MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta Values and
Quasi-Symmetric Functions
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
Michael E. Hoffman
U. S. Naval Academy
Workshop on
Grothendieck-Teichmüller Theory
and Multiple Zeta Values
Isaac Newton Institute
9 April 2013
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
Outline
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
1 Multiple Zeta Values
2 The Quasi-Symmetric Functions
3 ζ : QSym0 → R and its Extension ζu
4 u = γ and Generating Functions
ζ : QSym0 →
R and its
Extension ζu
5 Genera and Topology
u = γ and
Generating
Functions
6 Factoring ζγ
Genera and
Topology
7 MZVs of Even Arguments
Factoring ζγ
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
Multiple Zeta Values
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
For positive integers i1 , i2 , . . . ik with i1 > 1 (to ensure
convergence), we define the multiple zeta value (henceforth
MZV) ζ(i1 , i2 , . . . , ik ) as the sum of the k-fold series
X
1
.
ik
i1 i2
n
n
·
·
·
n
k
n1 >n2 ···>nk ≥1 1 2
We call k the depth of ζ(i1 , . . . , ik ), and i1 + · · · + ik its weight.
Of course, the MZVs of depth one are the values ζ(i) of the
Riemann zeta function at positive integers i > 1, for which
Euler proved the formula
(2π)2 i
|B2i |,
(1)
2(2i)!
but he also studied depth-two MZVs. Interest in MZVs of
general depth dates from about 1990. We will mention some
topological appearances of MZVs later in this talk.
ζ(2i) =
Michael E. Hoffman
MZVs & QSym
Symmetric and Quasi-Symmetric Functions
MZVs &
QSym
Michael E.
Hoffman
Let x1 , x2 , . . . be a countable sequence of indeterminates, each
of degree 1, and let
P ⊂ Q[[x1 , x2 , . . . ]]
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
be the set of formal power series in the xi of bounded degree:
P is a graded Q-algebra. Any f ∈ P is quasi-symmetric if the
coefficients in f of
xip11 xip22 · · · xipnn
and xjp11 xjp22 · · · xjpnn
agree whenever i1 < · · · < in and j1 < · · · < jn . The
quasi-symmetric functions QSym form an algebra, which
properly includes the algebra Sym of symmetric functions, e.g.,
X
xi2 xj = x12 x2 + x12 x3 + x22 x3 + · · ·
(2)
i<j
is quasi-symmetric but not symmetric.
Michael E. Hoffman
MZVs & QSym
Monomial (Quasi)symmetric Functions
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
For a composition (ordered sequence of positive integers)
I = (i1 , . . . , ik ), the corresponding monomial quasi-symmetric
function MI ∈ QSym is defined by
X
MI =
xni11 xni22 · · · xnikk
n1 <n2 <···<nk
(so (2) above is M(2,1) ). Evidently {MI |I is a composition} is
an integral basis for QSym. For any composition I , let π(I ) be
the partition given by forgetting the ordering. For any partition
λ, the monomial symmetric function mλ is the sum of the MI
with π(I ) = λ, e.g.,
m21 = M(2,1) + M(1,2) .
The set {mλ |λ is a partition} is an integral basis for Sym.
Michael E. Hoffman
MZVs & QSym
Other Bases for Sym
MZVs &
QSym
1
Michael E.
Hoffman
ek = M(1)k = mπ((1)k ) ,
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
2
ζ : QSym0 →
R and its
Extension ζu
where (1)k is the composition consisting of k 1’s. Then
{eλ |λ is a partition} is an integral basis for Sym, where
eλ = eλ1 eλ2 · · · for λ = π(λ1 , λ2 , . . . ).
The complete symmetric functions are
X
X
hk =
MI =
mλ .
|I |=k
u = γ and
Generating
Functions
Genera and
Topology
The elementary symmetric functions are
3
Then {hλ |λ is a partition} is a integral basis for Sym.
The power-sum symmetric functions are
pk = M(k) = mk .
Factoring ζγ
MZVs of
Even
Arguments
|λ|=k
Then {pλ |λ is a partition} is a rational basis for Sym.
