On the generators of a certain algebra of q-multiple zeta values - - PowerPoint PPT Presentation

On the generators of a certain algebra

• f q-multiple zeta values

Ulf Kühn - Universität of Hamburg Zeta Values, Modular Forms and Elliptic Motives II ICMAT Madrid, December 5, 2014 joint work with Henrik Bachmann

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2 / 34

Multiple zeta values

Definition

For natural numbers s1 ≥ 2, s2, ..., sl ≥ 1 the multiple zeta value (MZV) of weight s1 + ... + sl and length l is defined by

ζ(s1, ..., sl) =

• n1>...>nl>0

1 ns1

1 . . . nsl l

.

The rules for the product of infinite sums imply that the product of MZV can be expressed as a linear combination of MZV with the same weight (stuffle product). MZV can be expressed as iterated integrals. This gives another way (shuffle product) to express the product of two MZV as a linear combination of MZV. These two products give a large number of Q-linear relations (extended double shuffle relations) between MZV. Conjecturally these are all relations between MZV, e.g.

ζ(2, 3) + 3ζ(3, 2) + 6ζ(4, 1)

shuffle

= ζ(2) · ζ(3)

stuffle

= ζ(2, 3) + ζ(3, 2) + ζ(5) .

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Dimension conjectures for MZ

Broadhurst-Kreimer Conjecture

The Q-algebra MZ of multiple zeta values is a free polynomial algebra, which is graded for the weight and filtered for the depth ("depth drop for even zetas"). The numbers gk,l of generators in weight k ≥ 3 and depth l are determined by

BK(x, y) =

• k,l

dimQ

• grW,D

k,l

MZ

• xkyl =

1 + E(x)y 1 − O(x)y + S(x)y2 − S(x)y4 =

• 1 + E(x) y
• k≥3,l≥1

1

• 1 − xkylgk,l

with

E(x) = x2 1 − x2 = x2 + x4 + x6 + ... "even zetas", O(x) = x3 1 − x2 = x3 + x5 + x7 + ... "odd zetas", S(x) = x12 (1 − x4)(1 − x6) = x12 + x16 + x18 + ... "cusp forms".

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Dimension conjectures for MZ

Zagier’s Conjecture

The following identities hold:

Zag(x) =

• k

dimQ

• grW

k MZ

• xk =

1 1 − x2 − x3 .

Zagier’s conjecture is implied by Broadhurst-Kreimer’s conjecture. In order to neglect the depth we just have to set y = 1 and get

Zag(x) = BK(x, 1) = 1 + E(x) 1 − O(x) = 1 +

x2 1−x2

1 −

x3 1−x2

= 1 1 − x2 − x3 .

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Dimension conjectures for MZ

Theorem (Euler, Ihara-Kaneko-Zagier, Goncharov, Ihara-Ochiai, Brown, ...)

Let l ≤ 3, then the numbers gk,l of generators for MZ of weight k and length l are not bigger than implied by the Broadhurst-Kreimer conjecture, i.e. we have

• k

gk,1 xk ≤ x2 + x3 1 − x2 ,

• k

gk,2 xk ≤

• k>0

k even

k − 2 6

• xk =

x8 (1 − x2)(1 − x6),

• k

gk,3 xk ≤

• k>0

k odd

(k − 3)2 − 1 48

• xk =

x11(1 + x2 − x4) (1 − x2)(1 − x4)(1 − x6).

Idea of proof: For l = 1 this is a trivial consequence of Euler’s formula for even zetas. For

l = 2, 3 we bound the number of generators by the dimension of the double shuffle spaces DS(k − l, l).

