BREAK & SUMMARY

BREAK & SUMMARY

SEMIDISTRIBUTIVE ACYCLIC REORIENTATION LATTICE

D skeletal =

• D vertebrate = transitive reduction of any induced subgraph of D is a forest

• D filled = any directed path joining the endpoints of an arc in D induces a tournament

THM. AR_D semidistributive lattice ⇐⇒ D is skeletal

THM. If D skeletal, the canonical join representation of an acyclic reorienta- tion E of D is E = W

a E_a where

• a runs over the arcs of D reversed in the transitive reduction of E

• E_a is the acyclic reorientation of D where an arc is reversed iff it is the only arc reversed in E along a path in D joining the endpoints of a

= ∨ = ∨

CHORDS & NON-CROSSING CHORD DIAGRAMS

CHORDS

• (u, v) is an arc of D

• 5 t 4 partitions the transitive support of (u, v) minus {u, v}

LEM. E acyclic reorientation of D. If (u, v) or (v, u) is in the transitive reduction of E, then χ^E_u,v = (u, v,5^E_u,v, 4^E_u,v) is a chord of D, where

5^E_u,v = {w ∈ V | (u, w) ∈ D r ^E and (w, v) ∈ D ∩ E} and 4^E_u,v = {w ∈ V | (u, w) ∈ D ∩ E and (w, v) ∈ D r ^E}

CHORDS & JOIN IRREDUCIBLES

PROP. The maps sending

• a join irreducible J of AR_D to the chord χ(J) = χ^J_u,v of D where (u, v) is the only reversed arc of the transitive reduction of J

• a chord χ = (u, v,5,4) of D to the join irreducible J(χ) of AR_D where an arc (w, w⁰) of D is reversed iff w ∈ 4 ∪ {u} and w⁰ ∈ 5 ∪ {v}

are reverse bijections between join irreducibles of AR_D and chords of D

join irreducibles chords

NON-CROSSING CHORD DIAGRAMS & CANONICAL JOIN REPRESENTATIONS

PROP. The maps sending

• a join irreducible J of AR_D to the chord χ(J) = χ^J_u,v of D where (u, v) is the only reversed arc of the transitive reduction of J

• a chord χ = (u, v,5,4) of D to the join irreducible J(χ) of AR_D where an arc (w, w⁰) of D is reversed iff w ∈ 4 ∪ {u} and w⁰ ∈ 5 ∪ {v}

are reverse bijections between join irreducibles of AR_D and chords of D

PROP. The maps sending

• an acyclic reorientation E of D to the diagram δ(E) of chords χ^E_u,v where (u, v) runs over all reversed arcs of the transitive reduction of E,

• a non-crossing chord diagram δ to the acyclic reorientation E(δ) = W

χ∈δ J(χ)

are reverse bijections between the acyclic reorientations of D and the non-crossing chord diagrams of D

(u, v, 5, 4) and (u⁰, v⁰, 5⁰, 4⁰) are crossing if there are w 6= w⁰ such that

w ∈ (5 ∪ {u, v}) ∩ (4⁰ ∪ {u⁰, v⁰}) and w⁰ ∈ (4 ∪ {u, v}) ∩ (5⁰ ∪ {u⁰, v⁰})

NON-CROSSING CHORD DIAGRAMS & CANONICAL JOIN REPRESENTATIONS

PROP. The canonical join complex is isomorphic to the non-crossing chord complex

(u, v, 5, 4) and (u⁰, v⁰, 5⁰, 4⁰) are crossing if there are w 6= w⁰ such that

w ∈ (5 ∪ {u, v}) ∩ (4⁰ ∪ {u⁰, v⁰}) and w⁰ ∈ (4 ∪ {u, v}) ∩ (5⁰ ∪ {u⁰, v⁰})

CONGRUENCES & QUOTIENTS

SUBCHORDS & FORCING

(u, v, 5, 4) subchord of (u⁰, v⁰,5⁰,4⁰) = u, v ∈ {u⁰, v⁰} ∪ 5⁰ ∪ 4⁰ and 5 ⊆ 5⁰ and 4 ⊆ 4⁰

