Celebrating Loday’s associahedron. (English) Zbl 1534.52012
The associahedron is a polytope whose combinatorial structure indifferently encodes rotations between binary trees, flips in the triangulations of a convex polygon, or associativity in parenthesised words. This structure can be traced back to Dov Tamari’s PhD thesis and Jim Stasheff’s work on loop spaces, but at that time, it was described as a topological ball and it took decades before it was constructed as a euclidean polytope. A particularly elegant geometric construction of the associahedron has been given by J.-L. Loday in a seminal article that appeared in Archiv der Mathematik [Arch. Math. 83, No. 3, 267–278 (2004; Zbl 1059.52017)].
For the combined 20th anniversary of Jean-Louis Loday’s article and 75th anniversary of Archiv der Mathematik, Vincent Pilaud, Francisco Santos, and Günter M. Ziegler survey Loday’s construction and a number of the results it has inspired since then.
The survey shows in a first part how Loday’s construction influenced the study of the lattice quotients of the weak order from both a geometric and an algebraic perspective. Its second and third parts review related constructions that owe something to Loday’s and their generalizations that arise for instance from the theory of cluster algebra, from the study of subword complexes, or from the theory of quiver representation. The fourth part highlights how Loday’s construction is instrumental in the recent definition of a cellular diagonal for the associahedron. In the last part, a list of other generalizations of the associahedron is provided that can be obtained from Loday’s construction, further illustrating its depth and influence.
For the combined 20th anniversary of Jean-Louis Loday’s article and 75th anniversary of Archiv der Mathematik, Vincent Pilaud, Francisco Santos, and Günter M. Ziegler survey Loday’s construction and a number of the results it has inspired since then.
The survey shows in a first part how Loday’s construction influenced the study of the lattice quotients of the weak order from both a geometric and an algebraic perspective. Its second and third parts review related constructions that owe something to Loday’s and their generalizations that arise for instance from the theory of cluster algebra, from the study of subword complexes, or from the theory of quiver representation. The fourth part highlights how Loday’s construction is instrumental in the recent definition of a cellular diagonal for the associahedron. In the last part, a list of other generalizations of the associahedron is provided that can be obtained from Loday’s construction, further illustrating its depth and influence.
Reviewer: Lionel Pournin (Paris)
MSC:
| 52B11 | \(n\)-dimensional polytopes |
| 52B12 | Special polytopes (linear programming, centrally symmetric, etc.) |
| 52B15 | Symmetry properties of polytopes |
Citations:
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