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double shuffle space for MZ

There is a unique way to define ζ

∃

(s1, ..., sl) for all (s1, ..., sl) ∈ Nl ("shuffle regularised

MZV") such that the generating series

F

∃ l (x1, ..., xl) =

• (s1,...,sl)∈Nl

ζ

∃

(s1, ..., sl) xs1−1

1

· ... · xsl−1

l

,

with notation f ♯(x1, ..., xj) = f(x1 + ... + xj, ..., xj−1 + xj, xj) satisfies

• F

∃ j

♯(x1, ..., xj)

• F

∃ l−j

♯(xj+1, ..., xl) =

• F

∃ n

♯

• shj(x1, ..., xl),

(1) where shj denotes the set of all shuffles of type (j, l − j). If we consider (1) modulo products, then (assuming MZV are graded for the weight) we obtain for each generator of weight k and depth l a homogenous polynomial in the set

{f ∈ Q[x1, ..., xl]

• deg f = k − l and f ♯|shj(x1, ..., xl) = 0 ∀ 1 ≤ j ≤ l − 1}

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double shuffle space for MZ

Analogously there are stuffle regularised MZV and their generating series

F ∗

l (x1, ..., xl) =

• (s1,...,sl)∈Nl

ζ∗(s1, ..., sl) xs1−1

1

· ... · xsl−1

l

satisfies (ignoring the terms with MZV of depth less than l)

F ∗

j (x1, ..., xj)F ∗ l−j(xj+1, ..., xl−j) = F ∗ l

• shj(x1, ..., xl) + ...

(2) If we consider (2) modulo products and terms of lower depth, we obtain for each generator a homogenous polynomial in the set

{f ∈ Q[x1, ..., xl]

• deg f = k − l and f|shj(x1, ..., xl) = 0 ∀ 1 ≤ j ≤ l − 1}

Because of ζ

∃

(s1, ..., sl) ≡ ζ∗(s1, .., sl) modulo lower depth, we get Proposition

Let the double shuffle space be defined by

DS(k − l, l) = {f ∈ Q[x1, ..., xl]

• deg f = k − l,

f ♯|shj = f|shj = 0 ∀ j} ,

then we get for the number gk,l of algebra generators weight k and depth l for MZ the bound

gk,l ≤ dimQ DS(k − l, l).

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generators for MZ in l=2: double shuffle space

The spaces DS(k − 2, 2) are spanned by polynomials such that

f(x1 + x2, x2) + f(x1 + x2, x1) = 0 and f(x1, x2) + f(x2, x1) = 0 ⇐ ⇒ f(x1, x2) = −f(x1, x1 − x2) and f(x1, x2) = −f(x2, x1).

Let ptp−1 :=

1 0

1 −1

• and t = ( 0 1

1 0 ) and G = ptp−1, t. Then

DS(k − 2, 2) = Q[x1, x2]G

k−2,

where of the action of G on Q[x1, x2] is determined above. Now by Molien’s theorem the numbers of invariant polynomials of given degree is determined by

HQ[x1,x2]G(x) = 1 12

• g∈G

det(g) det(1 − g · x) = x6 (1 − x2)(1 − x6) = x6 + x8 + x10 + 2x12 + 2x14 + ...

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generators for MZ in l=3: double shuffle space

Let t =

0 0 1

0 1 0 1 0 0

• , p =

1 1 1

1 1 0 1 0 0

• and set H = t, ptp−1, −1, then

DS(k − 3, 3) ⊂ Q[x1, x2, x3]H

Let c2 =

0 1 0

1 0 0 0 0 1

• , c3 =

0 1 0

0 0 1 1 0 0

• and G = t, ptp−1, c3. The main idea is to interprete the

shuffle sh1 = 1 + c2 + c3, as the relative Reynolds operator from H to G, this is due to the key identity tc2 = c−1

3

. Then we get an exact sequence

0 → DS(·, 3) → Q[x1, x2, x3]H

|sh1⊕|p sh1 p−1 Q[x1, x2, x3]G ⊕ Q[x1, x2, x3]Gp → 0

where Gp = t, ptp−1, pc3p−1. Again by Molien’s theorem we get the result. Remark: The cases l ≥ 4 are only partially understood so far.

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bi-brackets

Definition

For n ∈ N and natural numbers r1, . . . , rl ≥ 0, s1, . . . , sl > 0 we call

σ s1,...,sl

r1,...,rl (n) :=

• u1v1+···+ulvl=n

u1>···>ul>0 v1,...,vl>0

ur1

1 vs1 1 . . . url l vsl l

the bi-multiple divisor sum of n with bi index s1,...,sl

r1,...,rl and with

κ := r1!(s1 − 1)! . . . rl!(sl − 1)! we define their generating series as s1, . . . , sl r1, . . . , rl

• := 1

κ ·

• n∈N

σ s1−1,...,sl−1

r1 ,..., rl

(n) qn .

