Comparing Diagonals on the Associahedra

Abstract

We prove that the formula for the diagonal approximation ${\Delta }_K$ on J. Stasheff's $n$-dimensional associahedron $K_{n+2}$ derived by the current authors in 2004 agrees with the "magical formula" for the diagonal approximation ${\Delta }_K^{\prime }$ derived by Markl and Shnider in 2006, by Loday in 2011, and by Masuda, Thomas, Tonks, and Vallette in 2021.

The initial posting defines the S-U diagonal ${\Delta }_K$ on the top dimensional cell of $K_n$; its value on proper faces is given by comultiplicative extension. Version 2 defines ${\Delta }_K$ on all faces of $K_n$ explicitly. Version 3 incorporates recommendations by the referee. Version 4 cites a paper by Loday. 8 pages, 3 Figures