Fuchsian type multipliers and shuffle algebras.

G.H.E. Duchamp,

LIPN – UMR 7030 CNRS, Institut Galillée – Université Paris 13.

September, 12th 2012, Séminaire Lotharingien de Combinatoire

Strobl (Austria)

(joint work with Matthieu Deneufchâtel and Hoang Ngoc Minh)

Summary

1. Introduction

2. Strings of primitives

3. Shuffle algebras

4. Duality Series/Polynomials

5. Abstract setting

6. Sketch of the proof

7. Application to polylogs

8. Concluding remarks and perspectives

8.1 Triangle condition applied to other entries and function fields 8.2 Constants of (alien ?) derivation 8.3 Better understanding of the combinatorics behind the differential

Galois group

Introduction

▶ Many discussions and interactions convinced us that a special care should be payed on the study of coefficients about polylogarithms

▶ Today we will show an abstract theorem about differential algebra

and some of its applications to polylogarithms. ▶ The road to polylogarithms is the following

▶ Riemann zeta values

▶ One wants to produce a tractable ℚ-algebra where these functions

live ▶ Euler-Zagier sums ▶ Polylogarithms (analytic function) ▶ Integral form of polylogarithms (today’s concern)

Strings of primitives 1/4

We begin by a classical example, the generation of polylogarithms. On the following domain,

0 1 Ω ℂ − (] − ∞, 0] ∪ [1, +∞[)

let’s play a game ...

G.H.E. Duchamp, LIPN – UMR 7030 CNRS, Institut Galillée – Université Paris 13. Fuchsian type multipliers and shuffle algebras.

Strings of primitives 2/4

Starting from 1 (the constant function on Ω), we read the words of {x_{0}, x_{1}}^{∗} and compute w.r.t. the following rule (using any “primitive-making” process).

\begin{matrix}∫?\frac{dz}{z} \\ ∫?\frac{dz}{1−z} \end{matrix}

We then get a collection of analytic functions (S_{w}(z))_{w∈{x_{0},x_{1}}^{∗}} satisfying

S′_{x_{0}w}(z) = \frac{1}{z}S_{w}(z) ; S′_{x_{1}w}(z) = \frac{1}{1 − z} S_{w}(z)  (1)

A classical method (Chen’s iterated integrals) to obtain solutions of (5) is to pick an element z_{0} ∈ Ω and compute the following iterated integrals

S_{x_{i_{1}}x_{i_{2}}···x_{in}}(z) = ∫_{z_{0}}^{z} dω_{i_{1}} ∫_{z_{0}}^{y_{1}} … ∫_{z_{0}}^{y_{n−1}} dω_{i_{n}} ; ω_{0} = \frac{dz}{z} ω_{1} = \frac{dz}{1 − z} (2)

One can also initialize the process (Drinfel’d) with a chain of well choosen primitives i.e.

S_{x_{0}^{n}}(z) := \frac{log(z)^{n}}{n!}  (3)

and integrate w.r.t. the other moves. In these two cases, the solution is a morphism w.r.t. the shuffle product (see below).

We then get a map w ↦ S_{w} from X^{∗} = {x_{0}, x_{1}}^{∗} to C^{ω}(Ω, ℂ) (a ring) which we note as a sum S = ∑_{w∈X^{∗}}⟨S|w⟩w (a noncommutative series). We recall that the algebra of noncommutative series is endowed with the convolution product (here also concatenation)

ST := ∑_{w∈X^{∗}} ( ∑_{uv=w}⟨S|u⟩⟨T |v⟩)w  (4)

Now, with the extension of \frac{d}{dz} to series, one has

\frac{d}{dz} (S) = MS  (5)

with M = \frac{1}{z}.x_{0} + \frac{1}{1−z}.x_{1}. This extension of the derivative to noncommutative series permits a closer study of solutions of equations of type (5). In particular, with certain initial conditions, solutions of (5) are morphisms of the shuffle algebra (this is the case of the two solutions given above).

G.H.E. Duchamp, LIPN – UMR 7030 CNRS, Institut Galillée – Université Paris 13. Fuchsian type multipliers and shuffle algebras.

Shuffle algebras

There is another product between series which is linked to the Hopf algebra structure of the noncommutative polynomials (viewed as the enveloping algebra of the free Lie algebra)

(k⟨X ⟩, conc, 1_{X^{∗}}, Δ , ϵ, S) .  (6)

The letters are primitive i.e. for all x ∈ X

Δ (x) = x ⊗ 1 + 1 ⊗ x .  (7)

Then the shuffle product can be defined either by duality with Δ or by the following recursion

1 w = w 1 = w ;
a.u b.v = a.(u b.v) + b(a.u v)  (8)