For short hand we refer to this q-series as bi-brackets of length l and of weight

w = s1 + · · · + sl + r1 + · · · + rl.

Its upper weight is given by s1 + · · · + sl and lower weight equals r1 + · · · + rl.

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bi-brackets

The bi-brackets can also be written as

s1, . . . , sl r1, . . . , rl

• = c ·
• n1>···>nl>0

nr1

1 Ps1−1(qn1) . . . nrl l Psl−1(qnl)

(1 − qn1)s1 . . . (1 − qnl)sl ,

where the Pk−1(t) are the Eulerian polynomials defined by

Pk−1(t) (1 − t)k = Li1−k(t) =

• d>0

dk−1td .

Examples:

P0(t) = P1(t) = t , P2(t) = t2 + t , P3(t) = t3 + 4t2 + t , 1, 1 0, 1

• =
• n1>n2>0

qn1n2qn2 (1 − qn1)(1 − qn2) , 4, 2, 1 2, 0, 5

• =

1 3! · 2! · 5!

• n1>n2>n3>0

n2

1(q3n1 + 4q2n1 + qn1) · qn2 · n5 3qn3

(1 − qn1)4 · (1 − qn1)2 · (1 − qn1)1 .

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multiple divisor sums and modular forms

For r1 = · · · = rl = 0 we also write

s1, . . . , sl 0, . . . , 0

• = [s1, . . . , sl] =:

1 (s1 − 1)! . . . (sl − 1)!

• n>0

σs1−1,...,sl−1(n)qn .

We call the coefficients σs1−1,...,sl−1(n) multiple divisor sums and their generating series

[s1, . . . , sl] will be called brackets. In the case l = 1 we get the classical divisor sums σk−1(n) =

d|n dk−1 and

[k] = 1 (k − 1)!

• n>0

σk−1(n)qn .

These function appear in the Fourier expansion of classical Eisenstein series which are (quasi)-modular forms for SL2(Z), for example

G2 = − 1 24 + [2] , G4 = 1 1440 + [4] , G6 = − 1 60480 + [6] .

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BD and MD

Definition

By BD we denote the Q-vector space spanned by all bi-brackets and 1 and by MD ⊆ BD we denote the Q-vector space spanned by all brackets and 1. The (bi)-brackets have a direct connection to multiple zeta values, since they are q-multiple zeta values:

Theorem (B.-K. , Zudilin)

Assume that s1 > r1 + 1 and sj ≥ rj + 1 for j = 2, .., l. Then

lim

q→1

• 1 − q

s1+...+sl s1, ..., sl r1, ..., rj

• =

1 r1! · ... · rl!ζ(s1 − r1, ..., sl − rl).

Remark: Another very interesting connection to MZV is given by the Fourier expansion of multiple Eisenstein series.

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bi-brackets - connections to mzv

In BD we have the sub algebra of admissible brackets given by

qMZ :=

• [s1, . . . , sl] ∈ MD
• s1 > 1
• Q.

It has a filtration given by the weight k = s1 + · · · + sl. For [s1, . . . , sl] ∈ FilW

k (qMZ)

define the map Zk by

Zk ([s1, . . . , sl]) = lim

q→1(1 − q)k[s1, . . . , sl] =

• ζ(s1, . . . , sl) if k = s1 + ... + sl

0 else , Proposition

The linear map Zk induces a surjection

Zk : FilW

k (qMZ) → grW k (MZ).

Therefore relations in qMZ give rise to relations between MZV and conversely all relations between MZV lift to qMZ. Example:

[4] = 2[2, 2] − 2[3, 1] + [3] − 1 3[2]

Z4

= ⇒ ζ(4) = 2ζ(2, 2) − 2ζ(3, 1) .

