Bijections for the deformations of the braid arrangement Olivier - PowerPoint PPT Presentation

Bijections for the deformations of the braid arrangement Olivier Bernardi - Brandeis University 1 2 3 1 2 3 1 3 1 2 1 2 3 2 3 1 2 3 1 2 3 Discrete Math Days of the North-East, May 2017

Hyperplane arrangements A hyperplane arrangement of dimension n is a finite collection of affine hyperplanes in R n . Example: x 2 x 1

Hyperplane arrangements A hyperplane arrangement of dimension n is a finite collection of affine hyperplanes in R n . Example: x 2 x 1 The complement of the hyperplanes is divided into regions .

Braid arrangement Def: The braid arrangement of dimension n has hyperplanes { x i − x j = 0 } for all 0 ≤ i < j ≤ n .

Braid arrangement Def: The braid arrangement of dimension n has hyperplanes { x i − x j = 0 } for all 0 ≤ i < j ≤ n . Example: n = 3 x 1 x 1 − x 2 = 0 x 1 − x 3 = 0 n ! regions x 2 x 3 x 2 − x 3 = 0

Deformations of the braid arrangement Def: Fix S ⊂ Z finite. The S -deformed braid arrangement A S ( n ) ⊂ R n has hyperplanes { x i − x j = s } for all 0 ≤ i < j ≤ n , and all s ∈ S .

Deformations of the braid arrangement Def: Fix S ⊂ Z finite. The S -deformed braid arrangement A S ( n ) ⊂ R n has hyperplanes { x i − x j = s } for all 0 ≤ i < j ≤ n , and all s ∈ S . Example: S = { 0 , 1 } and n = 3 . x 1 x 1 − x 2 = 1 x 1 − x 2 = 0 ( n + 1) n − 1 regions x 2 x 3

Known counting results for S ⊆ {− 1 , 0 , 1 } [Stanley, Postnikov, Athanasiadis, . . . ]

Known counting results for S ⊆ {− 1 , 0 , 1 } B ( n ) = set of rooted binary trees with n labeled nodes. 9 8 2 4 1 5 3 7 6

Known counting results for S ⊆ {− 1 , 0 , 1 } S = {− 1 , 0 , 1 } S = { 0 , 1 } S = {− 1 , 1 } S = { 1 } S = { 0 } Catalan Shi Semi-order Braid Linial T ∈B ( n ) T ∈B ( n ) s.t. T ∈B ( n ) s.t. T ∈B ( n ) s.t. T ∈B ( n ) s.t. . . . . . . . . . . . .

Known counting results for S ⊆ {− 1 , 0 , 1 } S = {− 1 , 0 , 1 } S = { 0 , 1 } S = {− 1 , 1 } S = { 1 } S = { 0 } Catalan Shi Semi-order Braid Linial T ∈B ( n ) T ∈B ( n ) s.t. T ∈B ( n ) s.t. T ∈B ( n ) s.t. T ∈B ( n ) s.t. w w v v v v v v v v v v v v u u u u u u u u u u u u u u ⇓ ⇓ ⇓ ⇓ ⇓ ⇓ u>v u>v u>v v >u u>v u>v

Known counting results for S ⊆ {− 1 , 0 , 1 } S = {− 1 , 0 , 1 } S = { 0 , 1 } S = {− 1 , 1 } S = { 1 } S = { 0 } Catalan Shi Semi-order Braid Linial T ∈B ( n ) T ∈B ( n ) s.t. T ∈B ( n ) s.t. T ∈B ( n ) s.t. T ∈B ( n ) s.t. w w v v v v v v v v v v v v u u u u u u u u u u u u u u ⇓ ⇓ ⇓ ⇓ ⇓ ⇓ u>v u>v u>v v >u u>v u>v “ Why? ” Ira Gessel

Known counting results for S ⊆ {− 1 , 0 , 1 } S = {− 1 , 0 , 1 } S = { 0 , 1 } S = {− 1 , 1 } S = { 1 } S = { 0 } Catalan Shi Semi-order Braid Linial T ∈B ( n ) T ∈B ( n ) s.t. T ∈B ( n ) s.t. T ∈B ( n ) s.t. T ∈B ( n ) s.t. w w v v v v v v v v v v v v v u u u u u u u u u u u u u u ⇓ ⇓ ⇓ ⇓ ⇓ ⇓ u>v u>v u>v v >u u>v u>v v v v u u u “ Why? ” ⇓ ⇓ ⇓ Ira Gessel u>v u>v u>v & u<v

