Conical zeta values and their double subdivision relations

0.33 MB·English

by  Li Guo

CONICAL ZETA VALUES AND THEIR DOUBLE SUBDIVISION RELATIONS
LIGUO,SYLVIEPAYCHA,ANDBINZHANG
3 Abstract. We introduce the concept of a conical zeta value as a geometric generalization of a
1 multiple zeta value in the context of convex cones. The quasi-shuffle and shuffle relations of
0 multiplezetavaluesaregeneralizedtoopenconesubdivisionandclosedconesubdivisionrelations
2
respectivelyforconicalzeta values. Inorderto achievethe closedconesubdivisionrelation, we

n also interpret linear relations among fractions as subdivisions of decorated closed cones. As a
a
generalizationofthedoubleshufflerelationofmultiplezetavalues,wegivethedoublesubdivision
J
relationofconicalzetavaluesandformulatethe extendeddoublesubdivisionrelationconjecture
5
forconicalzetavalues.
1

T] Contents
N
1. Introduction 1
.
h 1.1. Multiplezetavaluesand conicalzeta values 1
t
a 1.2. Doubleshuffleand doublesubdivisionrelations 2
m
1.3. Layoutofthepaper 3
[
2. Convexcones andconical zetavalues 4

1 2.1. Polyhedralcones 4
v
2.2. Subdivisionofcones 5
0
7 2.3. Conical zetavaluesand opensubdivisionrelations 7
3 3. Closed conesand simplefractions 9
3
3.1. Fromclosed cones tosimplefractions 10
.
1
3.2. Subdivisionsofcones andsimplefractions 13
0
3 4. Decorated cones and purefractions 17
1 4.1. Decorated closedcones 17
:
v 4.2. Subdivisionofdecorated cones and purefractions 19
i
X 5. Conical zetavaluesrevisited: doublesubdivisionrelationsand Shintanizetavalues 22
r 5.1. Closedsubdivisionrelation andshufflerelation 22
a
5.2. Doublesubdivisionrelationsand doubleshufflerelations 25
5.3. Conical zetavaluesand Shintanizetavalues 30
References 33
1. Introduction
1.1. Multiple zeta values and conical zeta values. Multiple zeta values (MZVs) are special
valuesofthemulti-variableanalyticfunction
1
(1) ζ(s ,··· ,s ) =
1 k ns1 ···nsk
n1>X···>nk>0 1 k
Date:January16,2013.
1
2 LIGUO,SYLVIEPAYCHA,ANDBINZHANG
at integers s ≥ 2,s ≥ 1,1 ≤ i ≤ k. Their study in the two variable case goes back to Goldbach
1 i
and Euler. The general concept was introduced in the early 1990s, leading to developments in
bothmathematics[17,25],whereMZVsconjecturallyspan(periodsof)mixedTatemotives,and
physics [5], where MZVs mysteriously appeared in Feynman integral computations. Since then
the subject has been studied intensively with interactions to a broad range of areas, including
arithmetic geometry, combinatorics, number theory, knot theory, Hopf algebra, quantum field
theoryand mirrorsymmetry[1, 3, 4, 6, 10, 12, 16, 14, 18, 20,23, 24, 26].
MZVs have several generalizations, such as Hurwitz multiple zeta values and multiple poly-
logarithms. In this paper we give a geometric generalization of MZVs in the context of convex
r
cones. Foran open convex conesC = R v spanned by vectors v ∈ Zk ,1 ≤ i ≤ r, we define
>0 i i ≥0
i=1
theconicalzeta functionassociatedwithC by
P
1
ζ(C;s ,··· ,s ) := ,s ∈ C,1 ≤ i ≤ k,
1 k ns1 ···nsk i
(n1,···,Xnk)∈C∩Zk≥0 1 k
where itconverges and definea conical zeta valueto be thevalueof thefunctionat nonnegative
integer arguments. Such values contain MZVs as special cases when the cones are taken to be
Chen cones {x > ··· > x > 0}.
1 k
1.2. Double shuffle and double subdivision relations. A major goal in the study of MZVs is
to determine all algebraic relations among the MZVs. According to the Double Shuffle Conjec-
ture [16, 18], all such relations come from the shuffle and quasi-shuffle relations (the extended
doubleshufflerelation)thatencodeproductsofMZVsfromtheirsummationandintegrationrep-
resentations[20]. Inthisspirit,wegeneralizethedoubleshufflerelationofMZVstoconicalzeta
valuesas subdivisionsofopen and closedcones.