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bi-brackets - connections to mzv - example I

To get the first relation ζ(2, 1) = ζ(3) between MZV by using bi-brackets one considers the double shuffle relation for [1] · [2]. It is:

[1, 2] + 2[2, 1] − [2] + 2 1

• sh

= [1] · [2]

st

= [1, 2] − 1 2[2] + [2, 1] + [3]

and therefore

[2, 1] + 2 1

• = [3] + 1

2[2] .

Since [2],

2

1

• ∈ ker Z3 one obtains this relation by applying Z3.

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bi-brackets - connections to mzv - example II

We also rediscover exotic relations related to cusp forms, e.g. the cusp form

∆ = q

n>0(1 − qn)24 can be written as

−1 26 · 5 · 691∆ = 168[5, 7] + 150[7, 5] + 28[9, 3] + 1 1408[2] − 83 14400[4] + 187 6048[6] − 7 120[8] − 5197 691 [12] .

Letting Z12 act on both sides one obtains the relation

5197 691 ζ(12) = 168ζ(5, 7) + 150ζ(7, 5) + 28ζ(9, 3) .

These type of relations can also be explained via the theory of period polynomials (Gangl-Kaneko-Zagier; Schneps; Baumard; Pollack) or via multiple modular values (Brown).

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bi-brackets - connections to mzv

To summarize we have the following objects in the kernel of Zk, i.e. ways of getting relations between multiple zeta values using brackets. Elements of lower weights, i.e. elements in FilW

k−1(MD).

Derivatives Modular forms, which are cusp forms Since 0 ∈ ker Zk, any linear relation between brackets in FilW

k (MD) gives an element in the

kernel. But these are not all elements in the kernel of Zk. There are elements in the kernel of Zk which can’t be "described" by just using elements of

MD in the list above.

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bi-brackets - connections to mzv

In weight 4 one has the following relation of MZV

ζ(4) = ζ(2, 1, 1) ,

i.e. it is [4] − [2, 1, 1] ∈ ker Z4. But this element can’t be written as a linear combination of cusp forms, lower weight brackets or derivatives. But one can show that

[4] − [2, 1, 1] = 1 2 (d[1] + d[2]) − 1 3[2] − [3] + 2, 1 1, 0

• and

2,1

1,0

• ∈ ker Z4.

Conjecture (rough version)

The kernel of Zk is spanned by the elements of the above list and (essentially) the bi-brackets with at least one rj = 0.

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bi-brackets - generating series

Many statements on bi-brackets are obtained by using their generating function.

Definition

For the generating function of the bi-brackets we write

• X1, . . . , Xl

Y1, . . . , Yl

• :=
• s1,...,sl>0

r1,...,rl>0

• s1 , . . . , sl

r1 − 1 , . . . , rl − 1

• Xs1−1

1

. . . Xsl−1

l

· Y r1−1

1

. . . Y rl−1

l

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bi-brackets - Key Lemma

Key Lemma

For n ∈ N set Ln(X) :=

eXqn 1−eXqn ∈ Q[[q, X]].

For all l ≥ 1 we have the following identities for the generating series of bi-brackets

• X1, . . . , Xl

Y1, . . . , Yl

• =
• u1>···>ul>0

l

• j=1

eujYjLuj(Xj)

The product of the function Ln is given by

Ln(X)·Ln(Y ) =

• k>0

Bk k! (X − Y )k−1 Ln(X) + (−1)k−1Ln(Y )

• + Ln(X) − Ln(Y )

X − Y

Proof: For the second statement one shows by direct calculation that

Ln(X) · Ln(Y ) = 1 eX−Y − 1Ln(X) + 1 eY −X − 1Ln(Y )

and then uses the gen. series

X eX−1 = n≥0 Bn n! Xn of the Bernoulli numbers.

