Michael E. Hoffman

Quasi-Shuffle Products

Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication * on the set of noncommutative polynomials in A which we call a quasi-shuffle product; it can be viewed as a generalization of the shuffle product III. We extend this commutative algebra structure to a Hopf algebra (U, *, Δ); in the case where A is the set of positive integers and the operation on A is addition, this gives the Hopf algebra of quasi-symmetric functions. If rational coefficients are allowed, the quasi-shuffle product is in fact no more general than the shuffle product; we give an isomorphism exp of the shuffle Hopf algebra (U, III, Δ) onto (U, *, Δ) the set L of Lyndon words on A and their images { exp(w) ∣ w ∈ L} freely generate the algebra (U, *). We also consider the graded dual of (U, *, Δ). We define a deformation *q of * that coincides with * when q = 1 and is isomorphic to the concatenation product when q is not a root of unity. Finally, we discuss various examples, particularly the algebra of quasi-symmetric functions (dual to the noncommutative symmetric functions) and the algebra of Euler sums.

Combinatorics of rooted trees and Hopf algebras

We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally associates a multiplicity to each pair of rooted trees. By using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the associated multiplicities. Now the symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators, as well as the inner product mentioned above, to Kreimers Hopf algebra. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this leads to the Hopf algebra of Grossman and Larson. We show that the inner product on rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with the graded dual of Kreimers Hopf algebra, correcting an earlier result of Panaite.

Relations of multiple zeta values and their algebraic expression

Abstract We establish a new class of relations, which we call the cyclic sum identities, among the multiple zeta values ζ(k1,…,kl)=∑n1>⋯>nl⩾11/(n1k1⋯nkkl). These identities have an elementary proof and imply the “sum theorem” for multiple zeta values. They also have a succinct statement in terms of “cyclic derivations” as introduced by Rota, Sagan, and Stein. In addition, we discuss the expression of other relations of multiple zeta values via the shuffle and “harmonic” products on the underlying vector space H of the noncommutative polynomial ring Q 〈x,y〉 , and also using an action of the Hopf algebra of quasi-symmetric functions on Q 〈x,y〉 .

Quasi-symmetric functions and mod p multiple harmonic sums

We present a number of results about (finite) multiple harmonic sums modulo a prime, which provide interesting parallels to known results about multiple zeta values (i.e., infinite multiple harmonic series). In particular, we prove a “duality” result for mod p harmonic sums similar to (but distinct from) that for multiple zeta values. We also exploit the Hopf algebra structure of the quasi-symmetric functions to do calculations with multiple harmonic sums mod p, and obtain, for each weight n ≤ 9, a set of generators for the space of weight-n multiple harmonic sums mod p.

Generalized Chebyshev polynomials associated with affine Weyl groups

We begin with a compact figure that can be folded into smaller replicas of itself, such as the interval or equilateral triangle. Such figures are in one-to-one correspondence with affine Weyl groups. For each such figure in n-dimensional Euclidean space, we construct a sequence of polynomials Pk: Rn -Rn so that the mapping Pk is conjugate to stretching the figure by a factor k and folding it back onto itself. If n = 1 and the figure is the interval, this construction yields the Chebyshev polynomials (up to conjugation). The polynomials Pk are orthogonal with respect to a suitable measure and can be extended in a natural way to a complete set of orthogonal polynomials.

Algebraic Aspects of Multiple Zeta Values

Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves “coding” the multiple zeta values by monomials in two noncommuting variables x and y. Multiple zeta values can then be thought of as defining a map ζ: ℌ0 → R from a graded rational vector space ℌ0 generated by the “admissible words” of the noncommutative polynomial algebra Q〈x,y〉. Now ℌ0 admits two (commutative) products making ζ a homomorphism-the shuffle product and the “harmonic” product. The latter makes ℌ0 a subalgebra of the algebra QSym of quasi-symmetric functions. We also discuss some results about multiple zeta values that can be stated in terms of derivations and cyclic derivations of Q〈x,y〉, and we define an action of QSym on Q〈x,y〉 that appears useful. Finally, we apply the algebraic approach to relations of finite partial sums of multiple zeta value series.

On multiple zeta values of even arguments

For k ≤ n, let E(2n,k) be the sum of all multiple zeta values with even arguments whose weight is 2n and whose depth is k. Of course E(2n, 1) is the value ζ(2n) of the Riemann zeta function at 2n, and it is well known that E(2n, 2) = 3 4ζ(2n). Recently Shen and Cai gave formulas for E(2n, 3) and E(2n, 4) in terms of ζ(2n) and ζ(2)ζ(2n − 2). We give two formulas for E(2n,k), both valid for arbitrary k ≤ n, one of which generalizes the Shen–Cai results; by comparing the two we obtain a Bernoulli-number identity. We also give explicit generating functions for the numbers E(2n,k) and for the analogous numbers E⋆(2n,k) defined using multiple zeta-star values of even arguments.

Endomorphisms of the cohomology of complex Grassmannians

For any complex Grassmann manifold C, we classify all endomorphisms of the rational cohomology ring of C which are nonzero on dimension 2. Some applications of this result are given.

The Hopf algebra structure of multiple harmonic sums

Multiple harmonic sums appear in the perturbative computation of various quantities of interest in quantum field theory. In this article we introduce a class of Hopf algebras that describe the structure of such sums, and develop some of their properties that can be exploited in calculations.

Periods of mirrors and multiple zeta values

In a recent paper, A. Libgober showed that the multiplicative sequence {Q i (c 1 ,…,c i )} of Chern classes corresponding to the power series Q(z) = Γ(1 + z) -1 appears in a relation between the Chern classes of certain Calabi-Yau manifolds and the periods of their mirrors. We show that the polynomials Q i can be expressed in terms of multiple zeta values.