Algebra of special derivations on a free algebra

The number of indecomposable eMZVs at given weight and length is conjectured to equal the number of iterated Eisenstein integrals of corresponding weight and length upon taking relations between special derivations on a free algebra into account. Here we provide all irreducible relations known to us.

It is in principle possible though computationally very laborious to test their action on the generators of the free algebra. Thus our new relations remain conjectural at this stage despite of passing numerous consistency checks.

From Pollack, Aaron: Relations between derivations..., these relations correspond to period polynomials of cusp forms.

Depth 2

For depth 2, a general formula is known from math/0606301:

0 = \sum_{i=1}^{2n+2p-1} \frac{ [\epsilon_{2p+2n-i+1},\epsilon_{i+1}] }{(2p+2n-i-1)!} \Bigg( \frac{ (2n-1)! B_{i-2p+1} }{(i-2p+1)!}+\frac{ (2p-1)! B_{i-2n+1} }{(i-2n+1)!} \Bigg)\,.

where B_i are Bernoulli numbers and n and p are positive integers.

Depth 3

Depth 4

Depth 5