Totally Odd Depth-graded Multiple Zeta Values and Period Polynomials

Charlotte Dietze Charlotte Dietze, Max-Planck-Institut für Mathematik,Vivatsgasse 7,53111 Bonn, Germany Ludwig-Maximilians-Universität München, Mathematisches Institut, Theresienstr. 39, 80333 München Charlotte.Dietze@campus.lmu.de , Chokri Manai Chorkri Manai, Max-Planck-Institut für Mathematik,Vivatsgasse 7,53111 Bonn, Germany , Christian Nöbel Christian Nöbel, Max-Planck-Institut für Mathematik,Vivatsgasse 7,53111 Bonn, Germany Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisches Institut, Endenicher Allee 60, 53115 Bonn and Ferdinand Wagner Ferdinand Wagner, Max-Planck-Institut für Mathematik,Vivatsgasse 7,53111 Bonn, Germany Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisches Institut, Endenicher Allee 60, 53115 Bonn ferdinand.wagner@uni-bonn.de

September 2016

Abstract.

Inspired by the paper of Tasaka [tasaka], we study the relations between totally odd, motivic depth-graded multiple zeta values. Our main objective is to determine the rank of the matrix $C_{N, r}$ defined by Brown [Brown]. We will give new proofs for (conjecturally optimal) upper bounds on $rank C_{N, 3}$ and $rank C_{N, 4}$ , which were first obtained by Tasaka [tasaka]. Finally, we present a recursive approach to the general problem, which reduces the evaluation of $rank C_{N, r}$ to an isomorphism conjecture.

Key words and phrases:

Multiple zeta values, period polynomials

2010 Mathematics Subject Classification:

Primary 11M32, Secondary 11F67

\addbibresource

mzv.bib

1. Introduction

In this paper we will be interested in $Q$ -linear relations among totally odd depth-graded multiple zeta values (MZVs), for which there conjecturally is a bijection with the kernel of a specific matrix $C_{N, r}$ connected to restricted even period polynomials (for a definition, see [schneps] or [gkz2006, Section 5]).

For integers $n_{1}, \dots, n_{r - 1} \geq 1$ and $n_{r} \geq 2$ , the MZV of $n_{1}, \dots, n_{r}$ is defined as the number

ζ(n1,…,nr)\coloneqq∑0<k1<⋯<kr1kn11⋯knrr.

We call the sum $n_{1} + \dots + n_{r}$ of arguments the weight and their number $r$ the depth of $ζ (n_{1}, \dots, n_{r})$ . One classical question about MZVs is counting the number of linearly independent $Q$ -linear relations between MZVs. It is highly expected, but for now seemingly out of reach that there are no relations between MZVs of different weight. Such questions become reachable when considered in the motivic setting. Motivic MZVs $ζ^{m} (n_{1}, \dots, n_{r})$ are elements of a certain $Q$ -algebra $H = ⨁_{N \geq 0} H_{N}$ which was constructed by Brown in [brownMixedMotives] and is graded by the weight $N$ . Any relation fulfilled by motivic MZVs also holds for the corresponding MZVs via the period homomorphism $p e r : H \to R$ .

We further restrict to depth-graded MZVs: Let $Z_{N, r}$ and $H_{N, r}$ denote the $Q$ -vector space spanned by the real respectively motivic MZVs of weight $N$ and depth $r$ modulo MZVs of lower depth. The depth-graded MZV of $n_{1}, \dots, n_{r}$ , that is, the equivalence class of $ζ (n_{1}, \dots, n_{r})$ in $Z_{N, r}$ , is denoted by $ζ_{D} (n_{1}, \dots, n_{r})$ . The elements of $H_{N, r}$ are denoted $ζ_{D}^{m} (n_{1}, \dots, n_{r})$ analogously. The dimension of $Z_{N, r}$ is subject of the Broadhurst-Kreimer Conjecture.

Conjecture 1.1 (Broadhurst-Kreimer).

The generating function of the dimension of the space $Z_{N, r}$ is given by

\sum N, r \geq 0 {dim}_{Q} Z_{N, r} \cdot x^{N} y^{r} ? = \frac{1 - E (x) y}{1 - O (x) y + S (x) y^{2} - S (x) y^{4}},

where we denote $E(x)\coloneqqx21−x2=x2+x4+x6+⋯$ , $O(x)\coloneqqx31−x2=x3+x5+x7+⋯$ , and $S(x)\coloneqqx12(1−x4)(1−x6)$ .