The double shuffle relation can be summarized in the following commutativediagram that we
willgeneralizeto conicalzeta values.
(2) (H∗,∗) oo oo oo (HX,X)
0 JJJζJ∗JJJJJ%% η xxqqqqqζqXqqqq 0
QMZV
Here
(a) H∗ := Q1 ⊕ z ···z is the quasi-shuffle algebra [15] with the quasi-shuffle
0 s1 sk
s1,···,sk≥1,s1≥2,k≥1
product ∗, encodLingtheMZVsbythealgebrahomomorphism
ζ∗ : H∗ → QMZV, z ···z 7→ ζ(s ,··· ,s );
0 s1 sk 1 k
(b) HX := Q1⊕ xs1−1x ···xsk−1x is the shuffle algebra with the shuffle product
0 0 1 0 1
s1,···,sk≥1,s1≥2,k≥1
X, encodingtheMLZVsbythealgebrahomomorphism
ζX : HX → QMZV, xs1−1x ···xsk−1x
0 0 1 0 1
and
(c) η : HX → H∗,xs1−1x ···xsk−1x → z ···z , istheobviouslinearbijection.
0 0 0 1 0 1 s1 sk
TheDoubleShuffle Conjecturestatesthatthekernel of ζ∗ istheideal ofH∗ generated by theset
1
{w ∗w −η−1(η(w )Xη(w )), |w ∈ {z }∪H∗,w ∈ H∗}.
1 2 1 2 1 1 0 2 0
CONICALZETAVALUESANDTHEIRDOUBLESUBDIVISIONRELATIONS 3
Thequasi-shuffle(stuffle)encodingζ∗ ofMZVsfollowsdirectlyfrom thedefinitionofMZVs.
Thisisgeneralized to CZVs as openconesubdivisions.
TheshuffleencodingζX islessdirect. Itisderivedbytheintegralrepresentation ofMZVs[20]
or, alternatively, from the integral representation of the multiple zeta fractions [11]. As shown
there, themultiplezetafractions
1
s ,··· ,s
(3) f 1 k := , s,u ≥ 1,1 ≤ i ≤ k
(cid:2)u1,··· ,uk(cid:3) (u1 +···+uk)s1(u2 +···+uk)s2 ···uksk i i
on theonehand givemultiplezetavalues
s ,···,s
(4) ζ(s ,··· ,s ) = f 1 k
1 k (cid:2)u ,···,u (cid:3)
1 k
u1,X···,uk≥1
and on the other hand satisfy the shuffle relation. The shuffle relation of MZVs then follows by
summingoverthethe us.
i
TogeneralizetheshufflerelationofMZVstoCZVs,weinterprettheshufflerelationofmultiple
zeta fractions geometrically, starting from the simple observation that relations among fractions
such as
1 1 1
+ =
u (u +u ) u (u +u ) u u
1 1 2 2 1 2 1 2
are related to subdivisionsof cones, here the cone x ≥ 0,x ≥ 0 is seen as a union of the cones
1 2
x ≥ x ≥ 0and x ≥ x ≥ 0. Moregenerally,weinterpretlinearrelationsamongsimplefractions
1 2 2 1
i.e., of the type 1 where L ,··· ,L are linear independent linear forms, to subdivisions of
L1L2···Lk 1 k
simplicial cones. A differentiation procedure then further relates fractions 1 with s ∈ N,
Ls1Ls2···Lsk i
1 2 k
towhatwecallalgebraicsubdivisionsofdecoratedconesobtainedfromdifferentiatinggeometric
subdivisions of the underlying geometric cones. Such fractions arise as Laplace transforms: for
example, if L = u + ···+ u , for s > 1,s ≥ 1,··· ,s ≥ 1, we observe that the multiple zeta
i i k 1 2 k
fractions
1
s ,··· ,s
f 1 k =
(cid:2)u1,··· ,uk(cid:3) Ls1Ls2 ···Lsk
1 2 k
can be written as differentiated Laplace transforms on the closed cone x ≥ ··· ≥ x ≥ 0 (see
1 k
Proposition(4.8)fornotations):
1 1
f(cid:2)us11,,······ ,,uskk(cid:3) = (s1 −1)!···(sk −1)!∂Ls1∗1−1···∂Lsk∗k−1L1 ···Lk
1
= (s1 −1)!···(sk −1)!∂Ls1∗1−1···∂Lsk∗k−1Zx1>···>xk>0e−Pki=1xiui dx1···dxk.
