Permutoassociahedron information

The permutoassociahedron of dimension $2$ and the correspondence between its vertices and the bracketed permutations of three terms $a$ , $b$ , and $c$ .

The four facets of the permutoassociahedron of dimension $3$ that share vertex $(ab)(cd)$ . Three of these facets are quadrilaterals and the fourth is a pentagon.

In mathematics, the permutoassociahedron is an $n$ -dimensional polytope whose vertices correspond to the bracketings of the permutations of $n+1$ terms and whose edges connect two bracketings that can be obtained from one another either by moving a pair of brackets using associativity or by transposing two consecutive terms that are not separated by a bracket.

The permutoassociahedron was first defined as a CW complex by Mikhail Kapranov who noted that this structure appears implicitly in Mac Lane's coherence theorem for symmetric and braided categories as well as in Vladimir Drinfeld's work on the Knizhnik–Zamolodchikov equations.^[1] It was constructed as a convex polytope by Victor Reiner and Günter M. Ziegler.^[2]

^ Kapranov, Mikhail M. (1993). "The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation". Journal of Pure and Applied Algebra. 85 (2): 119–142. doi:10.1016/0022-4049(93)90049-Y.
^ Reiner, Victor; Ziegler, Günter M. (1994). "Coxeter-associahedra". Mathematika. 41 (2): 364–393. doi:10.1112/S0025579300007452.

[1] Kapranov, Mikhail M. (1993). "The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation". Journal of Pure and Applied Algebra. 85 (2): 119–142. doi:10.1016/0022-4049(93)90049-Y.

[2] Reiner, Victor; Ziegler, Günter M. (1994). "Coxeter-associahedra". Mathematika. 41 (2): 364–393. doi:10.1112/S0025579300007452.

Permutoassociahedron information

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Permutoassociahedron

Permutohedron

Associahedron

Cyclohedron