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bi-brackets - partition relation

Theorem (partition relation)

For all l ≥ 1 we have

• X1, . . . , Xl

Y1, . . . , Yl

• =
• Y1 + · · · + Yl, . . . , Y1 + Y2, Y1

Xl, Xl−1 − Xl, . . . , X1 − X2

• Proof: Let Xl+1 := 0, then the Key Lemma implies
• X1, . . . , Xl

Y1, . . . , Yl

• =
• u1>···>ul>0

l

• j=1

euj Yj Luj (Xj ) =

• u1>···>ul>0

l

• j=1

euj (Xl+1−j −Xl+2−j )Luj (Y1 + · · · + Yl−j+1) =

• Y1 + · · · + Yl, . . . , Y1 + Y2, Y1

Xl, Xl−1 − Xl, . . . , X1 − X2

• The partition relation (Ecalle notation: swap invariance) gives linear relations between bi-brackets

in a fixed length, for example

s r

• =

r + 1 s − 1

• for all r, s ∈ N ,

2, 2 1, 1

• = −2

2, 2 0, 2

• +

2, 2 1, 1

• − 4

3, 1 0, 2

• + 2

3, 1 1, 1

• .

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bi-brackets - algebra structure

Theorem

The space BD is a sub algebra of Q[[q]]. It is filtered with respect to the weight W , the lower weight D and the length L, i.e.,

FilW,D,L

k1,d1,l1(BD) · FilW,D,L k2,d2,l2(BD) ⊂ FilW,D,L k1+k2,d1+d2,l1+l2(BD) .

and d := q d

dq is a derivation.

Examples:

[1] · [1] = 2[1, 1] + [2] − [1] [1] · 1 1

• =

1, 1 0, 1

• +

1, 1 1, 0

• −

1 1

• +

2 1

• d

1, 2 3, 4

• = 4

2, 2 4, 4

• + 10

1, 3 3, 5

• .

Proof: Either one shows that there is homomorphism of a quasi-shuffle algebra on infinitely many bi-words (zi, zj)i,j∈N onto BD, or one uses generating series as indicated in the next slide.

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bi-brackets - algebra structure - stuffle product

Proposition (stuffle product - special case of the algebra structure)

The product of the generating series in length one can be written as:

• X1

Y1

• ·
• X2

Y2

• st

=

• X1, X2

Y1, Y2

• +
• X2, X1

Y2, Y1

• +

1 X1 − X2

• X1

Y1 + Y2

• −
• X2

Y1 + Y2

• +

∞

• k=1

Bk k! (X1 − X2)k−1

• X1

Y1 + Y2

• + (−1)k−1
• X2

Y1 + Y2

• .

Proof sketch: Do the usual calculation

• X1

Y1

• ·
• X2

Y2

• =
• n1>0

en1Y1Ln(X1) ·

• n2>0

en2Y2Ln(X2) =

• n1>n2>0

· · · +

• n2>n1>0

· · · +

• n1=n2>0

. . . =

• X1, X2

Y1, Y2

• +
• X2, X1

Y2, Y1

• +
• n>0

en(Y1+Y2)Ln(X1)Ln(X2) .

and then use the second statement of the Key Lemma to rewrite Ln(X1)Ln(X2).

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bi-brackets - stuffle product

Comparing the coefficients in the stuffle product of the generating function we obtain:

Proposition (explicit stuffle product)

For s1, s2 > 0 and r1, r2 ≥ 0 we have

s1 r1

• ·

s2 r2

• st

= s1, s2 r1, r2

• +

s2, s1 r2, r1

• +

r1 + r2 r1 s1 + s2 r1 + r2

• +

r1 + r2 r1 s1

• j=1

(−1)s2−1Bs1+s2−j (s1 + s2 − j)! s1 + s2 − j − 1 s1 − j

• j

r1 + r2

• +

r1 + r2 r1 s2

• j=1

(−1)s1−1Bs1+s2−j (s1 + s2 − j)! s1 + s2 − j − 1 s2 − j

• j

r1 + r2

• Notice: If r1 = r2 = 0, i.e. when the two brackets are elements in MD, all elements on the

right hand side are also elements in MD.

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bi-brackets - stuffle & shuffle product

The partition relation induces an involution P on the bi-brackets. Therefore we get for bi-brackets

a and b the identity a · b = P

• P(a) · P(b)
• We call these linear relations in BD the double shuffle relations.