This is the starting point for our generalization of the double shuffle relation among zeta values
associatedwithmoregeneral cones. Eventuallyweobtainageometricinterpretationofthecom-
mutative diagram in Eq. (2) and generalize it to a commutative diagram in Eq. (30) of double
subdivisionrelationforCZVs.
1.3. Layoutofthepaper. Aftersummarizingconceptsandbasicfactsonconvexcones,wegive
in Section 2 thedefinitionofconical zetavaluesand theiropen subdivisionrelation as ageneral-
izationofthestuffle(quasi-shuffle)relationofMZVs. Inordertogeneralizetheshufflerelationof
MZVstoCZVs, wegeneralizetheshufflerelationofmultiplezetafractions toasuitablerelation
for a much larger class of fractions derived from CZVs by means of a differentiation procedure
4 LIGUO,SYLVIEPAYCHA,ANDBINZHANG
similar to the one described above in the case of MZVs. We achieve this in two steps. In Sec-
tion3,werelateviaabijectionclosedconesmodulosubdivisionstosimplefractions. Thuslinear
relations among simple fractions are precisely those coming from subdivisionsof closed simpli-
cial cones. By means of the natural differential structureon fractions, in Section 4 we inferfrom
thisbijectionaonetoonecorrespondencebetweendecoratedconesmodulosubdivisionandpure
fractions. This correspondence between cones and fractions is applied in Section 5 to provide
closed subdivision relations of CZVs when expressed as Shintani zeta values. In doing so, the
shuffle product ofmultiplezeta fractions seen as decompositions offractions withlinearpoles is
reflectedgeometricallyassubdivisionsoftheclosedChencones. Combiningtheopenandclosed
subdivisionrelationswiththeconceptofconepairsgivesthedoublesubdivisionrelationofCVZs
that generalizes the doubleshuffle relation ofMZVs. Finally it is shownthat CVZs and Shintani
zetavaluesspan thesamelinearspace.
In this paper we shall not touch on divergent conical zeta values, which will be the subject of
a forthcoming paper. Divergent MZVs, which in recent years have been studied in the algebraic
frameworkofConnesandKreimer[7]inspiredbythemethodofrenormalizationofquantumfield
theory,canbedefinedusingseveralapproachessuchas[2,12,21]. Inthisforthcomingpaper,we
construct a coalgebra structure on cones and as an application, we show that renormalization of
conical zetavaluesrecoversthelocalEuler-Maclaurin formula[2, 10].
2. Convex cones andconical zetavalues
2.1. Polyhedralcones. Wefirstcollectbasicnotationsandfacts(mostlyfollowing[8]and[27])
on cones that will be used in this paper. Let K ⊆ R be a field and let k ≥ 0 be an integer. In
practice K isusuallythefield Q ofrationalnumbers.
(a) A closed(polyhedral) cone(resp. An open (polyhedral)cone) in Kk istheconvexset
(5) hv ,··· ,v ic := K v +···+K v ,
1 n ≥0 1 ≥0 n
(resp. hv ,··· ,v io := K v +···+K v ), v ∈ Kk ,1 ≤ i ≤ n.
1 n >0 1 >0 n i ≥0
(b) A cone is always taken to be a closed or open polyhedral cone. In particular, the term
polyhedral will be omitted in this paper and we sometimes write hv ,··· ,v i when the
1 n
closednessortheopennessdoesnotplay anyrole.
(c) ForaconeC = hv ,··· ,v iin Kk and afield L ⊆ R, letC(L) denote L v +···+L v if
1 n ≥0 1 ≥0 n
C isclosedand L v +···+L v ifC isopen.
>0 1 >0 n
(d) Theset{v ,··· ,v }inthedefinitionofaconeiscalledthegeneratingsetorthespanning
1 n
set of the cone. The dimension of the K-linear subspace generated by the cone is called
itsdimension.
(e) A closed cone in Kk can also be described as the intersection ∩H of finitely many half
i ui
spaces H = {x ∈ Kk |u(x) ≥ 0}definedbylinearfunctionalsu with K-coefficients on Kk
ui i i
(seee.g. Theorem 1.3in[27]).