Remark.

It should be mentioned that $S (x) = \sum_{n > 0} dim S_{n} \cdot x^{n}$ , where $S_{n}$ denotes the space of cusp forms of weight $n$ , for which there is an isomorphism to the space of restricted even period polynomials of degree $n - 2$ (defined in [schneps] or [gkz2006, Section 5]).

In his paper [Brown], Brown considered the $Q$ -vector space $Z_{N, r}^{odd}$ (respectively $H_{N, r}^{odd}$ ) of totally odd (motivic) and depth-graded MZVs, that is, $ζ_{D} (n_{1}, \dots, n_{r})$ (respectively $ζ_{D}^{m} (n_{1}, \dots, n_{r})$ ) for $n_{i} \geq 3$ odd, and linked them to a certain explicit matrix $C_{N, r}$ , where $N = n_{1} + \dots + n_{r}$ denotes the weight. In particular, he showed that any right annihilator $(a_{n_{1}, \dots, n_{r}})_{(n_{1}, \dots, n_{r}) \in S_{N, r}}$ of $C_{N, r}$ induces a relation

\sum (n_{1}, \dots, n_{r}) \in S_{N, r} a_{n_{1}, \dots, n_{r}} ζ_{D}^{m} (n_{1}, \dots, n_{r}) = 0, hence also \sum (n_{1}, \dots, n_{r}) \in S_{N, r} a_{n_{1}, \dots, n_{r}} ζ_{D} (n_{1}, \dots, n_{r}) = 0

(see Section 2 for the notations) and conjecturally all relations in $Z_{N, r}^{odd}$ arise in this way. This led to the following conjecture (the uneven part of the Broadhurst-Kreimer Conjecture).

Conjecture 1.2 (Brown [Brown]).

The generating series of the dimension of $Z_{N, r}^{odd}$ and the rank of $C_{N, r}$ are given by

1 + \sum N, r > 0 rank C_{N, r} \cdot x^{N} y^{r} ? = 1 + \sum N, r > 0 {dim}_{Q} Z_{N, r}^{odd} \cdot x^{N} y^{r} ? = \frac{1}{1 - O (x) y + S (x) y^{2}} .

The contents of this paper are as follows. In Section 2, we explain our notations and define the matrices $C_{N, r}$ due to Brown [Brown] as well as $E_{N, r}$ and $E_{N, r}^{(j)}$ considered by Tasaka [tasaka]. In Section 3, we briefly state some of Tasaka’s results on the matrix $E_{N, r}$ . Section 4 is devoted to further investigate the connection between the left kernel of $E_{N, r}$ and restricted even period polynomials, which was first discovered by Baumard and Schneps [schneps] and appears again in [tasaka, Theorem 3.6]. In Section 5, we will apply our methods to the cases $r = 3$ and $r = 4$ . The first goal of Section 5 will be to show

Theorem 1.3.

Assume that the map from Theorem 3.2 is injective. We then have the lower bound

\sum N > 0 {dim}_{Q} ker C_{N, 3} \cdot x^{N} \geq 2 O (x) S (x),

where $\geq$ means that for every $N > 0$ the coefficient of $x^{N}$ on the right-hand side does not exceed the corresponding one on the left-hand side.

This was stated without proof in [tasaka]. Furthermore, we will give a new proof by the polynomial methods developed in Section 4 for the following result.

Theorem 1.4 (Tasaka [tasaka, Theorem 1.3]).

Assume that the map from Theorem 3.2 is injective. We then have the lower bound

\sum N > 0 {dim}_{Q} ker C_{N, 4} \cdot x^{N} \geq 3 O (x)^{2} S (x) - S (x)^{2} .

In the last two subsections of this paper, we will consider the case of depth 5 and give an idea for higher depths. For depth 5, we will prove that upon Conjecture 3.3 due to Tasaka ([tasaka, Section 3]), the lower bound predicted by Conjecture 1.2 holds, i.e.

\sum N > 0 {dim}_{Q} ker C_{N, 5} \cdot x^{N} \geq 4 O (x)^{3} S (x) - 3 O (x) S (x)^{2} .

These bounds are conjecturally sharp (i.e. the ones given by Conjecture 1.2). Finally, we will prove a recursion for value of ${dim}_{Q} ker C_{N, r}$ under the assumption of a similar isomorphism conjecture stated at the end of Section 4, which was proposed by Claire Glanois.

