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Comparing Diagonals on the Associahedra

Samson Saneblidze, Ronald Umble

Abstract

We prove that the formula for the diagonal approximation $Δ_{K}$ 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 $Δ_{K}^{\prime}$ derived by Markl and Shnider in 2006, by Loday in 2011, and by Masuda, Thomas, Tonks, and Vallette in 2021.

Comparing Diagonals on the Associahedra

Abstract

We prove that the formula for the diagonal approximation on J. Stasheff's -dimensional associahedron derived by the current authors in 2004 agrees with the "magical formula" for the diagonal approximation derived by Markl and Shnider in 2006, by Loday in 2011, and by Masuda, Thomas, Tonks, and Vallette in 2021.
Paper Structure (3 sections, 3 theorems, 32 equations)

This paper contains 3 sections, 3 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Diagonals Induced by $\Delta_P$
  3. Agreement of $\Delta_K$ and $\Delta^{\prime }_{K}$

Key Result

Proposition 1

If $a$ is an associahedral $k$-cell and $u$ is a subdivision $k$-cell of $a,$ then

Theorems & Definitions (14)

  • Example 1
  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • proof
  • Example 2
  • Proposition 2
  • proof
  • ...and 4 more