Explicit formulas for Grassmannian polylogarithms

Abstract

We give a new explicit formula for Grassmannian polylogarithms in terms of iterated integrals. We also explicitly reduce the Grassmannian polylogarithm in weight 4 and in weight 5 each to depth 2. Furthermore, using this reduction in weight 4 we obtain an explicit, albeit complicated, form of the so-called 4-ratio, which gives an expression for the Borel class in continuous cohomology of GL 4 in terms of Li 4 .

On the Goncharov depth conjecture and a formula for volumes of orthoschemes

“…This result is sharp as it is easy to show using a coproduct (discussed in §2.1) that a general multiple polylogarithm of weight n can not be expressed via multiple polylogarithms of depth strictly less than tn{2u . The previous best-known result states that a polylogarithm of depth n can be expressed via polylogarithms of depth at most n ´2; see [Cha17], [CGR19b], and [CGR19a] for further results on the depth reduction problem for multiple polylogarithms.…”