Congruence Normality of Simplicial Hyperplane Arrangements

Congruence Normality of Simplicial Hyperplane Arrangements

Sophia Elia

Joint work with Michael Cuntz and Jean-Philippe Labb´e DGeCo Seminar

November 12, 2020

Definitions

A (real, central) hyperplane H is a codimension-1 subspace in R^d: H := {x 2 R^d : n · x = 0 for some n 2 R^d}.

A (finite, real, central) hyperplane arrangement A is a finite non-empty set of hyperplanes.

Regions = closures of connected components R^d \ A.

A region is simplicial if the normal vectors of its facet-defining hyperplanes are linearly independent.

Every region is simplicial ) A is simplicial The rank of A = dim(span({normals})).

Gr¨ unbaum’s catalogue of rank-3 simplicial arrangements (1971, 2009)

near-pencils polygons & symmetries

1

2n-gons, symmetries, 1 Arrangements from the three infinite families of simplicial arrangements

Gr¨ unbaum’s catalogue: 90 sporadic arrangements

Gr¨unbaum 2009

Changes to the Catalogue

• 5 new sporadic arrangements (Cuntz 2011, 2020)

• The list is complete for up to 27 hyperplanes (Cuntz 2011)

• 53 sporadic arrangements come from finite Weyl groupoids (Cuntz–Heckenberger 2009)

• 37 hyperplanes still conjectured upper bound

Cuntz–E–Labb´e 2020: normals and invariants of the updated catalogue.

Goal: Add more structure to the catalogue

! test arrangements for congruence normality in general

Posets from hyperplane arrangements

Choose a base region B, orient all normals away from it:

n · v_B  0, v_B 2 B

The separating set S(R) of a region R is the set of hyperplanes that separate it from B.

The poset of regions P_B(A):

• elements: regions

• relations: R₁  R₂ if and only if S(R₁) ✓ S(R₂).

Poset of Regions

Other types of posets from hyperplane arrangements:

- Facial weak order (Dermenjian–Hohlweg–McConville–Pilaud 2019) - Intersection lattice

Congruence normality

What do we know about the poset of regions?

Theorem (Bj¨orner–Edelman–Ziegler 1990)

If A is simplicial, then P_B(A) is a lattice for all choices of base region.

Theorem (Caspard–Le Conte de Poly-Barbut–Morvan 2004)

Posets of regions of finite Coxeter arrangements are congruence normal.

Congruence normality

A subset S of (P,) is convex if x,y 2 S and x  z  y ) z 2 S. A lattice is congruence normal if it is obtainable from the one element lattice by a finite sequence of doublings of convex sets.

Congruence uniform lattices : convex set ! interval

Simplicial arrangements: congruence normal , congruence uniform

Motivation for studying congruence normality

• Congruence uniform lattices are well-behaved - easier to understand lattice congruences

• Simplicial arrangements $ normal fans of simple zonotopes

Open Question

Which lattice quotients are polytopal (normal fans of polytopes)?

Conjecture (Padrol–Pilaud–Ritter 2020)

If the poset of regions is congruence uniform, then any shard admits a shard polytope.

Each shard admits a “shard” polytope ! they construct polytope for quotient fan

Summary So Far

• Discussed current catalogue of simplicial rank-3 arrangements

• Goal: find out which ones are congruence normal

• Saw some motivation for looking at this property

• Now: look at how to check for congruence normality

Shards - Introduced by Reading (2003)

Shards are pieces of hyperplanes used to understand lattice congruences.

Shards - Introduced by Reading (2003)

Shards are pieces of hyperplanes used to understand lattice congruences.

Shards - Introduced by Reading (2003)

Shards are pieces of hyperplanes used to understand lattice congruences.

Shards - Introduced by Reading (2003)

Shards are pieces of hyperplanes used to understand lattice congruences.