Acknowledgments

This research was conducted as part of the Hospitanzprogramm (internship program) at the Max-Planck-Institut für Mathematik (Bonn). We would like to express our deepest thanks to our mentor, Claire Glanois, for introducing us into the theory of multiple zeta values. We are also grateful to Daniel Harrer, Matthias Paulsen and Jörn Stöhler for many helpful comments.

2. Preliminaries

2.1. Notations

In this section we introduce our notations and we give some definitions. As usual, for a matrix $A$ we define $ker A$ to be the set of right annihilators of $A$ . Apart from this, we mostly follow the notations of Tasaka in his paper [tasaka]. Let

SN,r\coloneqq{(n1,…,nr)∈Zr | n1+⋯+nr=N, n1,…,nr≥3 odd},

where $N$ and $r$ are natural numbers. Since the elements of the set $S_{N, r}$ will be used as indices of matrices and vectors, we usually arrange them in lexicographically decreasing order. Let

denote the vector space of restricted totally even homogeneous polynomials of degree $N - r$ in $r$ variables. There is a natural isomorphism from $V_{N, r}$ to the $Q$ -vector space ${V e c t}_{N, r}$ of $n$ -tuples $(a_{n_{1}, \dots, n_{r}})_{(n_{1}, \dots, n_{r}) \in S_{N, r}}$ indexed by totally odd indices $(n_{1}, \dots, n_{r}) \in S_{N, r}$ , which we denote

(2.1)

\begin{matrix} π : V_{N, r} & \sim ⟶ {V e c t}_{N, r} \sum (n_{1}, \dots, n_{r}) \in S_{N, r} a_{n_{1}, \dots, n_{r}} x_{1}^{n_{1} - 1} \dots x_{r}^{n_{r} - 1} & ⟼ {(a_{n_{1}, \dots, n_{r}})}_{(n_{1}, \dots, n_{r}) \in S_{N, r}} . \end{matrix}

We assume vectors to be row vectors by default.

Finally, let $W_{N, r}$ be the vector subspace of $V_{N, r}$ defined by

	$WN,r\coloneqq{P∈VN,r \| P(x1,…,xr)$	$= P (x_{2} - x_{1}, x_{2}, x_{3}, \dots, x_{r})$
		$- P (x_{2} - x_{1}, x_{1}, x_{3}, \dots, x_{r})} .$

That is, $P (x_{1}, x_{2}, x_{3}, \dots, x_{r})$ is a sum of restricted even period polynomials in $x_{1}, x_{2}$ multiplied by monomials in $x_{3}, \dots, x_{r}$ . More precisely, one can decompose

(2.2)

W_{N, r} = ⨁ \begin{matrix} 1 < n < N n even \end{matrix} W_{n, 2} \otimes V_{N - n, r - 2},

where $W_{n, 2}$ is the space of restricted even period polynomials of degree $n - 2$ . Since $W_{n, 2}$ is isomorphic to the space $S_{n}$ of cusp forms of weight $n$ by Eichler-Shimura correspondence (see [zagier]), (2.2) leads to the following dimension formula.

Lemma 2.1 (Tasaka [tasaka, equation 3.10]).

For all $r \geq 2$ ,

\sum N > 0 {dim}_{Q} W_{N, r} \cdot x^{N} = O (x)^{r - 2} S (x) .

2.2. Ihara action and the matrices $E_{N, r}$ and $C_{N, r}$

We use Tasaka’s notation (from [tasaka]) for the polynomial representation of the Ihara action defined by Brown [Brown, Section 6]. Let

	$\circ - : Q [x_{1}] \otimes Q [x_{2}, \dots, x_{r}]$	$⟶ Q [x_{1}, \dots, x_{r}]$
	$f \otimes g$	$⟼ f \circ - g,$

where $f \circ - g$ denotes the polynomial

(f∘–g)(x1,…,xr)\coloneqqf(x1)g(x2,…,xr)+r−1∑i=1(f(xi+1−xi)g(x1,…,^xi+1,…,xr)−(−1)degff(xi−xi+1)g(x1,…,^xi,…,xr)).

(the hats are to indicate, that $x_{i + 1}$ and $x_{i}$ resp. are omitted in the above expression).

