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Realization of the Stasheff polytope. (English) Zbl 1059.52017
The main result is a simple realization of the Stasheff polytope. By definition, the Stasheff polytope \(K^n\) of dimension \(n\) is a finite cell complex whose \(k\)-cells are in bijection with the planar trees having \(n-k+1\) internal vertices and \(n+2\) leaves. Different realizations of the Stasheff polytope were proposed earlier by J. D. Stasheff, S. L. Devadoss and I.M. Gelfand and others. The proposed new realization respects the symmetry, the faces have simple equations etc. The construction is the following. A planar binary tree \(T\) with \(n+1\) leaves being given, the leaves of \(T\) are labeled from left to right by \(0,1,\dots\). The internal vertices are labeled by \(1,2,\dots\) in such a way that the \(i\)th vertex falls in between the leaves \(i-1\) and \(i\). The number of leaves on the left side, resp. right side, of the \(i\)th vertex is denoted by \(a_i\), resp. \(b_i\). The product \(a_ib_i\) is called the weight of the \(i\)th vertex. Finally to the tree \(T\) the author associates a point \(M_T\) in \(R^n\) whose \(i\)th coordinate is the weight of the \(i\)th vertex of \(T\). It is demonstrated that the convex hull of all the points \(M_T\) in \(R^n\) is a realization of the Stasheff polytope \(K^{n-1}\). The constructed convex polytope lies in the affine hyperplane \(x^1+\cdots +x^n = \frac{1}{2}n(n+1)\) and has different interesting properties thoroughly analyzed by the author.

52B11 \(n\)-dimensional polytopes
55P47 Infinite loop spaces
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