Shards: Congruence Normality Criterion

⌃ ! ⇥ , H_⌃ cuts H_⇥ and dim(⌃ \ ⇥) = dim-2

Theorem (Reading 2004)

A simplicial arrangement is congruence normal i↵ the directed graph on shards is acyclic.

Motivation for an oriented matroid perspective

Find a way to avoid computing

( posets of regions polyhedral objects

dimensions of intersection

Shards ! shard covectors

Let P = {p_i}ⁱ2[m] be an ordered set of vectors. A covector on P is a vector of signs (c_i)_i2[m] 2 {0,+, }^m defined as

c := (sign(x· p_i +a))_i2[m], where x 2 R^d and a 2 R.

A restricted covector on U ✓ P has “*”s on all entries not in U.

Shards ! shard covectors

A shard covector is a restricted covector with exactly one zero.

H₁

H₂ H₃

H₄

H₅ H₆

⇥

The shard ⇥ has shard covector: ✓ = (+,⇤, ,+, ,0).

Translation of Shard Cycle Criterion

p₁

p₂

p₃

p₄ p₅

p₆

A subset of P consisting of all the points that lie on the affine hull of two distinct points of P is called a line - `.

A line covector h<sup>`</sup> of a line ` is a covector on {normals} s.t.

h_k = 0 , n_k 2 `

Translation of Shard Cycle Criterion

The intersection \ of two covectors is the coordinate-wise operation:

+\+ := +, +\ = \+ := 0, \ := ,

0\" = " \0 := 0, ⇤ \ " = "\ ⇤ := ",

where " 2 {0,+, ,⇤}.

Theorem (Cuntz–E–Labb´e)

Let ⌃,⇥ be shards on hyperplanes i,j respectively, let ,✓ be their shard covectors.

⌃ ! ⇥ if and only if

✓_i 2 {+, }

and

9 line covector h : h \ \ ✓ = h.

Image: Cuntz 2020

Results from the computation

Classification of the Catalogue

P_B(A) always CN P_B(A) sometimes CN P_B(A) never CN Finite Weyl Groupoids F²(m) (m 10) A(22,288)

A(22,288)

F²(m) (m  8) F³(m) (m 17) A(25,360)

A(25,360)

F³(m) (m  13) 41 arrangements A(35,680)

A(35,680)

A(15,120)

A(15,120)

A(31,480)

A(31,480)

55 arrangements 61 arrangements 3 arrangements

Classification of the Catalogue

P_B(A) always CN P_B(A) sometimes CN P_B(A) never CN Finite Weyl Groupoids F²(m) (m 10)

A(22,288)

A(22,288) F²(m) (m  8) F³(m) (m 17)

A(25,360)

A(25,360) F³(m) (m  13) 41 arrangements

A(35,680)

A(35,680)

A(15,120)

A(15,120)

A(31,480)

A(31,480)

55 arrangements 61 arrangements 3 arrangements

Always Congruence Normal

Theorem (Cuntz–E–Labb´e)

• Near-pencils are always congruence normal.

• Simplicial arrangements from finite Weyl groupoids of any rank are always congruence normal.

Two others are always CN:

• Coxeter Arrangement H₃ - 15 hyperplanes

• H₃’s point-line dual - 31 hyperplanes

Sometimes Congruence Normal

2nd family

1

3rd family Theorem (Cuntz–E–Labb´e)

• The second infinite family with at least 10 hyperplanes are sometimes congruence normal.

• The third infinite family with at least 17 hyperplanes are sometimes congruence normal.

41 other sporadic arrangements are sometimes congruence normal.

Sometimes Congruence Normal

2nd family

1

3rd family Theorem (Cuntz–E–Labb´e)

• The second infinite family with at least 10 hyperplanes are sometimes congruence normal.

• The third infinite family with at least 17 hyperplanes are sometimes congruence normal.

Never Congruence Normal

• Three hyperplane arrangements are never congruence normal.

• They have 22, 25, and 35 hyperplanes.

• No way to construct through doublings of convex sets

Thank you for listening