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A simple permutoassociahedron. (English) Zbl 1475.52018

The \(n\)-dimensional permutoassociahedron is a polytope, first introduced by M. M. Kapranov as a CW-complex [J. Pure Appl. Algebra 85, No. 2, 119–142 (1993; Zbl 0812.18003)] and then constructed as a polytope by V. Reiner and G. M. Ziegler [Mathematika 41, No. 2, 364–393 (1994; Zbl 0822.52007)]. Its vertices correspond to the bracketings of the permuted products of \(n+1\) terms. For instance, the expressions \((ab)c\), \(a(bc)\), and \((ba)c\) correspond to three different vertices of the \(2\)-dimensional permutoassociahedron.
The edges of the permutoassociahedron can be classified into two types. The first type of edges connect two bracketed products when they are obtained from one another by using the associativity rule once, as for instance \((ab)c\) and \(a(bc)\). The second type of edges connect two products obtained from one another by transposing two terms that are not separated by a bracket, as for example \((ab)c\) and \((ba)c\). By this description, the permutoassociahedron is not a simple polytope as soon as \(n\geq3\). Indeed, the number of possible transpositions depends on how the expression is bracketed.
An \(n\)-dimensional simple CW-complex analogue to the permutoassociahedron has been described by Z. Petrić [J. Algebr. Comb. 39, No. 1, 99–125 (2014; Zbl 1295.52015)]. Its vertices still correspond to the bracketings of permuted products of \(n+1\) terms, but its edges that correspond to transpositions are modified. Here, that simple permutoassociahedron is constructed as a polytope. More precisely, the polytope is obtained as an intersection of half-spaces that the authors describe explicitly. An alternative construction of the simple permutoassociahedron as a Minkowski sum has been given by J. Ivanović [Appl. Anal. Discrete Math. 14, No. 1, 055–093 (2020; Zbl 1474.52018)].

MSC:

52B11 \(n\)-dimensional polytopes
51M20 Polyhedra and polytopes; regular figures, division of spaces
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