(f) Let Cc(K) (resp. Co(K)) denote the set of closed (resp. open cones) in Kk, k ≥ 1. For
k k
k = 0 we set Cc(K) = {0} (resp. Co(K) = {0}) by convention. The natural inclusions
0 0
Cc(K) → Cc (K) (resp. Co(K) → Co (K)) induced by the natural inclusion Kk → Kk+1,
k k+1 k k+1
giveriseto thedirect limitset Cc(K) = limCc(K) (resp. Co(K) = limCo(K)).
k k
−→ −→
(g) A simplicialcone ora simplex coneisa conespannedbylinearly independentvectors.
(h) A cone in Qk is called a rational cone. Thus a rational cone is spanned by vectors in Qk
(equivalentlyinZk).
CONICALZETAVALUESANDTHEIRDOUBLESUBDIVISIONRELATIONS 5
(i) A smooth cone is a rational cone with a spanning set that is a part of a basis of Zk ⊆ Rk.
In thiscase, thespanningset isuniqueand is calledtheprimary setofthecone.
(j) A coneiscalled strongly convex ifitdoes notcontainanylinearsubspace.
(k) A face of a closed cone hv ,··· ,v ic in Kk is a subset of the form hv ,··· ,v ic ∩{u = 0},
1 n 1 n
where u : Kk → K is a linear function with K-coefficients which is non-negative on
hv ,··· ,v ic.
1 n
(l) A face of an open cone hv ,··· ,v io in Kk is an open cone of the form hv ,··· ,v io(K)
1 n i1 ir
where hv ,··· ,v ic(K) isaface ofhv ,··· ,v ic.
i1 ir 1 n
(m) Aface F ofaconeC isagainaconeandwewrite F ≤ C. IfF isaproperfaceofaconeC
we write F < C. A 1-dimensional face is called an edge. A codimension 1 face is called
afacet.
(n) For ~x = (x ,··· ,x ) and~y = (y ,··· ,y ) in Rk, let (~x,~y) denote the inner product x y +
1 k 1 k 1 1
···+ x y . Throughthisinnerproduct, Rk is identifiedwithitsowndualspace (Rk)∗.
k k
2.2. Subdivision of cones. In this subsection, we recall some facts about subdivisions of poly-
hedral cones. Forthesakeofcompleteness,weprovideproofs forsomeoftheresults.
Definition 2.1. (a) A subdivision of a closed cone C ∈ Cc(K) is a set {C ,··· ,C } ⊆ Cc(K)
k 1 r k
such that
(i) C = ∪r C,
i=1 i
(ii) C ,··· ,C havethesamedimensionasC and
1 r
(iii) intersect alongtheirfaces i.e., C ∩C is afaceofbothC andC .
i j i j
(b) A subdivisionofan openconeC is theset oftherelativeinteriorsoftheclosedcones in
(6) {∩t D | {i ,··· ,i} ⊆ {1,··· ,r},1 ≤ t ≤ r}
j=1 ij 1 t
where {D ,··· ,D} is a subdivision of the closure C of C. By convention, the relative
1 t
interiorof {0}is {0}.
(c) A subdivision of a rational cone is called smooth if all the cones in the subdivision are
smooth.
Proposition2.2. (a) Any cone in Kn can be subdivided into stronglyconvex simplicial cones
in Kn.
(b) Anystronglyconvexsimplicialrationalconecan besubdividedintosmoothcones.
Proof. (a) By taking the intersections with coordinate orthants, we can assume that the cone is
stronglyconvex.
Now for a strongly convex cone C in Kn, we take its barycenter type subdivision built as
follows. Inthefollowingweidentifyapoint M inKn withthevectorO~M = v. Foreachface F of
C,takeavectorv ∈ Kn intherelativeinteriorof F. Notethat,sinceF(Q) ⊇ F andF(Q)isdense
F
in F(R), such a vectoralways exists. Let n = dim(C). If theconeC is open, then the open cones
hv ,···v io ⊂ C with F < ··· < F ,0 ≤ ℓ ≤ n, are simplicialand intersect along theirfaces. If
F1 Fℓ 1 ℓ
theconeC is closed, then the closed cones hv ,··· ,v ic with F < ··· < F , are simplicialand
F1 Fn 1 n
intersectalongtheirfaces. Thustoprovethatthisgivesasubdivision,weonlyneed toprovethat
theunionofthesecones isC.
First we prove this for a closed cone C. We proceed by induction on the dimension n of C.