Bijection

Bijection for S ⊆ {− 1 , 0 , 1 } Trees: T S ( n ) = set of trees in B ( n ) such that: If − 1 / ∈ S If 0 / ∈ S If 1 / ∈ S v v v ⇒ u>v ⇒ u>v ⇒ u<v u u u

Bijection for S ⊆ {− 1 , 0 , 1 } Trees: T S ( n ) = set of trees in B ( n ) such that: If − 1 / ∈ S If 0 / ∈ S If 1 / ∈ S v v v ⇒ u>v ⇒ u>v ⇒ u<v u u u Map: Φ S : T S ( n ) �→ regions of A S ( n ) � � Φ S ( T ) = { x i − x j < s } { x i − x j > s } s ∈ S, 1 ≤ i<j ≤ n s ∈ S, 1 ≤ i<j ≤ n ( s,i,j ) ∈ T + ∈ T + ( s,i,j ) / where T + is . . .

Bijection for S ⊆ {− 1 , 0 , 1 } Trees: T S ( n ) = set of trees in B ( n ) such that: If − 1 / ∈ S If 0 / ∈ S If 1 / ∈ S v v v ⇒ u>v ⇒ u>v ⇒ u<v u u u q Tree order: s h m r p p a ≺ T b ≺ T c ≺ T d ≺ T e · · · c l g o o f b k k n j e i d a

Bijection for S ⊆ {− 1 , 0 , 1 } Trees: T S ( n ) = set of trees in B ( n ) such that: If − 1 / ∈ S If 0 / ∈ S If 1 / ∈ S v v v ⇒ u>v ⇒ u>v ⇒ u<v u u u Map: Φ S : T S ( n ) �→ regions of A S ( n ) � � Φ S ( T ) = { x i − x j < s } { x i − x j > s } s ∈ S, 1 ≤ i<j ≤ n s ∈ S, 1 ≤ i<j ≤ n ( s,i,j ) ∈ T + ∈ T + ( s,i,j ) / (0 , i, j ) ∈ T + where if i ≺ T j , ( − 1 , i, j ) ∈ T + if right-child ( i ) � T j , (1 , i, j ) ∈ T + if i ≺ T right-child ( j ) .

Bijection for S ⊆ {− 1 , 0 , 1 } Trees: T S ( n ) = set of trees in B ( n ) such that: If − 1 / ∈ S If 0 / ∈ S If 1 / ∈ S v v v ⇒ u>v ⇒ u>v ⇒ u<v u u u Map: Φ S : T S ( n ) �→ regions of A S ( n ) � � Φ S ( T ) = { x i − x j < s } { x i − x j > s } s ∈ S, 1 ≤ i<j ≤ n s ∈ S, 1 ≤ i<j ≤ n ( s,i,j ) ∈ T + ∈ T + ( s,i,j ) / (0 , i, j ) ∈ T + where if i ≺ T j , ( − 1 , i, j ) ∈ T + if right-child ( i ) � T j , (1 , i, j ) ∈ T + if i ≺ T right-child ( j ) . Thm: Φ S is a bijection between T S ( n ) and the regions of A S ( n ) .

Example: Linial S = { 1 } T S ( n ) : v ⇒ u>v u v ⇒ u>v u

Example: Linial S = { 1 } x 1 1 2 T S ( n ) : Φ S : 3 1 2 3 v 1 ⇒ u>v 2 u 3 2 1 3 1 3 2 v ⇒ u>v 1 u 2 3 1 2 3 x 2 x 3

Example: Linial S = { 1 } x 1 1 2 T S ( n ) : Φ S : 3 1 2 3 v 1 ⇒ u>v 2 u 3 2 1 3 1 3 2 v ⇒ u>v 1 u 2 3 x 1 − x 2 =1 1 2 3 x 2 x 3 � � Φ S ( T ) = { x i − x j < 1 } { x i − x j > 1 } 1 ≤ i<j ≤ n 1 ≤ i<j ≤ n i ≺ T right-child ( j ) i � T right-child ( j )

Generalization S ⊆ [ − m..m ] • T ( m ) = set of rooted ( m +1) -ary trees with labeled nodes. 7 1 9 13 3 10 4 4 11 1 12 8 5 2 6

