Signed associahedra

Signed associahedra

[categories]

Date: Tue, 8 Jul 1997 12:47:46 -0400 (EDT)
From: James Stasheff <jds@math.upenn.edu>

Reiner and Burgiel's recent paper provoked the questions
at the beginning and end of the following:


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---------- Forwarded message ----------
Date: Tue, 8 Jul 1997 11:33:31 -0500 (CDT)
From: Victor Reiner <reiner@math.umn.edu>
To: jds@math.upenn.edu
Cc: burgiel@math.uic.edu
Subject: Signed associahedra

Dear Jim,

> Only thing more I could ask is to have the tree descrition
> in addition to the triangulation description of the cells.

I think we can oblige.  A signed dissection of the (n+2)-gon
corresponds to a plane tree T having n+1 leaves in which
one assigns +,-, or 0 to every "nook" between two branches of
the tree (think of the +,-,0's as being like lint trapped 
between ones toes!).  For example, a vertex having 5 children
in the tree will have 4 nooks between its branches, and hence
require 4 choices of +,-,0. Furthermore, whenever any of the
nooks below some vertex are assigned 0, all of the nooks below
that vertex must be assigned 0.

Then a signed dissection of the first kind (which indexes
the cells in the simple signed associahedron) requires
that only the root vertex can have its nooks assigned 0.
And a signed dissection of the second kind (which indexes
the cells in the non-simple signed associahedron) requires
that only the vertices whose children are all leaves
can have their nooks assigned 0.

The partial order, roughly speaking, translates into contracting non-leaf
edges in the tree and comparing the +,-,0 assignments in the nooks.

Do you know of any category/homotopy theoretic applications for
either of these signed associahedra?

Best wishes,
              Vic Reiner