For integers $m_{1}, \dots, m_{r}, n_{1}, \dots, n_{r} \geq 1$ , let furthermore the integer $e (\frac{m_{1}, \dots, m_{r}}{n_{1}, \dots, n_{r}})$ denote the coefficient of $x_{1}^{n_{1} - 1} \dots x_{r}^{n_{r} - 1}$ in $x_{1}^{m_{1} - 1} \circ - (x_{1}^{m_{2} - 1} \dots x_{r - 1}^{m_{r} - 1})$ , i. e.

(2.3)

x_{1}^{m_{1} - 1} \circ - (x_{1}^{m_{2} - 1} \dots x_{r - 1}^{m_{r} - 1}) = \sum \begin{matrix} n_{1} + \dots + n_{r} = m_{1} + \dots + m_{r} n_{1}, \dots, n_{r} \geq 1 \end{matrix} e (\frac{m_{1}, \dots, m_{r}}{n_{1}, \dots, n_{r}}) x_{1}^{n_{1} - 1} \dots x_{r}^{n_{r} - 1} .

Note that $e (\frac{m_{1}, \dots, m_{r}}{n_{1}, \dots, n_{r}}) = 0$ if $m_{1} + \dots + m_{r} \neq n_{1} + \dots + n_{r}$ .

Remark.

One can explicitly compute the integers $e (\frac{m_{1}, \dots, m_{r}}{n_{1}, \dots, n_{r}})$ by the following formula: ([tasaka, Lemma 3.1])

e (\frac{m_{1}, \dots, m_{r}}{n_{1}, \dots, n_{r}}) = δ (\frac{m_{1}, \dots, m_{r}}{n_{1} \dots, n_{r}}) + r - 1 \sum i = 1 δ (\frac{{^m}_{1}, m_{2}, \dots, m_{i}, {^m}_{i + 1}, m_{i + 2}, \dots, m_{r}}{n_{1}, \dots, n_{i - 1}, {^n}_{i}, {^n}_{i + 1}, n_{i + 2}, \dots, n_{r}}) \cdot ((- 1)^{n_{i}} (\frac{m_{1} - 1}{n_{i} - 1}) + (- 1)^{m_{1} - n_{i + 1}} (\frac{m_{1} - 1}{n_{i + 1} - 1}))

(again, the hats are to indicate that $m_{1}, m_{i + 1}, n_{i}, n_{i + 1}$ are omitted), where

δ(m1,…,msn1,…,ns)\coloneqq{1if mi=ni for all i∈{1,…,s}0else

denotes the usual Kronecker delta.

Definition 2.2.

Let $N, r$ be positive integers.

We define the $| S_{N, r} | \times | S_{N, r} |$ matrix

$EN,r\coloneqq(e(m1,…,mrn1,…,nr))(m1,…,mr),(n1,…,nr)∈SN,r.$
For integers $r \geq j \geq 2$ we also define the $| S_{N, r} | \times | S_{N, r} |$ matrix

$E(j)N,r\coloneqq(δ(m1,…,mr−jn1,…,nr−j)e(mr−j+1,…,mrnr−j+1,…,nr))(m1,…,mr),(n1,…,nr)∈SN,r.$

Definition 2.3 ([tasaka, Definition 2.3 and Proposition 3.3]).

The $| S_{N, r} | \times | S_{N, r} |$ matrix $C_{N, r}$ is defined as

CN,r\coloneqqE(2)N,r⋅E(3)N,r⋯E(r−1)N,r⋅EN,r.

3. Known Results

Recall the map $π : V_{N, r} \to {V e c t}_{N, r}$ (equation (2.1)). Theorem 3.1 due to Baumard and Schneps [schneps] establishes a connection between the left kernel of the matrix $E_{N, 2}$ and the space $W_{N, 2}$ of restricted even period polynomials. This connection was further investigated by Tasaka [tasaka], relating $W_{N, r}$ and the left kernel of $E_{N, r}$ for arbitrary $r \geq 2$ .

Theorem 3.1 (Baumard-Schneps [schneps, Proposition 3.2]).

For each integer $N > 0$ we have

π (W_{N, 2}) = ker^{t} E_{N, 2} .

Theorem 3.2 (Tasaka [tasaka, Theorem 3.6]).

Let $r \geq 2$ be a positive integer and $F_{N, r} = E_{N, r} - {id}_{{V e c t}_{N, r}}$ . Then, the following $Q$ -linear map is well-defined:

(3.1)

\begin{matrix} W_{N, r} & ⟶ ker^{t} E_{N, r} P (x_{1}, \dots, x_{r}) & ⟼ π (P) F_{N, r} . \end{matrix}

Conjecture 3.3 (Tasaka [tasaka, Section 3.3]).