Michael E. Hoffman
MZVs & QSym
Multiplication of Monomial Quasi-Symmetric
Functions
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
Two monomial quasi-symmetric functions MI and MJ multiply
according to a “quasi-shuffle” rule in which the parts of I and
J are shuffled and also combined. For example,
M(1) M(1,2) = M(1,1,2) + M(1,1,2) + M(1,2,1) + M(2,2) + M(1,3)
= 2M(1,1,2) + M(1,2,1) + M(2,2) + M(1,3) .
In general the number of terms in the product MI MJ is
min{`(I ),`(J)} X
`(I ) + `(J) − i
i, `(I ) − i, `(J) − i
i=0
3
so, e.g., M(1,1) M(2,1) has 42 + 111
+ 22 = 13 terms:
M(1,1) M(1,2) = 3M(1,1,1,2) + 2M(1,1,2,1) + M(1,2,1,1)
+ M(1,2,2) + M(2,1,2) + M(2,2,1) + 2M(1,1,3) + M(1,1,3,1) + M(2,3) .
Michael E. Hoffman
MZVs & QSym
Algebraic Structure of QSym
MZVs &
QSym
Michael E.
Hoffman
Outline
Over the rationals, the algebraic structure of QSym can be
described as follows. Order compositions lexicographically:
(1) < (1, 1) < (1, 1, 1) < · · · < (1, 2) < · · ·
< (2) < (2, 1) < · · · < (3) < · · ·
Multiple Zeta
Values
The QuasiSymmetric
Functions
A composition I is Lyndon if I < K whenever I = JK for
nonempty compositions J, K .
ζ : QSym0 →
R and its
Extension ζu
Theorem (Reutenauer-Malevenuto, 1995)
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
QSym is a polynomial algebra on MI , I Lyndon.
Since the only Lyndon composition ending in 1 is (1) itself, the
subspace QSym0 of QSym generated by the MI such that I
ends in an integer greater than 1 is a subalgebra, and in fact
QSym = QSym0 [M(1) ].
Michael E. Hoffman
MZVs & QSym
The Function A− : QSym → QSym
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
Consider the linear function from QSym to itself defined by
(
M(a1 ,...,ak−1 ) , if ak = 1,
A− (M(a1 ,...,ak−1 ,ak ) ) =
0,
otherwise.
(Here M∅ is interpreted as 1, so A− (M(1) ) = 1.)
From the quasi-shuffle description of multiplication in QSym
the following result can be proved.
Proposition
A− : QSym → QSym is a derivation.
Note that ker A− = QSym0 . So if we think of QSym as
QSym0 [M(1) ], then A− is differentiation by M(1) .
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
MZVs as Homomorphic Images
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
There is a homomorphism ζ : QSym0 → R given by
ζ(MI ) = ζ(ik , ik−1 , . . . , i1 )
for I = (i1 , . . . , ik ), induced by sending xj → 1j . (This is
well-defined since ik > 1 for MI ∈ QSym0 .) Note
ζ(pi ) = ζ(M(i) ) = ζ(i) for i > 1. The intersection
Sym0 = QSym0 ∩ Sym
is the subspace of Sym generated by the mλ such that all parts
of λ are 2 or more, and is in fact the subalgebra of Sym
generated by the pi with i > 1.
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
Extending ζ to ζu : QSym → R[u]
MZVs &
QSym
Michael E.
Hoffman
Outline
Since QSym = QSym0 [M(1) ], we can extend ζ to a
homomorphism ζu : QSym → R[u] by defining ζu (w ) = ζ(w )
for w ∈ QSym0 and ζu (M(1) ) = u. In view of the Proposition
above, we have the following commutative diagram:
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
ζu
QSym −−−−→ R[u]



d 
A− y
du y
ζu
QSym −−−−→ R[u]
In the case u = γ ≈ 0.5572 (Euler’s constant), the
homomorphism ζu turns out to be especially useful. To explain
why we must digress a bit to introduce some generating
functions.
Michael E. Hoffman
MZVs & QSym
Generating Functions
MZVs &
QSym
1
Michael E.
