The 1/3-2/3 Conjecture for Coxeter groups
Yibo Gao Joint work with: Christian Gaetz
Massachusetts Institute of Technology
UW Combinatorics and Geometry Seminar, 2020
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 1 / 35 Overview
1 Linear extensions and the 1/3-2/3 Conjecture
2 Weak order and convex sets
3 The fully commutative case
4 Generalized semiorders
5 A uniform lower bound
6 More examples
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 2 / 35 Linear extensions of posets
A linear extension of a poset P is an order-preserving bijection:
λ : P → [n] = {1, 2,..., n}.
Let E(P) be the set of linear extensions of P and e(P) be the number of linear extensions of P.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 3 / 35 The 1/3-2/3 Conjecture
For p, q ∈ P, let δ(p, q) be the fraction of linear extensions such that λ(p) < λ(q): |λ ∈ E(P) | λ(p) < λ(q)| δ(p, q) = . e(P)
Conjecture (Kislicyn 1968; Fredman 1974; Linial 1984) For any finite poset P that is not a total order, there exists p, q such that 1 2 ≤ δ(p, q) ≤ . 3 3 Define the balance constant to be
b(P) = max min{δ(p, q), 1 − δ(p, q)} p,q∈P
Then the conjecture is saying that b(P) ≥ 1/3.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 4 / 35 Why 1/3?
Example
p2
p1 p3
This poset has 3 linear extensions p1p2p3, p1p3p2, p3p1p2, with δ(p1, p2) = 1, δ(p1, p3) = 2/3, δ(p2, p3) = 1/3, thus b(P) = 1/3.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 5 / 35 Known Results
It is nontrivial that b(P) is uniformly bounded away from zero: • b(P) ≥ 1/2e ≈ 0.1839 (Kahn and Linial 1991) • b(P) ≥ 3/11 ≈ 0.2727 (Kahn and Saks 1984) √ • b(P) ≥ (5 − 5)/10 ≈ 0.2764 (Brightwell, Felsner, and Trotter 1995)
The full conjecture is known for: • Width-2 posets • Semiorders (unit interval orders) • Height-2 posets • Series parallel posets (N-free posets)
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 6 / 35 Coxeter groups and the weak order
A Coxeter group W is generated by a set of simple reflections S = {s1,..., sr } such that
2 mi,j W = hs1,..., sr | si = id, (si sj ) = id, ∀i, ji
for some mij ∈ {2, 3,..., ∞}.
Let `(w) denote the Coxeter length of w ∈ W .
The (left) weak order ≤L is generated by
w l sw if `(w) + 1 = `(sw).
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 7 / 35 Coxeter groups and the weak order
Let (W , S) be a Coxeter system.
The reflections T are the conjugates of simple reflections.
The (right) inversion set of w ∈ W is
Inv(w) := {t ∈ T | `(wt) < `(w)}.
Proposition 0 0 0 For w, w ∈ W , w ≤L w if and only if Inv(w) ⊂ Inv(w ).
It’s not hard to see that `(w) = |Inv(w)|.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 8 / 35 Convex sets in Coxeter groups
Let H be the Hasse diagram of the weak order ≤L.
A subset C ⊂ W is called convex if for any x, y ∈ C, any shortest-length path between x and y remains in C.
Proposition (Tits 1974) Any convex subset C ⊂ W is of the form
B C = WA := {w ∈ W | A ⊂ Inv(w) ⊂ B}.
For our purposes, we usually translate a convex subset C by some element w ∈ W to assume that A = ∅.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 9 / 35 Example: the symmetric group Sn
The symmetric group Sn is a Coxeter group generated by S = {(12), (23),... (n−1 n)} with reflections T = {(ij) | 1 ≤ i < j ≤ n}. The reflection (ij) is an inversion of w if w(i) > w(j). {(13),(23),(24)} C = {1234, 1243, 2134, 2143, 3142} is convex and C = W∅ .
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 10 / 35 Posets as convex sets in Sn
For a poset P on {p1,..., pn}, we identify E(P) as a set of permutations
CP = {λ ∈ E(P) | λ(1)λ(2) ··· λ(n) ∈ Sn}.
Proposition
The map P 7→ CP gives a bijection between labeled poset on n elements and convex sets of Sn.