For all $r \geq 2$ , the map described in Theorem 3.2 is an isomorphism.

Remark.

For now, only the case $r = 2$ is known, which is an immediate consequence of Theorem 3.1. In [tasaka], Tasaka suggests a proof of injectivity, but it seems to contain a gap, which, as far as the authors are aware, couldn’t be fixed yet. However, assuming the injectivity of morphisms (3.1) one has the following relation.

Corollary 3.4 (Tasaka [tasaka, Corollary 3.7]).

For all $r \geq 2$ ,

\sum N > 0 {dim}_{Q} ker^{t} E_{N, r} \cdot x^{N} \geq O (x)^{r - 2} S (x) .

4. Main Tools

4.1. Decompositions of $E_{N, r}^{(j)}$

We use the following decomposition lemma:

Lemma 4.1.

Let $2 \leq j \leq r - 1$ and arrange the indices $(m_{1}, \dots, m_{r}), (n_{1}, \dots, n_{r}) \in S_{N, r}$ of $E_{N, r}^{(j)}$ in lexicographically decreasing order. Then, the matrix $E_{N, r}^{(j)}$ has block diagonal structure

E_{N, r}^{(j)} = diag (E_{3 r - 3, r - 1}^{(j)}, E_{3 r - 1, r - 1}^{(j)}, \dots, E_{N - 3, r - 1}^{(j)}) .

Proof.

This follows directly from Definition 2.2. ∎

Corollary 4.2.

We have

E_{N, r}^{(2)} E_{N, r}^{(3)} \dots E_{N, r}^{(r - 1)} = d i a g (C_{3 r - 3, r - 1}, C_{3 r - 1, r - 1}, \dots, C_{N - 3, r - 1}) .

Proof.

Multiplying the block diagonal representations of $E_{N, r}^{(2)}, E_{N, r}^{(3)}, \dots, E_{N, r}^{(r - 1)}$ block by block together with Definition 2.3 yields the desired result. ∎

Corollary 4.3.

For all $r \geq 3$ ,

\sum N > 0 {dim}_{Q} ker (E_{N, r}^{(2)} \dots E_{N, r}^{(r - 1)}) \cdot x^{N} = O (x) \sum N > 0 {dim}_{Q} ker C_{N, r - 1} \cdot x^{N} .

Proof.

According to Corollary 4.2, the matrix $E_{N, r}^{(2)} \dots E_{N, r}^{(r - 1)}$ has block diagonal structure, the blocks being $C_{3 r - 3, r - 1}, C_{3 r - 1, r - 1}, \dots, C_{N - 3, r - 1}$ . Hence,

	$\sum N > 0 {dim}_{Q} ker (E_{N, r}^{(2)} \dots E_{N, r}^{(r - 1)}) \cdot x^{N}$	$= \sum N > 0 (N - 3 \sum k = 3 r - 3 {dim}_{Q} ker C_{k, r - 1}) \cdot x^{N}$
		$= \sum N > 0 {dim}_{Q} ker C_{N, r - 1} (x^{N + 3} + x^{N + 5} + x^{N + 7} + \dots)$
		$= O (x) \sum N > 0 {dim}_{Q} ker C_{N, r - 1} \cdot x^{N},$

thus proving the assertion. ∎

4.2. Connection to polynomials

Motivated by Theorem 3.2, we interpret the right action of the matrices $E_{N, r}^{(2)}, \dots, E_{N, r}^{(r - 1)}, E_{N, r}^{(r)} = E_{N, r}$ on ${V e c t}_{N, r}$ as endomorphisms of the polynomial space $V_{N, r}$ . Having established this, we will prove Theorems 1.3 and 1.4 from a polynomial point of view.

Definition 4.4.

The restricted totally even part of a polynomial $Q (x_{1}, \dots, x_{r}) \in V_{N, r}$ is the sum of all of its monomials, in which each exponent of $x_{1}, \dots, x_{r}$ is even and at least 2. Let $r \geq j$ . We define the $Q$ -linear map

φ_{j}^{(r)} : V_{N, r} ⟶ V_{N, r},

which maps each polynomial $Q (x_{1}, \dots, x_{r}) \in V_{N, r}$ to the restricted totally even part of

Q (x_{1}, \dots, x_{r}) + r - 1 \sum i = r - j + 1 (Q (x_{1}, \dots, x_{r - j}, x_{i + 1} - x_{i}, x_{r - j + 1}, \dots, {^x}_{i + 1}, \dots, x_{r}) - Q (x_{1}, \dots, x_{r - j}, x_{i + 1} - x_{i}, x_{r - j + 1}, \dots, {^x}_{i}, \dots, x_{r})) .

