Skip to main content
MathOverflow

Return to Question

Changed n-3 to n-2.
Source Link
Tom Goodwillie
Tom Goodwillie
  • 55.9k
  • 7
  • 151
  • 240

Let $A_n$ be the dual simplicial complex to the associahedron on $n$ letters. The complex $A_n$ is thus a simplicial triangulation of an $(n-3)$-dimensional sphere. The vertices of $A_n$ correspond to the ways of inserting one nontrivial parentheses into the expression $a_1 a_2 \cdots a_n$, and the $(n-3)$-dimensional simplices correspond to ways of inserting the maximal number $(n-3)$($n-2$) of parentheses into this expression. For instance, the $1$-dimensional simplices of $A_4$ are precisely \begin{align*} &((a_1 a_2) a_3) a_4\\ &(a_1 (a_2 a_3)) a_4\\ &a_1 ((a_2 a_3) a_4)\\ &a_1 (a_2 (a_3 a_4))\\ &(a_1 a_2) (a_3 a_4) \end{align*} Since $A_n$ is a triangulation of an $(n-3)$-dimensional sphere, it can be oriented (in two ways, we pick one).

Question: This orientation should give an orientation on each $(n-3)$-dimensional simplex $\sigma$. Such an orientation corresponds to an ordering (defined up to the action of the alternating group) of the vertices of $\sigma$, i.e., to the pairs of parentheses in $\sigma$. How can we explicitly write down this orientation?

Let $A_n$ be the dual simplicial complex to the associahedron on $n$ letters. The complex $A_n$ is thus a simplicial triangulation of an $(n-3)$-dimensional sphere. The vertices of $A_n$ correspond to the ways of inserting one nontrivial parentheses into the expression $a_1 a_2 \cdots a_n$, and the $(n-3)$-dimensional simplices correspond to ways of inserting the maximal number $(n-3)$ of parentheses into this expression. For instance, the $1$-dimensional simplices of $A_4$ are precisely \begin{align*} &((a_1 a_2) a_3) a_4\\ &(a_1 (a_2 a_3)) a_4\\ &a_1 ((a_2 a_3) a_4)\\ &a_1 (a_2 (a_3 a_4))\\ &(a_1 a_2) (a_3 a_4) \end{align*} Since $A_n$ is a triangulation of an $(n-3)$-dimensional sphere, it can be oriented (in two ways, we pick one).

Question: This orientation should give an orientation on each $(n-3)$-dimensional simplex $\sigma$. Such an orientation corresponds to an ordering (defined up to the action of the alternating group) of the vertices of $\sigma$, i.e., to the pairs of parentheses in $\sigma$. How can we explicitly write down this orientation?

Let $A_n$ be the dual simplicial complex to the associahedron on $n$ letters. The complex $A_n$ is thus a simplicial triangulation of an $(n-3)$-dimensional sphere. The vertices of $A_n$ correspond to the ways of inserting one nontrivial parentheses into the expression $a_1 a_2 \cdots a_n$, and the $(n-3)$-dimensional simplices correspond to ways of inserting the maximal number ($n-2$) of parentheses into this expression. For instance, the $1$-dimensional simplices of $A_4$ are precisely \begin{align*} &((a_1 a_2) a_3) a_4\\ &(a_1 (a_2 a_3)) a_4\\ &a_1 ((a_2 a_3) a_4)\\ &a_1 (a_2 (a_3 a_4))\\ &(a_1 a_2) (a_3 a_4) \end{align*} Since $A_n$ is a triangulation of an $(n-3)$-dimensional sphere, it can be oriented (in two ways, we pick one).

Question: This orientation should give an orientation on each $(n-3)$-dimensional simplex $\sigma$. Such an orientation corresponds to an ordering (defined up to the action of the alternating group) of the vertices of $\sigma$, i.e., to the pairs of parentheses in $\sigma$. How can we explicitly write down this orientation?

Source Link
Dylan
Dylan
  • 53
  • 2

Orienting the dual of the associahedron

Let $A_n$ be the dual simplicial complex to the associahedron on $n$ letters. The complex $A_n$ is thus a simplicial triangulation of an $(n-3)$-dimensional sphere. The vertices of $A_n$ correspond to the ways of inserting one nontrivial parentheses into the expression $a_1 a_2 \cdots a_n$, and the $(n-3)$-dimensional simplices correspond to ways of inserting the maximal number $(n-3)$ of parentheses into this expression. For instance, the $1$-dimensional simplices of $A_4$ are precisely \begin{align*} &((a_1 a_2) a_3) a_4\\ &(a_1 (a_2 a_3)) a_4\\ &a_1 ((a_2 a_3) a_4)\\ &a_1 (a_2 (a_3 a_4))\\ &(a_1 a_2) (a_3 a_4) \end{align*} Since $A_n$ is a triangulation of an $(n-3)$-dimensional sphere, it can be oriented (in two ways, we pick one).

Question: This orientation should give an orientation on each $(n-3)$-dimensional simplex $\sigma$. Such an orientation corresponds to an ordering (defined up to the action of the alternating group) of the vertices of $\sigma$, i.e., to the pairs of parentheses in $\sigma$. How can we explicitly write down this orientation?