Example:

1, 2 3, 4

• +

2, 1 4, 3

• − 35

2 2 7

• + 35

3 7

• st

= 1 3

• ·

2 4

• sh

= −35 1, 2 0, 7

• + 15

1, 2 1, 6

• − 5

1, 2 2, 5

• +

1, 2 3, 4

• − 5

2, 1 1, 6

• + 5

2, 1 2, 5

• − 3

2, 1 3, 4

• +

2, 1 4, 3

• −

1 6048 2 2

• +

1 720 2 4

• +

2 8

• 26 / 34

bi-brackets - conjectures

Conjectures

(I) All linear relations between bi-brackets come from the partition relation and the double shuffle relations. (II) Every bi-bracket can be written as a linear combination of brackets, i.e. the algebras BD and

MD are equal.

Remark: In fact we expect the following refinement (II’) of Conjecture (II):

FilW,D,L

k,d,l

(BD) ⊂ FilW,L

k+d,l+d(MD) .

Theorem

The conjecture (I) and (II) hold for all weights k ≤ 7. Idea of Proof: Via the partition-shuffle-spaces PS(k − n, n) we get upper bounds for

dimQ FilW

k (BD) and computing sufficiently many coefficients of brackets gives us lower

bounds for dimQ FilW

k (MD). Happily both numbers coincide.

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partition shuffle space - definition

Set p =

1

. . . 1 1 1 . . . 1 . . . . . . . . . . . . 1 . . .

• ∈ Gll(Z) and P =
• p

p−1

• ∈ GL2l(Z), then the partition

relation reads

• X1, . . . , Xl

Y1, . . . , Yl

• =
• Y1 + · · · + Yl, . . . , Y1 + Y2, Y1

Xl, Xl−1 − Xl, . . . , X1 − X2

• =
• X1, . . . , Xl

Y1, . . . , Yl

• P

.

If we set Shj =

• shj

shj

• ∈ Q[GL2l(Z)], then up to terms of length less than l
• X1, . . . , Xj

Y1, . . . , Yj

• ·
• Xj+1, . . . , Xl

Yj+1, . . . , Yl

• =
• X1, . . . , Xl

Y1, . . . , Yl

• Shj

+ ... Proposition

Let the partion shuffle space given by

PS(k − l, l) = {f ∈ Q[x1, .., xl, y1, .., yl]| deg f = k − l, f

• P − f = f
• Shj = 0 ∀j},

then the number gk,l of generators of weight k and length l for BD is bounded by

gk,l ≤ dimQ PS(k − l, l).

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partition shuffle space - dimension calculated with maple

l \ k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 1 1 2 1 3 1 4 5 6 7 8 9 2

• 1

2 8 14 23 38 58 3

• 1

3 9 27 62 125 238 4

• 1

3 12 37 ? ? 5

• 1

4 15 ? ? ? Table : upper bounds for gk,l

Using this data we get the upper bounds

• k

dimQ FilW

k (BD)xk ≤

1 1 − x

• k,l

1 (1 − xk)gk,l ≤ 1 + 2 x + 4 x2 + 8 x3 + 15 x4 + 28 x5 + 51 x6 + 92 x7 + 165 x8 + ...

In addition we have since MD ⊆ BD

• k

dimQ FilW

k (MD)xk ≤

• k

dimQ FilW

k (BD)xk

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dimensions calculated through q-expansions

k\l

1 2 3 4 5 6 7 8 9 10 11 1 1 1 2 2 1 3 4 4 1 4 7 8 3 1 5 10 14 15 5 1 6 14 22 27 28 6 1 7 18 32 44 50 51 7 1 8 23 44 67 84 91 92 8 1 9 28 59 97 133 156 164 165 9 1 10 34 76 135 200 254 284 293 294 10 1 11 40 97 183 290 396 474 512 522 523 11 1 12 47 120 242 408 594 760 869 916 927 928 12 1 13 54 147 313 559 ? ? ? ? ? ? 13 1 14 62 177 398 ? ? ? ? ? ? ? 14 1 15 70 212 498 ? ? ? ? ? ? ? 15 1 16 79 249 ? ? ? ? ? ? ? ?