Example
p3 p4
p1 p2
corresponds to the convex set {1234, 2134, 2143, 1243, 2413}.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 11 / 35 Balance constants for Coxeter groups
Recall the definition and we have |{λ ∈ E(P) | λ(p ) < λ(p )}| |{w ∈ C | (ij) ∈ Inv(w)}| δ(p , p ) = j i = P . j i e(P) e(P)
This suggests the following definition. Let C ⊂ W be a convex set and t ∈ T be an inversion. Define |{w ∈ C | t ∈ Inv(w)}| δ(t) = . |C|
Define the balance constant to be
b(C) = max min{δ(t), 1 − δ(t)}. t∈T
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 12 / 35 The 1/3-2/3 Conjecture for finite Coxeter groups
Remarkly, the natural generalization of the 1/3-2/3 Conjecture still seems to hold for finite Coxeter group. Conjecture (Gaetz and G. 2020) Let C be a convex set in a finite Coxeter group with |C| > 1, then b(C) ≥ 1/3.
There are new equality cases in every type not coming from type A!
Convex sets in finite Coxeter groups are exactly convex sets of regions in the corresponding Coxeter arrangement. The conjecture says that there is a hyperplane that cuts the convex set into roughly equal parts.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 13 / 35 Fully commutative elements in Weyl groups
Definition (Stembridge 1996) An element w ∈ W is fully commutative if no reduced word of w contains a consecutive substring si sj si ··· with mij > 2. | {z } mij
m Recall that we have relations (si sj ) ij = id.
Example
In S4, s2s1s3s2 = 3412 is fully commutative but s2s1s2s3 = 3241 is not.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 14 / 35 Fully commutative elements and width-2 poset
Proposition
A permutation w ∈ Sn is fully commutative if and only if w avoids 321.
Definition A poset has width k if its longest antichain has size k.
Proposition Under the bijection between (naturally labeled) posets and convex sets in Sn, width-2 posets P correspond to convex sets CP = [id, w]L where w is fully commutative.
The 1/3-2/3 Conjecture is known for width-2 poset. We strengthen this result to other Coxeter group.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 15 / 35 Fully commutative elements in Coxeter group
Theorem (Gaetz and G. 2020) Let w 6= id ∈ W be a fully commutative element in any Coxeter group W whose Coxeter diagram is acyclic, then b([id, w]L) ≥ 1/3.
In particular, this theorem applies to all finite Coxeter groups.
Example
In the affine Weyl group Af3, w = s1s3s0s2 is fully commutative, but b([id, w]L) = 2/7. s1 • •s2 s0 • •s3
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 16 / 35 Heaps of fully commutative elements
The proof of this theorem uses the theory of heaps. Definition (Stembridge 1996)
Let s = si1 ··· si` be a reduced word of w. The Heap poset Hs is the partial order ([`], ), which is the transitive closure of
2 j k if j ≤ k and (sij sik ) 6= id.
In the case where w is fully commutative, the Heap poset does not depend on the choice of a reduced word. We write the Heap poset as Hw . Proposition (Stembridge 1996)
Let w ∈ W be fully commutative. Then [id, w]L ' J(Hw ).
Here J(Hw ) is the lattice of order ideals of Hw .
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 17 / 35 Balance constants for order ideals
Let Q be a poset and J(Q) be the order ideals of Q. Define
|{J ∈ J(Q) | x ∈ J}| δideal(x, Q) = , |J(Q)| b(Q) = max min{δideal(x, Q), 1 − δideal(x, Q)}. x∈Q
For fully commutative w, we have b([id, w]L) = b(Hw ). Example: non-example
Consider the following poset Q = Hw , w = s1s3s0s2 ∈ W (Af3). We see that |J(Q)| = 7 and b(Q) = 2/7. 0• •2
1• •3
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 18 / 35 Equality cases
These intervals C = [e, w]L give equality cases b(C) = 1/3:
type diagram w Hw • • • A s s s s 2 1 2 1 2 • • •s3 D4 • • s4s2s3s1 • s1 s2 •s 4 • • • • • • B3 s1 s2 s3 s3s2s3s1 • • •
•s6 • • E6 • • • • • s6s3s2s4s1s3s5 • • s1 s2 s3 s4 s5 • • •
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 19 / 35 Semiorders
Definition A poset P is a semiorder (unit-interval order) if there exists a function f : P → R such that p < q if and only if f (q) − f (p) > 1.