Duality Series/Polynomials

As k⟨X ⟩ = k^{(X^{∗})} and k⟨⟨X ⟩⟩ = k^{X^{∗}}, we get a natural pairing

k⟨⟨X ⟩⟩ ⊗ k⟨X ⟩ → k  (9)

by ⟨S|P⟩ = ∑_{w∈X^{∗}}⟨S|w⟩⟨P|w⟩. Here is the meaning of the statement (see above) that certain series S are morphisms (in fact rings-of-functions valued characters) for the shuffle. We have

⟨S|u v⟩ = ⟨S|u⟩⟨S|v⟩  (10)

Abstract setting 1/2

Let (A, d) be a k-commutative associative differential algebra with unit (ch(k) = 0 and ker(d) = k) and C be a differential subfield of A (i.e. d(C) ⊂ C). We suppose that S ∈ A⟨⟨X ⟩⟩ is a solution of the differential equation

d(S) = MS ; ⟨S|1⟩ = 1  (11)

where the multiplier M = ∑_{x∈X} u_{x}x ∈ C⟨⟨X ⟩⟩ is an homogeneous series (a polynomial in case X is finite) of degree 1.

Abstract setting 2/2

Theorem

The following condition are equivalent :

i) The family (⟨S|w⟩)_{w∈X^{∗}} of coefficients of S is free over C. ii) The family of coefficients (⟨S|y⟩)_{y∈X ∪{1_{X^{∗}}}} is free over C. iii) The family (u_{x})_{x∈X} is such that, for f ∈ C and α_{x} ∈ k

d(f ) = ∑_{x∈X} α_{x}u_{x} ⟹ (∀x ∈ X )(α_{x} = 0) .  (12)

Remark

(iii) can be rephrased as iv) The family (u_{x})_{x∈X} is free over k and

d(C) ∩ span_{k}((u_{x})_{x∈X}) = {0} .  (13)

Sketch of the proof 1/2

▶ (iii)⟹(i) Suppose a linear dependence ∑_{w∈X^{∗}} α_{w}⟨S|w⟩ = 0 (finite support), it can be rewritten as ⟨S|P⟩ with some polynomial P ∈ C⟨X ⟩ ; P ≠ 0. Take the polynomial with the least degree (lenght of the leading monomial) and least leading word among the non-trivial relations, normalize it (because C is a field, one can divide by the leading coefficient) and write it

P = w + ∑_{u<w}⟨P|u⟩u  (14)

Now

0 = ⟨S|P⟩′ = ⟨S′|P⟩ + ⟨S|P′⟩ = ⟨MS|P⟩ + ⟨S|P′⟩ =
⟨S|M^{†}P⟩ + ⟨S|P′⟩ = ⟨S|M^{†}P + P′⟩  (15)

which implies that P′ = −M^{†}P hence ⟨P|w⟩′ = −⟨M^{†}P|w⟩ = − ∑_{x∈X} u_{x}⟨P|xw⟩ From this and the fact that ker(d) = k, one obtains a contradiction (if we had deg(P) ≥ 1).

G.H.E. Duchamp, LIPN – UMR 7030 CNRS, Institut Galillée – Université Paris 13. Fuchsian type multipliers and shuffle algebras.

Sketch of the proof 2/2

▶ (i)⟹(ii) obvious ▶ (ii)⟹(iii) An equality d(f ) = ∑_{x∈X} α_{x}u_{x} for a function f ∈ C can

be rewritten d(f − ∑_{x∈X} α_{x}S_{x}) = 0 and thus f − ∑_{x∈X} α_{x}S_{x} = c ∈ k which implies that f = c and all the α_{x} are zero.

Application to polylogs

▶ In order to get a field of function we have to withdraw the poles. We then consider fields of germs of functions. For simplicity, let’s take the field of rational functions ℂ(z), realized on Ω as functions defined on some (Ω − F )_{F finite}. This field fulfils the conditions of the theorem w.r.t. d = \frac{d}{dz} and then we have the independance of the polylogarithms w.r.t. the rational functions.

▶ One can even go further (larger fields of functions, and/or other

“entries” u_{x}) ... in particular ▶ all other multipliers of “Fuchsian type” (by this we mean that all

entries have distinct singularities of order one) yield the same conclusion.

Concluding remarks and perspectives

1. We did not touch the “shuffle” condition which is a easy consequence of the primitivity of the multiplier (Δ (M) = M ⊗ 1 + 1 ⊗ M)

2. Constants of (alien ?) derivatives The operators x_{0}^{−1} and x_{1}^{−1} act as (directional ?, alien ?) derivatives on A_{C} = C ⊗_{ℂ} A_{ℂ} . They coincide respectively with z \frac{d}{d z} and (1 − z)\frac{d}{d z} on A_{ℂ}, but their constant field is much larger.

What is the maximal constant field for these derivatives ? (presumably the field of germs of analytic functions which are inessential at 0 and 1).

3. Better understanding of the combinatorics behind the differential Galois group.

S′ = MS ; (SG)′ = S′G = M(SG)