Table : dimQ FilW,L

k,l (MD): exact value, lower bound

We read of the series

• k

dimQ FilW

k (MD)xk ≥ 1+2 x+4 x2+8 x3+15 x4+28 x5+51 x6+92 x7+165 x8+...

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generators for l=1: partition relation & quasi-modularity

We have

s

r

• =

r+1

s−1

• and d

s

r

• ∼

s+1

r+1

• for all r, s ∈ N and

M = Q

• [2], [4], [6]
• .

w nw,1

coeff

1 1 [1] 2 1 [2] 3 2 [3] , d[1] 4 1 [4] 5 3 [5], d[3], d2[1] 6 1 [6] 7 4 [7], d[5], d2[5], d3[1] . . . 2k − 1 k [2k − 1],d[2k − 3],...

( k ≥ 4)

2k

(k ≥ 4)

Table : upper bounds for gw,1

Proposition

• gk,1xk = x2+x4+x6+

x (1 − x2)2 = x+x2+2x3+x4+3x5+x6+4x7+5x9+...

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generators for l=2: partition shuffle space

Let p = ( 1 1

1 0 ), t = ( 0 1 1 0 ), P =

• p−1

p

• and T = ( t

t ).

The partition-shuffle relations in length l = 2 are then

f(x1, x2, y1, y2) = f(y1 + y2, y1, x2, x2 − x1) = f(x1, x2, y1, y2)

• P

f(x1, x2, y1, y2) = −f(x2, x1, y2, y1) = −f(x1, x2, y1, y2)

• T

This corresponds to an action of G = T, P with the character χ : G → ±1 given by

χ(P) = 1 and χ(T) = −1. The number of invariant polynomials can by determined by the

generalised Molien theorem.

Proposition

We have

• k

gk,2xk−2 ≤ x2 + 5 x6 − x8 + 2 x10 + x12 − x14 − 2 x16 + x18 − 5 x20 − x24 (1 − x2)2(1 − x6)(1 − x8)(1 − x12) = x2 + 2 x4 + 8 x6 + 14 x8 + 23 x10 + 38 x12 + 58 x14 + 78 x16 + ...

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generators for l=3: partition shuffle space

Let p =

1 1 1

1 1 0 1 0 0

• , t =

0 0 1

0 1 0 1 0 0

• , c3 =

0 1 0

0 0 1 1 0 0

• , P =
• p−1

p

• , T = ( t

t ) and

C3 = ( c3 c3 ) . On A := Q[x1, x2, x3, y1, y2, y3] we consider actions by the groups H′ := T, P, −1 , H := T, PTP, −1 and G := T, PTP, C3.

With Sh = 1 + C2 + C3 and since TC2 = C−1

3

we get similar as for MZV maps

AH

• Sh AG → 0

PS(·, 3) AH′

• .

Observation

Consider the Molien series HAG(x) and HAH′ (x), then modulo x16 we have

• k

dimQ PS(k, 3)xk = HAH′ (x) − HAG(x) = x2 + 3x4 + 9x6 + 27x8 + 62x10 + 125x12 + 238x14 + ...

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summary

bi-brackets are q-series whose coefficients are rational numbers given by sums over partitions. The space BD spanned by all bi-brackets form a differential Q-algebra and there are two different ways to express a product of bi-brackets. This give rise to a lot of linear relations between bi-brackets and conjecturally every element in

BD can be written as a linear combination of elements in MD.

This setup can also be seen as a combinatorial theory of modular forms. For example it follows directly by the double shuffle relations that G2

4 is a multiple of G8.

The elements in MD have a connection to multiple zeta values and elements in the kernel of

Zk give rise to relations between them.

Conjecturally the elements in the kernel of Zk can be described by using bi-brackets. In a recent joint work H. Bachmann and K. Tasaka studies "shuffle regularized multiple Eisenstein series", here bi-brackets also occur. There is a rich algebraic structure, e.g. a Lie algebra structure on PS follows from Ecalle’s

• theory. Need more data to formulate a Broadhurst-Kreimer type conjecture for BD.

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