Hoffman
Outline
The generating function of the elementary symmetric
functions is
X
Y
E (t) =
ei t i =
(1 + txi ).
i≥0
Multiple Zeta
Values
The QuasiSymmetric
Functions
2
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
That for the complete symmetric functions is
X
Y 1
= E (−t)−1 .
H(t) =
hi t i =
1 − txi
i≥0
3
i≥1
The logarithmic derivative of H(t) is the generating
function P(t) for the power sums:
Genera and
Topology
Factoring ζγ
i≥1
P(t) =
X d
X
d
log H(t) = −
log(1 − txi ) =
pi t i−1 .
dt
dt
i≥1
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
i≥1
Generating Functions Cont’d
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
The power series expansion of ψ(1 − t), where ψ is the
logarithmic derivative of the gamma function, is
X
ψ(1 − t) = −γ −
ζ(i)t i−1 .
i≥2
If u = γ, then ζu (p1 ) = γ and so ζu (P(t)) = −ψ(1 − t).
d
log H(t)) we have
Hence (since P(t) = dt
Theorem (Generating Function)
ζγ (H(t)) = Γ(1 − t) .
From E (t) = H(−t)−1 follows ζγ (E (t)) = Γ(1 + t)−1 .
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
Generating Functions Cont’d
MZVs &
QSym
Michael E.
Hoffman
Now it can be shown analytically that the generating function
X
F (s, t) =
ζ(n + 1, (1)m−1 )s n t m
n,m≥1
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
can be written
F (s, t) = 1 −
Γ(1 − t)Γ(1 − s)
.
Γ(1 − t − s)
Applying the Generating Function Theorem, this is
H(t)H(s)
F (s, t) = ζγ 1 −
.
H(t + s)
Now




X pi t i
X pi t i
 = e p1 t exp 
,
H(t) = exp 
i
i
i≥1
Michael E. Hoffman
i≥2
MZVs & QSym
Generating Functions Cont’d
MZVs &
QSym
Michael E.
Hoffman
so it follows that

1−
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
H(t)H(s)
= 1 − exp 
H(t + s)
X
pi
ti
+
si
i≥2
− (t +
i
s)i


and hence, applying ζγ ,
X
ζ(n + 1, (1)m−1 )s n t m =
n,m≥1

1 − exp 
X
ζ(i)
i≥2
ti
+
si
− (t +
i
s)i

.
Note that any MZV of the form ζ(n + 1, (1)m−1 ) is a
polynomial in the ζ(i), i ≥ 2, with rational coefficients, though
/ Sym for m > 1.
M((1)m−1 ,n+1) ∈
Michael E. Hoffman
MZVs & QSym
Hirzebruch’s Theory of Genera
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
A genus is a function φ that assigns to any manifold M in
some class (say those with an almost complex structure) a
number φ(M), subject to the requirement that
φ(M × N) = φ(M)φ(N).
For an almost complex M of dimension 2n, a genus can be
given by
φ(M) = hKn (c1 , . . . , cn ), [M]i,
where the ci ∈ H 2i (M; Z) are the Chern classes of M,
[M] ∈ H2n (M; Z) is the fundamental class of M, and {Ki } is a
multiplicative sequence: i.e., a sequence of homogeneous
polynomials (with deg Ki = i) so that, if
K (1 + c1 + c2 + · · · ) = 1 + K1 (c1 ) + K2 (c1 , c2 ) + · · · ,
then K (a)K (b) = K (ab) for any formal series a, b having
constant term 1.
Michael E. Hoffman
MZVs & QSym
Multiplicative Sequences
MZVs &
QSym
Michael E.
Hoffman
In fact, a multiplicative sequence {Ki } is determined by the
power series
K (1 + t) = 1 + r1 t + r2 t 2 + r3 t 3 + . . . ,
(3)
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
since (by the algebraic independence of the elementary
symmetric functions ei ) we can think of the homogeneous
terms of an arbitrary formal series as the elementary symmetric
functions ei in new variables x1 , x2 , . . . :
1 + e1 + e2 + e3 · · · = (1 + x1 )(1 + x2 ) · · ·
and thus
K (1 + e1 + e2 + · · · ) = K (1 + x1 )K (1 + x2 ) · · · .
