Poset Associahedra

Patrick Showers

October 18, 2012

3 a 3 0 c c a 2 c a a f 1AB 2 1 3 0 b 1 2 b a b 2 b b 2 A B d d 1 1 4

a b c a c d f a b b a b c d a b

0 1 2 3 0 1 2 3 1 2 1 2 1 2 3 4

Patrick Showers Poset Associahedra But first...

Preview

Polytope KP is a lattice of tubings, which are in turn comprised of tubes.

Patrick Showers Poset Associahedra Preview

Polytope KP is a lattice of tubings, which are in turn comprised of tubes.

But first...

Patrick Showers Poset Associahedra Predefinitions

I A lower set S of a poset contains all elements {x | x ≤ s, s ∈ S}.

Patrick Showers Poset Associahedra Lower Sets

Lower sets contain everything ‘beneath’ their members

Patrick Showers Poset Associahedra Predefinitions

I A lower set S of a poset contains all elements {x | x ≤ s, s ∈ S}.

I Important example of lower set:

I The boundary ∂p of a poset element p consisting of all elements strictly less than p.

Patrick Showers Poset Associahedra Boundaries

The boundary of an element is everything strictly below that element

Patrick Showers Poset Associahedra Utilizing Boundaries

I If we fix a poset element p and ask what other elements q satisfy ∂q = ∂p, we have the bundle bp = {q | ∂q = ∂p}

Patrick Showers Poset Associahedra Bundles

A B C ∂A = ∂B = ∂C

a b ∂a = ∂b

1 2 ∂1 = ∅ = ∂2

Patrick Showers Poset Associahedra Utilizing Boundaries

I If we fix a poset element p and ask what other elements q satisfy ∂q = ∂p, we have the bundle bp = {q | ∂q = ∂p}

I With boundaries, we call a subset S of poset elements filled if bp ∩ S 6= ∅ whenever ∂p ⊆ S

Patrick Showers Poset Associahedra Filled Sets

Filled sets have at least one p if they include ∂p

Patrick Showers Poset Associahedra Filled Sets

Filled sets have at least one p if they include ∂p

Patrick Showers Poset Associahedra Tubes and Tubings

A tube of a poset is a filled, connected, lower set

a b c d

Figure 1. Here is a Hasse diagram of a poset with various lower sets circled.(a) and (b) are not valid tubings, since both fail to be lled. (c) and (d) are tubings.

Patrick Showers Poset Associahedra and

I the union of any subset of tubes is filled

Tubes and Tubings

I A tube of a poset is a filled, connected, lower set

) a ( a ) ( b ) ( c ) ( d )

I A tubing is a set of tubes such that

I all tubes are pairwise nested or pairwise disjoint,

Patrick Showers Poset Associahedra Tubes and Tubings

I A tube of a poset is a filled, connected, lower set

) a ( a ) ( b ) ( c ) ( d )

I A tubing is a set of tubes such that

I all tubes are pairwise nested or pairwise disjoint, and

I the union of any subset of tubes is filled

Patrick Showers Poset Associahedra The Poset Associahedron

Given poset P:

The poset associahedron KP is the lattice of tubings, ordered by reverse inclusion

Patrick Showers Poset Associahedra Examples of KP

Associahedra

Patrick Showers Poset Associahedra Examples of KP

Permutohedra

Patrick Showers Poset Associahedra Abstract Polytope

An abstract polytope is a poset P satisfying four axioms (i) P has a least and greatest (ii) Flags of P are the same length (iii) P is strongly-connected (iv) 1-sections of P are isomorphic to line segments

A strongly-connected lattice whose 1-sections are line-segments

Patrick Showers Poset Associahedra KP as Abstract Polytope

I Theorem KP is an abstract polytope. Further,

I KP is simple P I The of KP is (|ba| − 1)

Patrick Showers Poset Associahedra Conjecture

KP is a

12 bx 12 bcx

12 bc a b c x

2x

1 2 12 ax

2c 2cx

12 abc 12 ac

Patrick Showers Poset Associahedra More Examples!

f = 32, 64, 38, 12

Patrick Showers Poset Associahedra More Examples!

f = 68, 136, 88, 20 3d facet with 2 octagons

Patrick Showers Poset Associahedra a

Patrick Showers Poset Associahedra