Sincethecasen = 1istrivial,weassumethatn ≥ 2. ForanyvectorvinC,ifvisamultipleofthe
vector v chosen in the relative interior of C as above, then we have the conclusion. Otherwise
C
the vectors v and v span a 2-dimensional linear space V. Let K v and K v be the outmost
C ≥0 1 ≥0 2
6 LIGUO,SYLVIEPAYCHA,ANDBINZHANG
intersections. Thenbyconvexity,hv ,v iisinC(K). Bythechoiceofv andv ,hu ,u i = V ∩C.
1 2 1 2 1 2
Thustheintersectionof V withtheboundaryofC arethetworays K u and K u . Thevectors
≥0 1 ≥0 2
v , v lie in some facets since the boundary of C is the union of its facets. Then v is a non-
1 2
negativeK-linearcombinationofv withoneofthetwovectorsv orv ,sayv . Bytheinduction
C 1 2 1
hypothesis,v isinhv ,··· ,v ic withF < ··· < F where F isafacetofC. Thenvisin
1 F1 Fn−1 v1 vn−1 vn−1
oneofthesimplicialsubdivisionshv ,··· ,v ,v ic(K).
F1 Fn−1 C
For an open cone, the proof is similar. The only difference is that the intersection rays K v
>0 1
and K v may be in the interior of some lower dimensional faces of C. Let v be a positive K-
>0 2
linearcombinationofv andv asintheclosedconecase. Thenbytheinductionhypothesis,v is
C 1 1
inhv ,··· ,v io withF < ··· < F anddimF ≤ n−1. Thusvisinthesimplicialsubdivisions
F1 Fr v1 vr vr
hv ,··· ,v ,v io.
F1 Fr C
(b)See thesecond exerciseon page48of[8]. (cid:3)
Lemma 2.3. (a) Forafamilyofclosedcones{C}in Kk,1 ≤ i ≤ m,thatspanthesamelinear
i
subspaceof Kn, there is a simplicialsubdivision{C } in Kk ofC such that any two ofC
ij i ij
eithercoincideoronlyintersectalongtheirfaces.
(b) For a family of rationalclosed cones {C}, 1 ≤ i ≤ m, that span the same linear subspace
i
of Qn, there is a smooth subdivision {C } of C such that every two of C either coincide
ij i ij
oronlyintersectalongtheirfaces.
Proof. (a) Each closed cone C can be written as an intersection of half hyperplanes H , j =
i ui
j
1,··· ,s
j
si
C = H ,1 ≤ i ≤ m.
i ui
j
j=1
\
Denote
S := {(i, j)|1 ≤ i ≤ m, 1 ≤ j ≤ s},
i
and
F := {ν : S → {1,−1}|∃i ,ν(i , j) = 1,∀j}.
0 0
Each elementν ∈ F defines a set
m si
C := H ,
ν ν((i,j))ui
j
i=1 j=1
\\
whichis aconethoughitmaybetrivial.
Foreach 1 ≤ i ≤ m, considertheset
0
F := {ν ∈ F|ν(i , j) = 1 forall j and dimC = n}.
i0 0 ν
ThenC = C . Furtherfor ν,µ ∈ ∪m F ,we have
i0 ν∈Fi0 ν i=1 i
S
C ∩C = H ∩H = H ui = 0 .
µ ν µ(i,j)ui ν(i,j)ui µ(i,j)ui j
j j j! !
(i,j)∈S µ(i,j)=ν(i,j) µ(i,j),ν(i,j)
\ \ \ \ (cid:8) (cid:9)
WenotethatthefacesofaclosedconeC := ∩k H areoftheform H ∩ {u = 0}
ℓ=1 uℓ ℓ∈K′ uℓ ℓ∈K′′ ℓ
forapartition K′ ⊔K′′ = [k]. ThusC ∩C is afaceofbothC andC .
µ ν ν µ
Therefore, for 1 ≤ i ≤ m, the set {C |ν ∈ F } is a subdivisionof(cid:0)CTand any(cid:1)tw(cid:0)To µ,ν ∈ ∩m F(cid:1),
ν i i i=1 i
C andC either coincide oronly intersect along their faces. By subdividingeach C ,ν ∈ ∪m F
µ ν ν i=1 i
into simplicial cones applying Proposition 2.2.(a), we obtain the simplicial subdivisionsofC in
i
theproposition.