PROP. J forces J⁰ ⇐⇒ χ(J) is a subchord of χ(J⁰)

CORO. congruence lattice of AR_D ' lower ideal lattice of subchord order

CORO. ≡ lattice congruence of AR_D

• E minimal in its ≡-class ⇐⇒ δ(E) ⊆ I^≡

• quotient AR_D/≡ ' subposet of AR_D induced by {E ∈ AR_D | δ(E) ⊆ I^≡}

COHERENT CONGRUENCES

(f^{, Ω)} = two of arbitrary subsets of V

I⁽_f^,Ω) = lower ideal of chords (u, v, 5, 4) of D such that 5 ⊆ f and 4 ⊆ Ω coherent congruence ≡_(f,Ω) = congruence with subchord ideal I⁽_f^,Ω)

QUOTIENT FANS & QUOTIENTOPES

GRAPHICAL ARRANGEMENT & GRAPHICAL ZONOTOPE

D directed acyclic graph

graphical arrangement H_D = arrangement of hyperplanes x_u = x_v for all arcs (u, v) ∈ D graphical zonotope Z_D = Minkowski sum of [e_u,e_v] for all arcs (u, v) ∈ D

hyperplanes of H_D ←→ summands of Z_D ←→ arcs of D

regions of H_D ←→ vertices of Z_D ←→ acyclic reorientations of D poset of regions of H_D ←→ oriented graph of Z_D ←→ acyclic reorientation poset of D

SHARDS & QUOTIENT FAN

Σ_χ =

x ∈ R^V

_xw ≤ x_u = x_v ≤ x_w⁰ for any w ∈ 5 and w⁰ ∈ 4

THM. A lattice congruence ≡ of AR_D defines a quotient fan F_≡ where

• the chambers of F_≡ are obtained by glueing the chambers of H_D corresponding to acyclic reorientations in the same equivalence class of ≡

• the union of the walls of F_≡ is the union of the shards Σ_χ for χ in the chord ideal X_≡

SHARDS & QUOTIENT FAN

Σ_χ =

x ∈ R^V

_xw ≤ x_u = x_v ≤ x_w⁰ for any w ∈ 5 and w⁰ ∈ 4

THM. A lattice congruence ≡ of AR_D defines a quotient fan F_≡ where

• the chambers of F_≡ are obtained by glueing the chambers of H_D corresponding to acyclic reorientations in the same equivalence class of ≡

• the union of the walls of F_≡ is the union of the shards Σ_χ for χ in the chord ideal X_≡

QUOTIENTOPES

THM. The quotient fan of any lattice congruence of AR_D is the normal fan of

• a Minkowski sum of associahedra of Hohlweg – Lange, and

• a Minkowski sum of shard polytopes of Padrol – P. – Ritter

χ-alternating matching = pair (M₅, M₄) with M₅ ⊆ {u} ∪ 5 and M₄ ⊆ 4 ∪ {v} s.t.

M₅ and M₄ are alternating along the transitive reduction of D shard polytope of χ = convex hull of signed charact. vectors of χ-alternating matchings

QUOTIENTOPES

THM. The quotient fan of any lattice congruence of AR_D is the normal fan of

• a Minkowski sum of associahedra of Hohlweg – Lange, and

• a Minkowski sum of shard polytopes of Padrol – P. – Ritter

PROP. For a coherent congruence ≡_(f,Ω), all facets defining inequalities of the quotien- tope of ≡_(f,Ω) are facet defining inequalities of the graphical zonotope of D

SOME OPEN QUESTIONS

OPEN QUESTIONS

1. Characterize hyperplane arrangements with a lattice of regions 2. Characterize simplicial quotient fans / simple quotientopes 3. Do all quotientopes admit a Hamiltonian cycle?

4. Describe the type cone of

• graphical zonotopes,

• coherent congruences,

• all congruences 5. Hopf algebras

6. Facial order

THANKS