We say {Ki } is the multiplicative sequence belonging to the
power series (3).
Michael E. Hoffman
MZVs & QSym
Mirror Symmetry and the Γ-Genus
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
The Γ-genus is given by
Γ(M 2n ) = hQn (c1 , . . . , cn ), [M 2n ]i,
where {Qi } is the multiplicative sequence belonging to the
series
1
Γ(1 + t)−1 = 1 + γt + (γ 2 − ζ(2))t 2 + · · · .
2
The polynomials Qi occur in the context of mirror symmetry,
which involves Calabi-Yau manifolds (for our purposes, simply
connected Kähler manifolds whose first Chern class vanishes).
Factoring ζγ
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
Mirror Symmetry and the Γ-Genus Cont’d
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
String theorists are interested in Calabi-Yau manifolds of (real)
dimension 6, since they occur in supersymmetry theory.
(10-dimensional spacetime looks locally like R4 × M 6 , where
M 6 is a Calabi-Yau manifold.) They found that Calabi-Yau
manifolds seemed to come in “mirror pairs,” i.e. distinct
manifolds that gave the same physics. At present there are
mathematically rigorous constructions of mirrors only for
certain classes of Calabi-Yau manifolds (or orbifolds), e.g.,
hypersurfaces or complete intersections in toric varieties.
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
Mirror Symmetry Cont’d
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
For one such class, Hosono, Klemm, Theisen and Yau (1995)
found relations between the Chern classes of a manifold and
period functions on its mirror: e.g., for a Calabi-Yau 3-fold X
that is a complete intersection in a product of weighted
projective spaces,
hc3 , [X ]i =
∂3c
1
Kijk
(0, . . . , 0).
6ζ(3)
∂ρi ∂ρj ∂ρk
(4)
Here c(ρ1 , ρ2 , . . . ) are coefficients of a generalized
hypergeometric series for the period function on a mirror of X
(with gamma functions replacing the factorials so it can be
differentiated), and Kijk is the Yukawa coupling (B-model
correlation function).
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
Libgober’s Formula
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
A. Libgober (1999) generalized these relations to higher
dimensions: e.g., if X is a Calabi-Yau d-fold which is a
hypersurface in a toric Fano manifold satisfying a technical
condition,
hQd (c1 , . . . , cd ), [X ]i =
∂ d c(0, . . . , 0)
1
Ki1 ,i2 ,...,id
d!
∂ρi1 ∂ρi2 · · · ∂ρid
where Ki1 ,...,id is the suitably normalized d-point function
corresponding to a mirror X̃ of X and c(ρ1 , . . . , ) are the
coefficients of the hypergeometric series for the holomorphic
period of X̃ at a maximal degeneracy point (of a partial
compactification of the deformation space of X̃ ).
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
(5)
Libgober’s Formula Cont’d
MZVs &
QSym
Michael E.
Hoffman
Outline
We can write the coefficients of the Qi in terms of MZVs.
Theorem
The coefficient of eλ in Qi (e1 , . . . , ei ) is ζγ (mλ ) for any
partition λ of i.
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Proof.
We use Generating Function Theorem, together with the
symmetry of the transition matrices between the bases {eλ }
and {mλ } of Sym:
u = γ and
Generating
Functions
X
Genera and
Topology
j≥0
Factoring ζγ
MZVs of
Even
Arguments
1
=
Γ(1 + xi )
i≥1
YX
X
X
ζγ (ej )xij =
ζγ (eλ )mλ =
ζγ (mλ )eλ .
Qj (e1 , . . . , ej ) =
Y
i≥1 j≥0
Michael E. Hoffman
λ
MZVs & QSym
λ
Libgober’s Formula Cont’d
MZVs &
QSym
Thus, for example,
1
Q2 (c1 , c2 ) = ζ(2)c2 + (γ 2 − ζ(2))c12
2
Michael E.
