Timeline for Orienting the dual of the associahedron
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 25 at 15:36 | history | edited | Andy Putman | CC BY-SA 4.0 |
Fixed a typo
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| Aug 7, 2023 at 18:00 | comment | added | Dylan | This is very helpful, thank you very much! | |
| Aug 7, 2023 at 18:00 | vote | accept | Dylan | ||
| Aug 7, 2023 at 17:16 | comment | added | David E Speyer | Remark: Loday's formula for the normal vector in Section 2.1 looks messy, because he takes the representative which is orthogonal to $e_1+e_2+\cdots +e_{n-1}$. No need! The normal vector naturally lives in the quotient by $\mathbb{R}(e_1+e_2+\cdots +e_{n-1})$; choose the lift which makes the simplest formula. | |
| Aug 7, 2023 at 17:11 | comment | added | David E Speyer | I originally thought of this in terms of binary trees, and found a recursive description there. I was pleasantly surprised to realize, when I translated back to parenthesizations, that my recursive description was just "write the multiplication symbols in order"! | |
| Aug 7, 2023 at 17:10 | comment | added | David E Speyer | I found that, in each case, there was only one nonzero term when I expanded the determinant! So the sign of the determinant was easy to compute. My original definition of the ordering was the ordering which made this unique nonzero term correspond to the identity permutation. | |
| Aug 7, 2023 at 17:09 | comment | added | David E Speyer | Loday wrote down those normal vectors explicitly: The vertex which I have called $v_{ij}$ corresponds to the vector $e_i + e_{i+1} + \cdots + e_{j-1}$ (see Section 2.1). So it is natural to take the normal fan to the associahedron and lift the ray $v_{ij}$ to the vector $e_i + e_{i+1} + \cdots + e_{j-1}$ in $\mathbb{R}^{n-1}$, and to add an additional ray in the direction $e_1+e_2+\cdots +e_{n-1}$ (the direction which we quotiented by). I took binary planar trees with $n$-leaves, turned them into $(n-1)$-tuples of rays by Loday's recipe, and computed the determinant. | |
| Aug 7, 2023 at 17:07 | comment | added | David E Speyer | How I originally thought about this. Loday arxiv.org/abs/math/0212126 realized the associahedron as a polytope in the affine plane $x_1+x_2+\cdots+x_{n-1} = \binom{n}{2}$ in $\mathbb{R}^{n-1}$. So the normal vectors to the facts of the associahedron naturally live in $\mathbb{R}^{n-1}/\mathbb{R} (e_1+e_2+\cdots + e_{n-1})$. | |
| Aug 7, 2023 at 16:54 | history | answered | David E Speyer | CC BY-SA 4.0 |