Theorem (Brightwell 1989) If P is a semiorder which is not a chain, then b(P) ≥ 1/3.
We will be generalizing this result to finite Weyl groups.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 20 / 35 Root systems and Weyl groups
Definition (Root system) n Let E = R .A root system Φ ⊂ E is a finite set of vectors, such that Φ spans E; for α ∈ Φ, kα ∈ Φ iff k ∈ {±1}; for α, β ∈ Φ, 2hα, βi/hα, αi ∈ Z; for α, β ∈ Φ, 2hα, βi σ (β) := β − α ∈ Φ. α hα, αi
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 21 / 35 Root systems and Weyl groups
Let Φ ⊂ E be a root system. The Weyl group W (Φ) ⊂ GL(E) is generated by reflections {σα | α ∈ Φ}. It’s a Coxeter group. We can partition Φ into positive roots Φ+ and negative roots Φ−. Given Φ = Φ+ t Φ−, there is a unique choice of simple roots + + ∆ = {α1, . . . , αn} ⊂ Φ such that each α ∈ Φ can be written as an integral linear combination of ∆. There is a partial order on Φ+ given by α ≤ β if β − α is a nonnegative linear combinationn of ∆. The minimal elements of Φ+ are the simple roots and the unique maximum of Φ+ is the highest root ξ.
e1−e5
e2−e5
e3−e5
e1−e2 e2−e3 e3−e4 e4−e5 Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 22 / 35 Generalized semiorders
Let W be a finite crystallographic Weyl group. Recall
B WA = {w ∈ W | A ⊆ Inv(w) ⊆ B}.
Definition (Gaetz and G. 2020) B A convex set C = W∅ is a generalized semiorder if B is an order ideal of the root poset Φ+.
It recovers the definition of a semiorder in type A. Recall that a semiorder on P is defined by p < q if f (q) − f (p) > 1 for some f : P → R. Say P = {p1,..., pn} where f (p1) < ··· < f (pn). Then
{ei − ej | i < j, f (j) − f (i) ≤ 1}
+ is an order ideal of Φ in type An−1.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 23 / 35 Generalized semiorders
Theorem (Gaetz and G. 2020) Let C be a non-singleton generalized semiorder, then b(C) ≥ 1/3.
This theorem relies on the following purely root-theoretic fact. Lemma (Gaetz and G. 2020) Let J ⊂ Φ+ be a nonempty order ideal. Then there exists a simple root + α ∈ J such that we cannot find β1 6= β2 ∈ J with sαβ1, sαβ2 ∈ Φ \ J.
Our proof is very bad. It’s very technical and type dependent.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 24 / 35 A uniform lower bound
It is nontrivial that b(C) is bounded away from 0. Let W be a finite crystallographic Weyl group of rank r, highest root ξ, ∨ ∨ and fundamental coweights ω1 , . . . , ωr . Define
∨ m0 = mini hωi , ξi, ∨ m1 = maxi hωi , ξi, m = r/m0 + 1/m1 − ht(ξ)/m0m1.
Theorem (Gaetz and G. 2020) Let C be a non-singletone convex set in W , then 1 1 b(C) ≥ ≥ . 2emm1 2e12
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 25 / 35 A uniform lower bound
Theorem (Gaetz and G. 2020) Let C be a non-singletone convex set in W , then 1 b(C) ≥ . 2emm1
Type m0 m1 ht(Φ) m mm1 b(C) ≥ Ar 1 1 r 1 1 1/2e 2 Br 1 2 2r−1 1 2 1/2e , 1/2e 2 Cr 1 2 2r−1 1 2 1/2e , 1/2e 4 Dr 1 2 2r−3 2 4 1/2e 8 E6 1 3 11 8/3 8 1/2e 12 E7 1 4 17 3 12 1/2e 10.5 E8 2 6 29 7/4 21/2 1/2e 3.5 F4 2 4 11 7/8 7/2 1/2e 2.5 G2 2 3 5 1/2 5/2 1/2e , 1/3
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 26 / 35 Generalized order polytope
Let W ⊂ GL(E) be a finite crystallographic Weyl group. The fundamental alcove is
Qid := {v ∈ E | hv, αi ≥ 0 ∀α ∈ ∆, hv, ξi ≤ 1}.
For each w ∈ W , its corresponding alcove is
−1 Qw := w Qid.