Generalization S ⊆ [ − m..m ] • T ( m ) = set of rooted ( m +1) -ary trees with labeled nodes. • The last node among the children of u is denoted cadet( u ) . 7 1 9 13 3 10 4 4 11 1 12 8 5 2 6

Generalization S ⊆ [ − m..m ] • T ( m ) = set of rooted ( m +1) -ary trees with labeled nodes. • The last node among the children of u is denoted cadet( u ) . Def: T S = set of trees in T ( m ) such that for all v = cadet ( u ) , • # left-siblings ( v ) / ∈ S ∪ { 0 } ⇒ u < v , • − # left-siblings ( v ) / ∈ S ⇒ u > v . # left-siblings(v) v u

Generalization S ⊆ [ − m..m ] Def: S is transitive if it satisfies: • if a, b / ∈ S , with ab > 0 , then a + b / ∈ S , • if a, b / ∈ S , with ab < 0 , then a − b / ∈ S , • if 0 , a / ∈ S , with a > 0 , then − a / ∈ S .

Generalization S ⊆ [ − m..m ] Def: S is transitive if it satisfies: • if a, b / ∈ S , with ab > 0 , then a + b / ∈ S , • if a, b / ∈ S , with ab < 0 , then a − b / ∈ S , • if 0 , a / ∈ S , with a > 0 , then − a / ∈ S . Examples of transitive sets: • Any subset of {− 1 , 0 , 1 } . • Any interval of integers containing 1 . • S such that [ − k..k ] ⊆ S ⊆ [ − 2 k.. 2 k ] for some k .

Generalization S ⊆ [ − m..m ] Def: S is transitive if it satisfies: • if a, b / ∈ S , with ab > 0 , then a + b / ∈ S , • if a, b / ∈ S , with ab < 0 , then a − b / ∈ S , • if 0 , a / ∈ S , with a > 0 , then − a / ∈ S . Examples of transitive sets: • Any subset of {− 1 , 0 , 1 } . • Any interval of integers containing 1 . • S such that [ − k..k ] ⊆ S ⊆ [ − 2 k.. 2 k ] for some k . Thm: If S is transitive, then Φ S is a bijection between T S ( n ) and the regions of A S ( n ) .

Direct proof for S ⊆ {− 1 , 0 , 1 }

Warm up: Braid arrangement x 1 x 1 − x 2 = 0 x 1 − x 3 = 0 x 2 x 3 x 2 − x 3 = 0

Warm up: Braid arrangement x 1 x 1 − x 2 = 0 x 1 − x 3 = 0 ( x 1 , x 2 , x 3 ) x 2 x 3 x 2 − x 3 = 0 x 2 x 1 x 3

Warm up: Braid arrangement x 1 x 1 − x 2 = 0 x 1 − x 3 = 0 ( x 1 , x 2 , x 3 ) x 2 x 3 x 2 − x 3 = 0 x 2 x 1 x 3 n ! 2 1 3

Catalan arrangement S = {− 1 , 0 , 1 } x 1 x 1 − x 2 = 1 x 1 − x 2 = 0 x 1 − x 2 = − 1 ( x 1 , x 2 , x 3 ) x 2 x 3 x 2 +1 x 1 +1 x 3 +1 x 2 x 1 x 1 x 3

Catalan arrangement S = {− 1 , 0 , 1 } x 1 x 1 − x 2 = 1 x 1 − x 2 = 0 x 1 − x 2 = − 1 ( x 1 , x 2 , x 3 ) x 2 x 3 x 2 +1 x 1 +1 x 3 +1 x 2 x 1 x 1 x 3 non-nesting parentheses 2 1 3 n ! Cat ( n )

Catalan schemes = labeled non-nesting parenthesis systems x 1 3 2 1 2 3 1 3 2 1 2 3 1 3 2 1 2 3 1 3 2 1 2 3 1 3 1 2 2 1 3 321 231 3 1 2 2 1 3 3 1 2 312 213 2 1 3 132 123 3 1 2 2 1 3 1 3 2 1 2 3 1 3 2 1 2 3 1 3 2 1 2 3 x 2 x 3 1 3 2 1 2 3

Shi/SO/Linial regions as equivalence classes of schemes Definition: • Shi moves ( S = { 0 , 1 } ): j i i j if i < j • Semi-order moves ( S = {− 1 , 1 } ): j i j i • Linial moves ( S = { 1 } ) = Shi moves + semi-order moves