Remark.

Note that $φ_{1}^{(r)} \equiv {id}_{V_{N, r}}$ .

The following lemma shows that the map $φ_{j}^{(r)}$ corresponds to the right action of the matrix $E_{N, r}^{(j)}$ on ${V e c t}_{N, r}$ via the isomorphism $π$ .

Lemma 4.5.

Let $r \geq j$ . Then, for each polynomial $Q \in V_{N, r}$ ,

π (φ_{j}^{(r)} (Q)) = π (Q) E_{N, r}^{(j)} .

or equivalently, the following diagram commutes:

Proof.

We proceed by induction on $r$ . Let $r = j$ and

Q (x_{1}, \dots, x_{j}) = \sum (n_{1}, \dots, n_{j}) \in S_{N, j} q_{n_{1}, \dots, n_{j}} x_{1}^{n_{1} - 1} \dots x_{r}^{n_{j} - 1} .

Then, $E_{N, j}^{(j)} = E_{N, j}$ and thus

π (Q) E_{N, j} = {⎛ ⎜ ⎝ \sum (m_{1}, \dots, m_{j}) \in S_{N, j} q_{n_{1}, \dots, n_{j}} e (\frac{m_{1}, \dots, m_{j}}{n_{1}, \dots, n_{j}}) ⎞ ⎟ ⎠}_{(n_{1}, \dots, n_{j}) \in S_{N, j}} .

By (2.3) and linearity of the Ihara action $\circ -$ , the row vector on the right-hand side corresponds to $π$ applied to the restricted totally even part of the polynomial

(4.1)

\sum (n_{1}, \dots, n_{j}) \in S_{N, j} q_{n_{1}, \dots, n_{j}} x_{1}^{n_{1} - 1} \circ - (x_{1}^{n_{2} - 1} \dots x_{j - 1}^{n_{j} - 1}) .

On the other hand, plugging $r = j$ into Definition 4.4 yields that $φ_{j}^{(j)} (Q (x_{1}, \dots, x_{j}))$ corresponds to the restricted totally even part of some polynomial, which by definition of the Ihara action $\circ -$ coincides with the polynomial defined in (4.1). Thus, the claim holds for $r = j$ .

Now suppose that $r \geq j + 1$ and the claim is proven for all smaller $r$ . Let us decompose

Q (x_{1}, \dots, x_{r}) = N - (3 r - 3) \sum \begin{matrix} n_{1} = 3 n_{1} odd \end{matrix} x_{1}^{n_{1} - 1} \cdot Q_{N - n_{1}} (x_{2}, \dots, x_{r}),

where the $Q_{k}$ are restricted totally even homogeneous polynomials in $r - 1$ variables. In particular, $Q_{k} \in V_{k, r - 1}$ for all $k$ . Arrange the indices of $π (Q)$ in lexicographically decreasing order. Then, by grouping consecutive entries, $π (Q)$ is the list-like concatenation of $π (Q_{3 r - 3}), \dots, π (Q_{N - 3})$ , which we denote by

Since we have lexicographically decreasing order of indices, the block diagonal structure of $E_{N, r}^{(j)}$ stated in Lemma 4.1 yields

	$π (Q) E_{N, r}^{(j)}$	$= (π (Q_{3 r - 3}) E_{3 r - 3, r - 1}^{(j)}, π (Q_{3 r - 1}) E_{3 r - 1, r - 1}^{(j)}, \dots, π (Q_{N - 3}) E_{N - 3, r - 1}^{(j)})$
		$= (π (φ_{j}^{(r - 1)} (Q_{3 r - 3})), π (φ_{j}^{(r - 1)} (Q_{3 r - 1})), \dots, π (φ_{j}^{(r - 1)} (Q_{N - 3})))$
		$= π (φ_{j}^{(r)} (Q))$

by linearity of $φ_{j}^{(r)}$ and the induction hypothesis. This shows the assertion. ∎

Corollary 4.6.

For all $r \geq 2$ ,

Im^{t} (E_{N, r}^{(2)} \dots E_{N, r}^{(r - 1)}) \cap ker^{t} E_{N, r} ≅ ker φ_{r}^{(r)} \cap Im (φ_{r - 1}^{(r)} \circ \dots \circ φ_{2}^{(r)}) .