CONICALZETAVALUESANDTHEIRDOUBLESUBDIVISIONRELATIONS 7
(b)Taking K = QinItem(a),andapplyingProposition2.2.(b)tofurthersubdivideeachC there
ij
intosmoothcones, weachievethedesiredsubdivision. (cid:3)
2.3. Conical zeta values and open subdivisionrelations. We now introduceourmain concept
ofstudyinthispaper.
Definition2.4. LetC beanopenconeinRk andlet~s ∈ Ck. Definetheconicalzeta functionby
≥0
1
(7) ζ(C;~s) = ,
ns1 ···nsk
(n1,···X,nk)∈C∩Zk 1 k
ifthesumconverges. Herewehaveusedtheconventionthat 0s = 1 forany s.
Note that with this convention, the definition of ζ(C;~s) does not depend on the integer k such
that C ⊆ Rk and ~s ∈ Ck. Thus we can use ζ(C;~s) without referring to k. When s ,··· ,s are
≥0 1 k
taken to be integers, we call ζ(C;~s) a conical zeta value (CZV). Sometimes we use the name
open conical zeta value also. Let CZVo denotetheset of convergent conical zetavalues. Define
thespaceQCZVo tobethespaceofconvergent conical zeta values overQ.
Lemma 2.5. LetC bean openconein Rk . For~s ∈ Zn with s ≥ 2,ζ(C;~s)is convergent.
≥0 i
Proof. LetC bethefirst coordinateorthant Rn . Then
≥0
k
ζ(C;~s) = ζ(s)
i
i=1
Y
and hence is convergent if s ≥ 2 for 1 ≤ i ≤ n. Then the statement holds for any open cone
i
C ⊆ Rn sinceζ(C;~s) ≤ ζ(Zn ;~s). (cid:3)
≥0 ≥0
An (open orclosed) Chen cone ofdimensionk is a (open orclosed)coneC spanned by the
k,σ
vectors{e ,e +e ,··· ,e +···+e }where{e ,··· ,e }isthestandardbasisofZn and
σ(1) σ(1) σ(2) σ(1) σ(k) 1 n
σ ∈ S , S is the symmetric group on {1,··· ,n}. Let C denote the standard (open or closed)
n n k
Chen conespanned by{e ,e +e ,··· ,e +···+e }.
1 1 2 1 k
Proposition2.6. Foranyopen Chen coneC , k ≥ 1,σ ∈ S , we have
k,σ n
ζ(C ;s ,··· ,s ) = ζ(s ,··· ,s ),
k,σ 1 n σ(1) σ(k)
where therighthandsideisthemultiplezetavalue.
Thereforethespace QMZVspanned byMZVsoverQ is asubspaceofQCZVo.
Proof. An elementofC ∩Zn isoftheform
k,σ
a e +a (e +e )+···+a (e +···+e )
1 σ(1) 2 σ(1) σ(2) k σ(1) σ(k)
=(a +···+a )e +(a +···+a )e +···+a e ,
1 k σ(1) 2 k σ(2) k σ(k)
where a ∈ (0,∞),1 ≤ i ≤ k. Hence
i
1
ζ(C ;s ,··· ,s ) = = ζ(s ,··· ,s ).
k,σ 1 n a1,X···,ak≥1 (a1 +···+ak)sσ(1)···aksσ(k) σ(1) σ(k)
(cid:3)
FromthedefinitionofCZVs, wederivethefollowinglemma.
8 LIGUO,SYLVIEPAYCHA,ANDBINZHANG
Lemma 2.7. Let {C} bea familyof opencones thatforma subdivisionof anopen coneC, then
i i
(8) ζ(C;~s) = ζ(C;~s).
i
i
X
Thisis calledan opensubdivisionrelationof CZVs.
We next show that open subdivision relations of open Chen cones recover the quasi-shuffle
relationsofMZVs. Firstrecall thequasi-shuffleencodingofMZVs. Define
(9) H∗ := Q1⊕ z ···z ,
0 s1 sk
s1,···,skM≥1,s1≥2,k≥1
withthequasi-shuffleproduct∗whichisdefinedrecursivelybutcanalsobedefinedbythestuffle
productas follows. Define
ϕ : [k] → [m],ψ : [ℓ] → [m]areorderpreserving,
St = (ϕ,ψ)
k,ℓ injectiveand im(ϕ)∪im(ψ) = [m]
(cid:26) (cid:27)
(cid:12)
and ϕ−1(i) = 0 whenϕ−1(i) =(cid:12)∅. Then
(cid:12)
(10) z ···z ∗z ···z = z ···z .
s1 sk sk+1 sk+ℓ sϕ−1(1)+sk+ψ−1(1) sϕ−1(m)+sk+ψ−1(m)
(ϕ,ψX)∈Stk,ℓ
Then thequasi-shuffleencodingofMZVsisgivenby thealgebrahomomorphism
ζ∗ : H∗ → QMZV, z ···z 7→ ζ(s ,··· ,s ),
0 s1 sk 1 k
namely,
ζ(s ,··· ,s )ζ(s ,··· ,s ) = ζ∗(z ···z ∗z ···z ).