Hoffman
(6)
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
and
Q3 (c1 , c2 , c3 ) = ζ(3)c3 + (γζ(2) − ζ(3))c1 c2 +
1 3
(γ − 3γζ(2) + 2ζ(3))c13 (7)
6
In the Calabi-Yau case we have c1 = 0 and terms involving γ
don’t appear: e.g.,
Q3 (0, c2 , c3 ) = ζ(3)c3
which shows that (5) generalizes (4).
Michael E. Hoffman
MZVs & QSym
Lu’s Γ̂-genus
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
Rongmin Lu (2008) considered the genus belonging to the
power series
e −γt Γ(1 + t)−1 ,
(8)
which he calls the Γ̂-genus, and related it to a regularized
S 1 -equivariant Euler class.
The logarithmic derivative of (8) is
X
−γ − ψ(1 + t) =
ζ(i)(−t)i−1 ,
i≥2
which is just the corresponding one for the Γ-genus with γ
removed. If we write {Pn } for the multiplicative sequence
corresponding to the Γ̂-genus, then Pn is Qn with γ set to zero.
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
Lu’s Genus Cont’d
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
That is, the coefficient of eλ in Pn (e1 , . . . , en ) is ζ0 (mλ ). For
example,
1
P2 (c1 , c2 ) = ζ(2)c2 − ζ(2)c12
2
from equation (6), and
1
P3 (c1 , c2 , c3 ) = ζ(3)c3 − ζ(3)c1 c2 + ζ(3)c13
3
from equation (7).
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
Hopf Algebra Characters
MZVs &
QSym
Michael E.
Hoffman
Now QSym is a graded Hopf algebra, with coproduct
X
∆(MI ) =
MJ ⊗ MK
I =JK
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
where the sum is over all decompositions of I as a juxtaposition
(including the cases J = ∅ and K = ∅). Also, ζγ is a real
character of the Hopf algebra QSym (that is, an algebra
homomorphism QSym → R). A character χ of QSym is even if
χ(w ) = (−1)|w | χ(w )
(9)
for homogeneous elements w of QSym, and odd if
χ(w ) = (−1)|w | χ(S(w )),
where S is the antipode of QSym.
Michael E. Hoffman
MZVs & QSym
(10)
The ABS Theorem
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
Theorem (Aguiar, Bergeron and Sottille, 2006)
Any real character of a Hopf algebra has a unique
decomposition in the convolution algebra into an even
character times an odd one.
Thus, there is an even character ζ+ and an odd character ζ− so
that ζγ = ζ+ ζ− in the convolution algebra, i.e.,
X
ζγ (w ) =
ζ+ (w(1) )ζ− (w(2) )
(11)
(w )
using Sweedler’s notation for coproducts:
X
∆(w ) =
w(1) ⊗ w(2) .
(w )
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
Primitives
MZVs &
QSym
The power sums pi = M(i) are primitive, i.e.
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
∆(pi ) = 1 ⊗ pi + pi ⊗ 1,
and so
ζ(i) = ζ(pi ) = ζ− (pi ) + ζ+ (pi ).
(12)
Equation (9) implies that ζ+ (w ) = 0 for any odd-dimensional
w , particularly w = pi , i odd. On the other hand, equation
(10) gives ζ− (pi ) = 0 for i even, since S(pi ) = −pi . Together
with equation (12), this says

(

i even,
0,
ζ(i), i even,
ζ+ (pi ) =
and ζ− (pi ) = ζ(i), i > 1 odd,

0,
i odd,

γ,
i = 1.
Michael E. Hoffman
MZVs & QSym
Even Parts Theorem
MZVs &
QSym
Michael E.
Hoffman
Outline
In general, computing ζ+ (MI ) and ζ− (MI ) for an arbitrary
monomial quasi-symmetric function MI is difficult, apart from
the general fact that
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
ζ+ (MI ) = 0
if |I | is odd.
Nevertheless, there is the following result, which can be proved
from the general theory of Aguiar-Bergeron-Sottille together
with a specific result of Aguiar-Hsiao (2004).
u = γ and
Generating
Functions
Theorem
Genera and
Topology
If all parts of I are even, then ζ− (MI ) = 0.
Factoring ζγ
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
Computing ζ+ and ζ− on Sym
MZVs &
QSym
Michael E.
