the coefficient of xsj1 xsj2 ···xsjkj in M is 1. Since α are different from each
ij1+1 ij1+ij2+2 rj αj j
other, the coefficient of xs11 xs12 ···xs1k1 in M is δ , where δ is the Kronecker delta. Thus,
the coefficient of the aboi1v1+e1mi1o1+nio12m+2ial inr1c M +j ···1+j c M isijc and hence c must be 0, a
1 α1 n αn 1 1
contradiction. Thiscompletestheproof. (cid:3)
Analogousto Eq. (4) we define the quasi-shuffleproduct, also denoted by ∗, for two left weak
compositions,anddenoteitbyα∗β = γwiththenotationinConvention2.2. Thenwehave
γ∈(α∗β)
P
thefollowingmultiplicationformulaforLWCmonomialquasi-symmetricfunctions.
10 LIGUO,HOUYIYU,ANDJIANQIANGZHAO
Theorem 3.7. Let α and β be left weak compositions. Then M M = M with thenotation
α β γ
γ∈(α∗β)
P
inConvention 2.2. In otherwords, defining
(12) φ : kLWC → LWCQSym, α 7→ M ,
α
extendingthelinearbijectionφ in Eq.(5), then M M = φ(α∗β).
α β
Proof. Let α = (α ,··· ,α ) ∈ Nk−1 × P and β = (β ,··· ,β) ∈ Nl−1 × P be two left weak
1 k 1 l
compositions. Itiseasytoseethateverysummandofα∗βisalsoaleftweak composition. Here
we use the convention that M = 1 if ℓ(α) = 0. We apply the induction on the sum k + l of the
α
lengths of α and β. If k + l = 1, then either M or M is 1, so the desired result is clearly true.
α β
Assumethatithas been verifiedfork+l 6 t foragivent > 1and considerthecase k+l = t+1.
Then, byEq. (10),
M M = xα1 ···xαkxβ1 ···xβl.
α β n1 nk m1 ml
X
16n1<n2<···<nk
16m1<m2<···<ml
Thesumontherighthand can bedividedintothreeparts:
(a)ifn < m , then,by theinductionhypothesis,
1 1
xα1 xα2 ···xαk xβ1 ···xβl = M ,
n1  n2 nk m1 ml (α1,(α2,···,αk)∗β)
(b)ifm <1Xn6n,1then,n1b<ynX2t<h··e·<niknductionhynp1<omt1h<Xems2i<s·,··<ml 
1 1
xβ1 xα1 ···xαk xβ2 ···xβl = M ,
m1 n1 nk m2 ml (β1,α∗(β2,···,βl))
(c)ifm = n1X6,mt1hen,bmy1<thnX1e<·i··n<dnkuctionhypomt1h<meXs2<is···,<ml 
1 1
xαn11+β1  xαn22 ···xαnkk xβm22 ···xβmll = M(α1+β1,(α2,···,αk)∗(β2,···,βl)).
Themul1t6iXpn1l=icma1tionoftnw1<onX2l<e··f·<tnwk eak compon1s<imtXi2o<n···s<mslatisfies thesamerecursion as inEq. (4). Thus
combiningthethreeparts, weobtain M M = M , as desired. (cid:3)
α β γ
γ∈(α∗β)
P
By Theorem3.7, thespace ofLWCquasi-symmetricfunctionsisa graded algebra
LWCQSym := LWCQSym ,
n
Mn>0
where LWCQSym is spanned by M = 1 and LWCQSym for n > 1 consists of homogeneous
0 ∅ n
LWC quasi-symmetric functions of degree n. Here the degree of an LWC monomial quasi-
symmetric function M is defined to be the size of α. Note that since we allow the entries of
α
αtobe0, therank ofthehomogeneouscomponentLWCQSym isinfiniteforeach n > 1.
n
Foravectorα := (α ,··· ,α ) ∈ Nk ×P,denoteα′ := (α ,··· ,α ) ∈ Nk−1 ×Pandα = (α ,α′).
0 k 1 k 0
Let X = {x }∪X. In thepowerseries algebrak[[X]], define
0
(13) Mα := xα00Mα′ = xα00 xαn11 ···xαnkk.
16nX1<···<nk