Proof.

By the previous Lemma 4.5, the following diagram commutes:

From this, we have $Im^{t} (E_{N, r}^{(2)} \dots E_{N, r}^{(r - 1)}) ≅ Im (φ_{r - 1}^{(r)} \circ \dots \circ φ_{2}^{(r)})$ and $ker^{t} E_{N, r} ≅ ker φ_{r}^{(r)}$ . Thereby, the claim is established. ∎

Lemma 4.7.

Let $j \leq r - 1$ . Then,

ker φ_{j}^{(r)} ≅ ⨁ n < N V_{N - n, r - j} \otimes ker E_{n, j} .

Proof.

Let $Q \in V_{N, r}$ . We may decompose

Q (x_{1}, \dots, x_{r}) = \sum n < N \sum (n_{1}, \dots, n_{r - j}) \in S_{N - n, r - j} x_{1}^{n_{1} - 1} \dots x_{r - j}^{n_{r - j} - 1} R_{n_{1}, \dots, n_{r - j}} (x_{r - j + 1}, \dots, x_{r}),

where $R_{n_{1}, \dots, n_{r - j}} \in V_{n, j}$ is a restricted totally even homogeneous polynomial. Note that we have $Q \in ker φ_{j}^{(r)}$ if and only if $φ_{j}^{(j)} (R_{n}) = 0$ holds for each $R_{n}$ in the above decomposition. By Lemma 4.5, $φ_{j}^{(j)} (R_{n}) = 0$ if and only if $π (R_{n}) \in ker E_{n, j}$ . Now, the assertion is immediate. ∎

Corollary 4.8.

Let $2 \leq j \leq r - 2$ . The restricted map

φ_{j}^{(r)} {∣ ∣}_{W_{N, r}} : W_{N, r} ⟶ W_{N, r}

is well-defined and satisfies

ker φ_{j}^{(r)} {∣ ∣}_{W_{N, r}} ≅ ⨁ n < N W_{N - n, r - j} \otimes ker E_{n, j} .

Proof.

Since $j \leq r - 2$ , for each $Q \in V_{N, r}$ the map $Q (x_{1}, \dots, x_{r}) \mapsto φ_{j}^{(r)} (Q)$ does not interfere with $x_{1}$ or $x_{2}$ and thus not with the defining property of $W_{N, r}$ . Hence, $φ_{j}^{(r)} {∣ ∣}_{W_{N, r}}$ is well-defined. The second assertion is done just like in the previous Lemma 4.7. ∎

Lemma 4.9.

Let $r \geq 3$ . For all $P \in W_{N, r}$ ,

π (- P) E_{N, r}^{(r - 1)} = π (P) F_{N, r} .

Proof.

Recall that by Lemma 4.5,

	$π (- P) E_{N, r}^{(r - 1)}$	$= π (φ_{r - 1}^{(r)} (- P (x_{1}, \dots, x_{r})))$

since $- P$ is antisymmetric with respect to $x_{1} \leftrightarrow x_{2}$ . In the same way we compute

	$π (P) F_{N, r}$	$= π (P) (E_{N, r} - {id}_{{V e c t}_{N, r}}) = π (φ_{r}^{(r)} (P (x_{1}, \dots, x_{r}))) - π (P)$

Now the desired result follows from

P (x_{1}, x_{2}, \dots, x_{r}) + P (x_{2} - x_{1}, x_{1}, x_{3}, \dots, x_{r}) - P (x_{2} - x_{1}, x_{2}, \dots, x_{r}) = 0,

since $P$ is in $W_{N, r}$ . ∎

Corollary 4.10.

Assume that the map from Theorem 3.2 is injective. Then, for all $r \geq 3$ ,

Proof.

This is immediate by the previous Lemma 4.9. ∎

Lemma 4.11.

For all $r \geq 3$ ,

Im (φ_{r - 1}^{(r)} \circ φ_{r - 2}^{(r)} {∣ ∣}_{W_{N, r}} \circ \dots \circ φ_{2}^{(r)} {∣ ∣}_{W_{N, r}}) \subseteq ker φ_{r}^{(r)} \cap Im (φ_{r - 1}^{(r)} \circ \dots \circ φ_{2}^{(r)}) .

Proof.