Hoffman
The situation is dramatically different if ζγ is restricted to Sym
(which is all we need for the Γ- and Γ̂-genera). The pi generate
Sym, so for every partition λ
Outline
mλ = Pλ (p1 , p2 , p3 , p4 , . . . )
Multiple Zeta
Values
The QuasiSymmetric
Functions
for some polynomial Pλ with rational coefficients. Since ζ+ and
ζ− are homomorphisms,
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
ζ− (mλ ) = Pλ (γ, 0, ζ(3), 0, . . . )
and
ζ+ (mλ ) = Pλ (0, ζ(2), 0, ζ(4), . . . ).
In view of Euler’s identity (1), the latter formula means that
ζ+ (mλ ) is a rational multiple of π |λ| when |λ| is even (and of
course ζ+ (mλ ) = 0 when |λ| is odd).
Michael E. Hoffman
MZVs & QSym
Generating Functions Yet Again
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
Since the involution of Sym that exchanges the ei and the hi
leaves the odd pi fixed, it follows that ζ− (en ) = ζ− (hn ), i.e.,
H(t)
ζ−
= 1.
E (t)
On the other hand, ζ+ (E (t)) is an even function, so
ζ+ (E (t)) = ζ+ (E (−t)) = ζ+ (H(t))−1 and thus
H(t)
ζ+
= ζ+ (H(t))2 .
E (t)
Hence, using ζ+ ζ− = ζγ and the Generating Function Theorem,
H(t)
H(t)
H(t)
ζ+ (H(t))2 = ζ+
ζ−
= ζγ
=
E (t)
E (t)
E (t)
Γ(1 − t)Γ(1 + t).
Michael E. Hoffman
MZVs & QSym
Factored Generating Function Theorem
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Because of the reflection formula for the gamma function, this
is
πt
.
ζ+ (H(t))2 =
sin πt
Thus we have the following result.
Theorem (Factored Generating Function)
The ABS factors of ζγ are given by
r
−1
ζ+ (E (t)) = ζ+ (H(t))
ζ− (E (t)) = ζ− (H(t)) =
Factoring ζγ
=
r
sin πt
πt
sin πt
Γ(1 − t).
πt
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
Explicit Formula for ζ− (en )
MZVs &
QSym
There is also the explicit formula for en in terms of power sums:
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
en =
X
i1 +2i2 +···+nin =n
p in
(−1)n
n
(−p1 )i1 · · · −
,
i1 !i2 ! · · · in !
n
which follows from
E (t) = H(−t)
−1
Z
= exp −
−t
P(s)ds .
0
If we apply ζ− to this, the minus signs cancel nicely to give
X
ζ− (en ) =
i1 +3i3 +5i5 ···=n
γ i1 ζ(3)i3 ζ(5)i5 · · ·
.
i1 !3i3 i3 !5i5 i5 ! · · ·
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
(13)
An Example
MZVs &
QSym
Michael E.
Hoffman
To see how to put these results together, we note that
r
sin πt
π2t 2
π4t 4
=1−
+
− ···
πt
12
1440
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
Then since the ei are divided powers,
∆(e4 ) = 1 ⊗ e4 + e1 ⊗ e3 + e2 ⊗ e2 + e3 ⊗ e1 + e4 ⊗ 1
and from the Factored Generating Function Theorem together
with equations (11) and (13) it follows that
ζγ (e4 ) = ζ− (e4 ) + ζ+ (e2 )ζ− (e2 ) + ζ+ (e4 )
π4
γ 4 γζ(3) π 2 γ 2
+
−
·
+
4!
3
12 2
1440
γ 4 γ 2 ζ(2) γζ(3) ζ(4)
=
−
+
+
.
24
4
3
16
=
Michael E. Hoffman
MZVs & QSym
MZVs of Even Arguments
MZVs &
QSym
Michael E.
