Permutoassociahedron Explained
In mathematics, the permutoassociahedron is an
-dimensional
polytope whose vertices correspond to the bracketings of the
permutations of
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]
Examples
When
, the vertices of the permutoassociahedron can be represented by bracketing all the permutations of three terms
,
, and
. There are six such permutations,
,
,
,
,
, and
, and each of them admits two bracketings (obtained from one another by associativity). For instance,
can be bracketed as
or as
. Hence, the
-dimensional permutoassociahedron is the
dodecagon with vertices
,
,
,
,
,
,
,
,
,
,
, and
.
When
, the vertex
is adjacent to exactly three other vertices of the permutoassociahedron:
,
, and
. The first two vertices are reached from
via associativity and the third via a transposition. The vertex
is adjacent to four vertices. Two of them,
and
, are reached via associativity, and the other two,
and
, via a transposition. This illustrates that, in dimension
and above, the permutoassociahedron is not a
simple polytope.
[3] Properties
The
-dimensional permutoassociahedron has
vertices. This is the product between the number of permutations of
terms and the number of all possible bracketings of any such permutation. The former number is equal to the
factorial
and the later is the
th
Catalan number.
By its description in terms of bracketed permutations, the 1-skeleton of the permutoassociahedron is a flip graph with two different kinds of flips (associativity and transpositions).
See also
Notes and References
- Mikhail M. . Kapranov . Mikhail Kapranov. The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation. Journal of Pure and Applied Algebra. 85. 2. 1993. 119–142. 10.1016/0022-4049(93)90049-Y .
- Reiner . Victor. Ziegler . Günter M. . Günter M. Ziegler. Coxeter-associahedra. Mathematika. 41. 2. 1994. 364–393. 10.1112/S0025579300007452.
- Baralić . Djordje. Ivanović . Jelena. Petrić . Zoran. A simple permutoassociahedron. Discrete Mathematics. 342. 12. December 2019. 111591. 10.1016/j.disc.2019.07.007. 1708.02482.