We may replace the right-hand side by just $ker φ_{r}^{(r)}$ . Note that by Corollary 4.8 the composition of restricted $φ_{j}^{(r)} {∣ ∣}_{W_{N, r}}$ on the left-hand side is well-defined. Moreover, each $Q \in Im (φ_{r - 1}^{(r)} \circ φ_{r - 2}^{(r)} {∣ ∣}_{W_{N, r}} \circ \dots \circ φ_{2}^{(r)} {∣ ∣}_{W_{N, r}})$ can be represented as $Q = φ_{r - 1}^{(r)} (P)$ for some $P \in W_{N, r}$ and thus $Q \in ker φ_{r}^{(r)}$ according to Lemma 4.9 and Theorem 3.2. ∎

Similar to Conjecture 3.3 we expect a stronger result to be true, which is stated in the following conjecture due to Claire Glanois:

Conjecture 4.12.

For all $r \geq 3$ ,

Im (φ_{r - 1}^{(r)} \circ φ_{r - 2}^{(r)} {∣ ∣}_{W_{N, r}} \circ \dots \circ φ_{2}^{(r)} {∣ ∣}_{W_{N, r}}) = ker φ_{r}^{(r)} \cap Im (φ_{r - 1}^{(r)} \circ \dots \circ φ_{2}^{(r)}) .

Remark.

Note that intersecting $ker φ_{r}^{(r)} \cap Im (φ_{r - 1}^{(r)} \circ \dots \circ φ_{2}^{(r)})$ , Conjecture 4.12 does not need the injectivity from Conjecture 3.3. However, we haven’t been able to derive Conjecture 4.12 from Conjecture 3.3, so it is not necessarily weaker.

5. Main Results

Throughout this section we will assume that the map from Theorem 3.2 is injective, i.e. the injectivity part of Conjecture 3.3 is true. This was also the precondition for Tasaka’s original proof of Theorem 1.4.

5.1. Proof of Theorem 1.3.

By Corollary 4.3, Remark 3 and the fact that $E_{N, 2} = C_{N, 2}$ we obtain

(5.1)

\sum N > 0 {dim}_{Q} ker E_{N, 3}^{(2)} \cdot x^{N} = O (x) \sum N > 0 {dim}_{Q} ker C_{N, 2} \cdot x^{N} = O (x) S (x) .

We use Corollary 4.10 and Lemma 2.1 to obtain

(5.2)

Now observe that since $C_{N, 3} = E_{N, 3}^{(2)} E_{N, 3}$ , we have

{dim}_{Q} ker C_{N, 3} = {dim}_{Q} ker^{t} E_{N, 3}^{(2)} + {dim}_{Q} (Im^{t} E_{N, 3}^{(2)} \cap ker^{t} E_{N, 3}) .

By (

Totally Odd Depth-graded Multiple Zeta Values and Period Polynomials

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

Conjecture 1.1 (Broadhurst-Kreimer).

Remark.

Conjecture 1.2 (Brown [Brown]).

Theorem 1.3.

Theorem 1.4 (Tasaka [tasaka, Theorem 1.3]).

Acknowledgments

2. Preliminaries

2.1. Notations

Lemma 2.1 (Tasaka [tasaka, equation 3.10]).

2.2. Ihara action and the matrices EN,r and CN,r

Remark.

Definition 2.2.

Definition 2.3 ([tasaka, Definition 2.3 and Proposition 3.3]).

3. Known Results

Theorem 3.1 (Baumard-Schneps [schneps, Proposition 3.2]).

Theorem 3.2 (Tasaka [tasaka, Theorem 3.6]).

Conjecture 3.3 (Tasaka [tasaka, Section 3.3]).

Remark.

Corollary 3.4 (Tasaka [tasaka, Corollary 3.7]).

4. Main Tools

4.1. Decompositions of E(j)N,r

Lemma 4.1.

Proof.

Corollary 4.2.

Proof.

Corollary 4.3.

Proof.

4.2. Connection to polynomials

Definition 4.4.

Remark.

Lemma 4.5.

Proof.

Corollary 4.6.

Proof.

Lemma 4.7.

Proof.

Corollary 4.8.

Proof.

Lemma 4.9.

Proof.

Corollary 4.10.

Proof.

Lemma 4.11.

Proof.

Conjecture 4.12.

Remark.

5. Main Results

5.1. Proof of Theorem 1.3.

2.2. Ihara action and the matrices $E_{N, r}$ and $C_{N, r}$

4.1. Decompositions of $E_{N, r}^{(j)}$