Hoffman
Outline
Suppose now we consider the sum E (2n, k) of all
even-argument MZVs of weight 2n and depth k, i.e.,
X
E (2n, k) =
ζ(2i1 , . . . , 2ik ).
i1 +···+ik =n
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
Then of course E (2n, 1) = ζ(2n). It follows easily from results
of Euler that E (2n, 2) = 34 ζ(2n), though no one seems to have
pointed this out in print until Gangl, Kaneko and Zagier did in
2006. Last year Shen and Cai published the formulas
5
1
E (2n, 3) = ζ(2n) − ζ(2)ζ(2n − 2)
8
4
35
5
E (2n, 4) = ζ(2n) − ζ(2)ζ(2n − 2)
64
16
Michael E. Hoffman
MZVs & QSym
A Formula for E (2n, k)
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
Inspired by these results, I proved the general formula
2k − 1
1
ζ(2n)
E (2n, k) = 2(k−1)
k
2
b k−1
c
2
X
2k − 2j − 1
1
ζ(2j)ζ(2n − 2j).
−
k
22k−3 (2j + 1)B2j
j=1
My proof used the following explicit generating-function result.
Theorem (Even MZV Sums)
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
p
sin(π (1 − s)t)
√ .
E (t, s) := 1 +
E (2n, k)t s = √
1 − s sin(π t)
n,k≥1
X
n k
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
More Generating Functions
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Let Nn,k be the sum of all monomial symmetric functions
corresponding to partitions of n of length k. Then Nn,k = 0
unless k ≤ n, and Nk,k = ek . We have ζD(Nn,k ) = E (2n, k),
where D : QSym → QSym is the degree-doubling function that
sends xi to xi2 , and thus M(i1 ,...,ik ) to M(2i1 ,...,2ik ) . If we let
F(t, s) = 1 +
X
Nn,k t n s k
n,k≥1
then E (t, s) = ζD(F(t, s)). (Note there is no need to use an
extension of ζ here, since D(QSym) ⊂ QSym0 .)
Factoring ζγ
MZVs of
Even
Arguments
Michael E. Hoffman
MZVs & QSym
More Generating Functions Cont’d
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
There is the formal factorization
∞
Y
F(t, s) =
(1 + stxi + st 2 xi2 + . . . )
i=1
∞
Y
1 + (s − 1)txi
=
1 − txi
i=1
= E ((s − 1)t)H(t)
To apply the homomorphism ζD to both sides of this equation,
we recall the formula
ζD(en ) = ζ((2)n ) =
Factoring ζγ
MZVs of
Even
Arguments
π 2n
(2n + 1)!
(which appeared in my 1992 paper), so that
Michael E. Hoffman
MZVs & QSym
More Generating Functions Cont’d
MZVs &
QSym
√
sinh(π t)
√
ζD(E (t)) =
.
π t
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Hence
√
√
π −t
π t
√ ,
√
ζD(H(t)) =
=
sinh(π −t)
sin(π t)
from which follows
p
√
sinh(π (s − 1)t) π t
p
√
ζD(F(s, t)) =
sin(π t)
π (s − 1)t
p
sin(π (1 − s)t)
√ ,
=√
1 − s sin(π t)
Factoring ζγ
MZVs of
Even
Arguments
and the Even MZV Sums Theorem is proved.
Michael E. Hoffman
MZVs & QSym
An Extension
MZVs &
QSym
Michael E.
Hoffman
Outline
Multiple Zeta
Values
The QuasiSymmetric
Functions
ζ : QSym0 →
R and its
Extension ζu
u = γ and
Generating
Functions
Genera and
Topology
Factoring ζγ
MZVs of
Even
Arguments
Note that D2 : QSym → QSym is the degree-quadrupling
homomorphism that sends xi to xi4 . Then it is known (See
Borwein, Bradley, and Broadhurst, Electron. J. Combin. 1997)
that
√
√
sinh(π 4 t) sin(π 4 t)
2
√
,
ζD (E (−t)) =
π4t
from which follows
p
p
sinh(π 4 (1 − s)t) sin(π 4 (1 − s)t)
√
√
√
ζD (F(t, s)) =
.
1 − s sinh(π 4 t) sin(π 4 t)
2
From this H. Yuan and J. Zhao obtained a (somewhat
complicated) formula for the sum of all MZVs of depth k and
weight 4n having all arguments divisible by 4.
Michael E. Hoffman
MZVs & QSym