Definition (Gaetz and G. 2020) For a convex set C ⊂ W , the generalized order polytope is [ O(C) := Qw . w∈C
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 27 / 35 Generalized order polytope
Definition (Gaetz and G. 2020) For a convex set C ⊂ W , the generalized order polytope is [ O(C) := Qw . w∈C
These are special cases of alcoved polytopes (Lam and Postnikov 2005).
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 28 / 35 A geometric approach
For a reflection t ∈ T , let Ht be its corresponding hyperplane. + − The hyperplane Ht cuts O(C) into two parts O(C)t and O(C)t . + The goal is to show that there exists t ∈ T such that both O(C)t and − O(C)t have volumes at least Vol(O(C)). Lemma Let C be a non-singleton convex set of a finite crystallographic Weyl group + of rank r and let oC be the centroid of O(C). Then there exists t ∈ Φ with alcoves on both sides of Ht , such that m |ho , ti| ≤ . C r + 1
In other words, there is a hyperplane Ht that is close to the centroid.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 29 / 35 A geometric approach
We now have a hyperplane Ht close to the centroid oC .
·
Proposition n Let Q ⊂ R be a full-dimensional compact convex body with centroid cQ . n Let m ≥ 1 and v ∈ R such that hv, cQ i ≥ −m/(n + 1), minx∈Q hv, xi ≤ −1, maxx∈Q hv, xi ≥ 1, then
Vol(Q+) 1 v ≥ . Vol(Q) 2em
This proposition says that we can obtain a bound if the centroid cQ is close to the hyperplane Hv , and the hyperplane is relatively in the middle of Q.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 30 / 35 A geometric approach
The proof generalizes the argument by Kahn and Linial, which uses the following immediate corollary of Brunn-Minkowski.
Lemma (Brunn-Minkowski)
1 λ n−1 n The function λ 7→ Vol(Qv ) is concave on R, where Q ⊂ R is a convex λ body and Qv := {x ∈ Q | hv, xi = λ}.
The proof of the proposition explicitly constructs the convex body Q that + minimizes Vol(Qv )/Vol(Q), which is a double cone.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 31 / 35 A twist in type Bn
2 We can achieve a better bound in type Bn, 1/2e instead of 1/2e , with an alternative definition of generalized order polytopes. Define the alcoves
η Qid := {v ∈ E | hv, αi ≥ 0 ∀α ∈ ∆, hv, ηi ≤ 1} η −1 η and Qw := w Qid, where η is the highest short root. Definition (Gaetz and G. 2020)
For a convex set C ⊂ W , the short-root order polytope in type Bn is S η O(C) := w∈C Qw .
The standard realization of the type Bn root system
Φ(Bn) = {±ei ± ej | 1 ≤ i < j ≤ n} ∪ {±ei | 1 ≤ i ≤ n},
simple roots ∆ = {e1 − e2,..., en−1 − en, en},
highest root ξ = e1 + e2, highest short root η = e1.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 32 / 35 More examples
Recall that b(C) ≥ for finite Weyl groups and an absolute constant . Example: no universal bound for Coxeter groups
Let Wn be the Coxeter group whose Dynkin diagram is a complete graph (with mij ≥ 3) on n generators {s1,..., sn}, where n ≥ 3. Consider the convex set
C = conv(s1, s2,..., sn) = {id, s1, s2,..., sn}.
Then δ(si ) = 1/(n + 1) and b(C) = 1/(n + 1). So there are no lower bounds for b(C) in arbitrary Coxeter groups.
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 33 / 35 More examples
In the original 1/3-2/3 Conjecture on posets, it is believed (Brightwell 1999) that width-2 posets come closest to violating the conjecture. This heuristics is not true for fully commutative elements. Example: heuristic on fully commutative elements • • • • Let W be with m12 ≥ 4, m23 ≥ 7, m34 ≥ 4. Since W is acyclic, our theorem says b([id, w]L) ≥ 1/3 for w 6= id fully commutative. Consider C = conv(id, u, v). We have b(C) = 3/10 < 1/3. s1s3s2s3 v = s1s4s2s3 u = s2s3s2s3
s4s2s3 s3s2s3 s1s2s3 s4s3 s2s3
s3
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 34 / 35 Thanks
Thank you for listening!
Yibo Gao (MIT) The 1/3-2/3 Conjecture for Coxeter groups December 2, 2020 35 / 35