1 k k+1 k+ℓ s1 sk sk+1 sk+ℓ
Welikewisegivean openconeencodingofCZVs.
Definition 2.8. (a) Let DCo be the set of decorated open cones consisting of pairs (C;~s)
whereC ⊆ Qk isan open rationalconeand ~s ∈ Zk .
≥0 ≥0
(b) Let {C} be an open subdivision of C. Then we also call {(C;~s)} an open subdivisionof
i i i i
(C;~s)and denoteit by
(C;~s) ≺ (C;~s).
i
i
X
(c) DenoteDCo forthesubsetof DCo such thatζ(C;~s)isconvergent.
0
(d) Define thelinearmap
(11) ζo : QDCo → QCZVo, (C;~s) 7→ ζ(C;~s).
0
(e) Let DCHo (resp. DCHo) denote the subset of DCo of decorated cones (C,~s) with under-
0
lyingconeC an open Chen cone(resp. thatgiveconvergentCZVs).
By definition, ζo(C;~s) = ζ(C;~s). Thus the two notations will be used interchangably; the
notationζo(C;~s)willbeusedto stressthemap ζo.
Thenwehavethebijection
H∗ → QDCHo,z ···z 7→ (he ,··· ,e +···+e io;s ,··· ,s ),
0 0 s1 sk 1 1 k 1 k
completingthefollowingcommutativediagramoflinearmaps.
CONICALZETAVALUESANDTHEIRDOUBLESUBDIVISIONRELATIONS 9
(12) H∗ // // // QDCHo(cid:31)(cid:127) // QDCo
0 0 o
ζ∗ ζo
(cid:15)(cid:15) (cid:15)(cid:15)
QMZV(cid:31)(cid:127) // QCZVo
In thiscontext,weobtainthefollowing
Theorem 2.9. (a) The quasi-shuffleproductin Eq.(10)correspondstotheopensubdivision
(13) he ,··· ,e +···+e ,e ,··· ,e +···+e io
1 1 k k+1 k+1 k+ℓ
= he +e ,··· ,e +e +···+e +e io.
ϕ−1(1) k+ψ−1(1) ϕ−1(1) k+ψ−1(1) ϕ−1(m) k+ψ−1(m)
(ϕ,ψa)∈Stk,ℓ
inthesensethatthecompositioninthetopofEq.(12)sendstherighthandsideofEq.(10)
totherighthandsideof Eq.(13).
(b) Thequasi-shuffleproductofMZVs:
ζ(s ,··· ,s )ζ(s ,··· ,s ) = ζ∗(z ···z ∗z ···z )
1 k k+1 k+ℓ s1 sk sk+1 sk+ℓ
coincideswith thesubdivisionof CZVs:
ζo(he ,··· ,e +···+e ,e ,··· ,e +···+e io;s ,··· ,s ,s ,··· ,s )
1 1 k k+1 k+1 k+ℓ 1 k k+1 k+ℓ
= ζo(he +e ,··· ,e +e +···+e +e io;s ,··· ,s ).
ϕ−1(1) k+ψ−1(1) ϕ−1(1) k+ψ−1(1) ϕ−1(m) k+ψ−1(k) 1 m
(ϕ,ψX)∈Stk,ℓ
Forexample,theclosedconesubdivision
he ,e ic = he ,e +e ic ∪he ,e +e ic
1 1 1 1 2 2 1 2
by Chen cones givestheopen conesubdivision
(he ,e io;s ,s ) = (he ,e +e io;s ,s )⊔(he ,e +e io;s ,s )⊔(he +e io;s ,s )
1 2 1 2 1 1 2 1 2 2 1 2 1 2 1 2 1 2
whichrecovers thequasi-shufflerelation
z ∗z = z z +z z +z .
s1 s2 s1 s2 s2 s1 s1+s2
Indeed wehave
ζ(s )ζ(s ) = ζo(he ,e io,s ,s )
1 2 1 2 1 2
= ζo(he ,e +e io,s ,s )+ζo(he ,e +e io,s ,s )+ζo(he +e io,s ,s )
1 1 2 1 2 2 1 2 1 2 1 2 1 2
= ζ(s ,s )+ζ(s ,s )+ζ(s + s ).
