On the Broadhurst-Kreimer generating series for multiple zeta values

Abstract

Let F denote the free polynomial algebra F = Q〈s3, s5, s7, . . .〉 on non-commutative variables si for odd i ≥ 3. The algebra F is weight-graded by letting sn be of weight n; we write Fn for the weight n part. In this paper we put a “special” decreasing depth filtration F = F ⊃ F ⊃ · · · ⊃ F ⊃ F · · · on F , based on the period polynomials associated to cusp forms on SL2(Z). We define a lattice L of particular combinatorially defined subspaces of F , and conjecture that this lattice is distributive. Assuming this conjecture, we show that the dimensions of the weight n filtered quotients Fd n/F d+1 n are given by the coefficients of the well-known Broadhurst-Kreimer generating series, defined by them to predict dimensions for the algebra of multiple zeta values. We end by explaining the expected relationship between F equipped with the special depth filtration and the algebras of formal and motivic multiple zeta values.