Multiple polylogarithms and the Steinberg module

We establish a connection between multiple polylogarithms on a torus and the Steinberg module of $\mathbb{Q}$, and show that multiple polylogarithms of depth $d$ and weight $n$ can be expressed via a single function ${\mathrm{L}\mathrm{i}}_{n-d+1,1,\dots ,1}(x_1,x_2,\dots ,x_d)$. Using this connection, we give a simple proof of the Bykovskiĭ theorem, explain the duality between multiple polylogarithms and iterated integrals, and provide a polylogarithmic interpretation of the conjectures of Rognes and Church-Farb-Putman.