1 2 2 1 1 2
3. Closedcones andsimplefractions
As noted in the introduction, we will establish a class of relations of CZVs that generalizes
the shuffle relation of MZVs. Motivated by the approach of multiple zeta fractions outlined in
the introduction, we first relate CZVs to a class of fractions and generalize the shuffle relation
of multiple zeta fractions to this class of fractions. Our geometric approach of generalizing the
shuffle relation consists in encoding all linear relations of these CZV fractions as subdivision
relations of closed cones from the CZVs. Thus we now make a digression of our discussion of
CZVs to relate closed cones with a family of rational functions, which we call simple fractions.
Under this correspondence we show that linear relations among simple fractions have a natural
10 LIGUO,SYLVIEPAYCHA,ANDBINZHANG
geometricinterpretationassubdivisionsofclosedcones. Thiscorrespondencewillbegeneralized
todecorated conesin thenextsection.
3.1. From closedcones to simplefractions.
Definition 3.1. Let K be a subfield of R. Let z,i ≥ 1 be a countable set of variables and let
i
~z = (z) . A simple fraction with coefficients in K is a fraction of the form 1 , where
L ,··· ,iLi≥1∈ K[~z]arelinearlyindependentlinearfunctions. Let S(K) betheK-linearLs1·u··Lbkspaceof
1 k
thequotientfield K(~z)of K[~z]generated by simplefractions withcoefficients in K.
Let KCc(K) denote the K-vector space spanned by Cc(K). We will define a natural map from
KCc(K)toS(K). Lawrencein[19](seealso[9])constructedasimilarmapbasedonthevaluation
property. Our map generalizes Lawrence’s map in so far as it takes non-zero values for lower
dimensionalcones inlargedimensionalspaces.
Let C be a closed simplicial cone in Kn with linearly independent generators v ,···v . Let
≥0 1 k
n
e,1 ≤ i ≤ n, be the standard basis of Kn. For 1 ≤ i ≤ k, let v = a e ,a ∈ K. Define linear
i i ji j ji
j=1
n P
functions L = L = a z andlet A = [a ]denotetheassociatedmatrixin M (K) withv as
i vi ji j C ij n×k i
j=1
columnvectors. LetPw(v ,··· ,v ) or w(C) denotethesum of absolutevalues ofthe determinants
1 k
ofallminorsof A ofrank k. Thendefine
C
w(v ,··· ,v )
(14) Φ (C) := 1 k .
n L ···L
1 k
Thisdefines amap Φ fromtheset Cc(K) ofclosedsimplicialcones in Kn toS(K).
n ≥0
Lemma 3.2. (a) LetC = hv ,··· ,v ic bea closedconeof ranknin Kn,then
1 n
Φ (C) = (−1)n ··· exp(x z +···+ x z )dx ···dx ,
n 1 1 n n 1 n
Z ZC(R)
whereC(R) istheR -linearspanofC and~zis anyelement in
≥0
Cˇ− := {~y| (~y,c) < 0,∀c ∈ C(R)}.
(b) Let C be a closed simplicial cone in Kk and let {C ,··· ,C } be a closed subdivision of C
1 r
r
intoclosedsimplicialconesC ,··· ,C in Kk. Then Φ (C) = Φ (C ).
1 r n n i
i=1
P
Proof. (a) SinceC isstronglyconvex,Cˇ− isa coneofrank n. Withournotation,wehave
(v ,··· ,v ) = (e ,··· ,e )A, (L ,··· ,L ) = (z ,··· ,z )A.
1 n 1 n 1 n 1 n
Anypoint xe inC can beuniquelyexpressedas yv throughachangeofvariables
i i i i
(x ,··· ,x )T = A(y ,··· ,y )T.
P 1 n 1P n
Thusweobtain
··· exp(x z +···+ x z )dx ···dx
1 1 n n 1 n
Z ZC(R)
+∞ +∞
= ··· exp (z ,··· ,z )A(y ,··· ,y )T |det(A)|dy ···dy
1 1 1 n 1 n
Z0 Z0
(cid:0) (cid:1)