The algebra of multiple harmonic series

Recently there has been much interest in multiple harmonic series [GRAPHICS] (which converge when the exponents i(j) are positive integers and i(1) > 1), also known as multiple zeta values or Euler/Zagier sums. Starting with the noncommutative polynomial algebra Q[x,y], we define a second multiplication which is commutative and associative, and call the resulting structure the harmonic algebra h. As a graded commutative algebra, I) turns out to be a free polynomial algebra with the number of generators in degree n given by the Witt formula for the number N(n) of basis elements of degree rt in the free Lie algebra on two generators. Multiple harmonic series can be thought of as images under a map zeta:h(0) --> R which is a homomorphism with respect to the commutative multiplication, where h(0) is an appropriate subalgebra of h. If we call i(1) + ... +i(k) the weight of the series zeta(i(1),..., i(k)), then (for n > 1) there are at most N(n) series of weight n that are irreducible in the sense that they are not sums of rational multiples of products of multiple harmonic series of lower weight. In fact, our approach gives an explicit set of ''algebraically'' irreducible multiple harmonic series. We also show that there is a subalgebra h(0) subset of h(1) subset of h related to the shuffle algebra which contains the algebra G of symmetric functions: in fact, zeta maps the elements of G n h(0) to algebraic combinations of zeta values zeta(i), for integer i much less than 2. The map zeta is not injective; we show how several results about multiple harmonic series can be recast as statements about the kernel of zeta, and propose some conjectures on the structure of the algebra h(0)/ker zeta. (C) 1997 Academic Press.