Background Facial Weak Order Lattice and properties Facial Weak Order Aram Dermenjian Joint work with: Christophe Hohlweg (LACIM) and Vincent Pilaud (CNRS & LIX) Universit´ e du Qu´ ebec ` a Montr´ eal 18 June 2017 A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 1/23
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties History and Background Finite Coxeter System ( W , S ) such that W : = � s ∈ S | ( s i s j ) m i , j = e for s i , s j ∈ S � where m i , j ∈ N ⋆ and m i , j = 1 only if i = j . A Coxeter diagram Γ W for a Coxeter System ( W , S ) has S as a vertex set and an edge labelled m i , j when m i , j > 2. m i , j s i s j Example 3 = ( s 1 s 2 ) 4 = ( s 2 s 3 ) 3 = ( s 1 s 3 ) 2 = e � W B 3 = � s 1 , s 2 , s 3 | s 2 1 = s 2 2 = s 2 4 Γ B 3 : s 1 s 2 s 3 A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 2/23
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties History and Background Finite Coxeter System ( W , S ) such that W : = � s ∈ S | ( s i s j ) m i , j = e for s i , s j ∈ S � where m i , j ∈ N ⋆ and m i , j = 1 only if i = j . A Coxeter diagram Γ W for a Coxeter System ( W , S ) has S as a vertex set and an edge labelled m i , j when m i , j > 2. m i , j s i s j Example W A n = S n +1 , symmetric group. Γ A n : s 1 s 2 s 3 s n − 1 s n A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 2/23
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties History and Background Finite Coxeter System ( W , S ) such that W : = � s ∈ S | ( s i s j ) m i , j = e for s i , s j ∈ S � where m i , j ∈ N ⋆ and m i , j = 1 only if i = j . A Coxeter diagram Γ W for a Coxeter System ( W , S ) has S as a vertex set and an edge labelled m i , j when m i , j > 2. m i , j s i s j Example W I 2 ( m ) = D ( m ), dihedral group of order 2 m . m Γ I 2 ( m ) : s 1 s 2 A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 2/23
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties History and Background Let ( W , S ) be a Coxeter system. Let w ∈ W such that w = s 1 . . . s n for some s i ∈ S . We say that w has length n , ℓ ( w ) = n , if n is minimal. Example Let Γ A 2 : s t . ℓ ( stst ) = 2 as stst = tstt = ts . Let the (right) weak order be the order on the Cayley graph where w ws and ℓ ( w ) < R ℓ ( ws ). A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 3/23
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties History and Background Theorem (Bj¨ orner [1984]) Let ( W , S ) be a finite Coxeter system. The weak order is a lattice graded by length. For finite Coxeter systems, there exists a longest element in the weak order, w ◦ . Example Let Γ A 2 : s t . sts = w ◦ = tst ts st s t e A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 4/23
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties Motivation In 2001, Krob, Latapy, Novelli, Phan, and Schwer extended the weak order to an order on all faces for type A using inversion tables. They 1 gave a local definition of this order using covers, 2 gave a global definition of this order combinatorially, and 3 showed that the poset for this order is a lattice. In 2006, Ronco and Palacios extended this new order to Coxeter groups of all types using cover relations. A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 5/23
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties Motivation In 2001, Krob, Latapy, Novelli, Phan, and Schwer extended the weak order to an order on all faces for type A using inversion tables. They 1 gave a local definition of this order using covers, � 2 gave a global definition of this order combinatorially, and 3 showed that the poset for this order is a lattice. In 2006, Ronco and Palacios extended this new order to Coxeter groups of all types using cover relations. Problems: Can we find a global definition for this poset, and is it a lattice? A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 5/23
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties Motivation In 2001, Krob, Latapy, Novelli, Phan, and Schwer extended the weak order to an order on all faces for type A using inversion tables. They 1 gave a local definition of this order using covers, � 2 gave a global definition of this order combinatorially, and � 3 showed that the poset for this order is a lattice. � In 2006, Ronco and Palacios extended this new order to Coxeter groups of all types using cover relations. Problems: Can we find a global definition for this poset, and is it a lattice? A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 5/23
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties Motivation In 2001, Krob, Latapy, Novelli, Phan, and Schwer extended the weak order to an order on all faces for type A using inversion tables. They 1 gave a local definition of this order using covers, � 2 gave a global definition of this order combinatorially, and � 3 showed that the poset for this order is a lattice. � In 2006, Ronco and Palacios extended this new order to Coxeter groups of all types using cover relations. Problems: Can we find a global definition for this poset, and is it a lattice? A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 6/23
Local Definition Background Global Definition Facial Weak Order Root Inversion Set Lattice and properties Equivalence Parabolic Subgroups Let I ⊆ S . W I = � I � is the standard parabolic subgroup with long element denoted w ◦ , I . W I : = { w ∈ W | ℓ ( w ) ≤ ℓ ( ws ), for all s ∈ I } is the set of minimal length coset representatives for W / W I . Any element w ∈ W admits a unique factorization w = w I · w I with w I ∈ W I and w I ∈ W I . By convention in this talk xW I means x ∈ W I . Coxeter complex - P W - the abstract simplicial complex whose faces are all the standard parabolic cosets of W . sts tsW { t } stW { s } ts st tW { s } sW { t } W s t W { t } W { s } e A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 7/23
Local Definition Background Global Definition Facial Weak Order Root Inversion Set Lattice and properties Equivalence Facial Weak Order Let ( W , S ) be a finite Coxeter system. Definition (Krob et.al. [2001, type A ], Palacios, Ronco [2006]) The (right) facial weak order is the order ≤ F on the Coxeter complex P W defined by cover relations of two types: ∈ I and x ∈ W I ∪{ s } , (1) xW I < · xW I ∪{ s } if s / (2) xW I < · xw ◦ , I w ◦ , I � { s } W I � { s } if s ∈ I , where I ⊆ S and x ∈ W I . A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 8/23
Local Definition Background Global Definition Facial Weak Order Root Inversion Set Lattice and properties Equivalence Facial weak order example ∈ I and x ∈ W I ∪{ s } (1) xW I < · xW I ∪{ s } if s / (2) xW I < · xw ◦ , I w ◦ , I � { s } W I � { s } if s ∈ I sts tsW { t } stW { s } (2) (2) ts st (1) (1) (2) (2) (2) (2) tW { s } sW { t } W (1) (1) (1) (1) s t (2) (2) W { t } W { s } (1) (1) e A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 9/23
Local Definition Background Global Definition Facial Weak Order Root Inversion Set Lattice and properties Equivalence Motivation In 2001, Krob, Latapy, Novelli, Phan, and Schwer extended the weak order to an order on all faces for type A using inversion tables. They 1 gave a local definition of this order using covers, � 2 gave a global definition of this order combinatorially, and � 3 showed that the poset for this order is a lattice. � In 2006, Ronco and Palacios extended this new order to Coxeter groups of all types using cover relations. Problems: Can we find a global definition for this poset, and is it a lattice? A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 10/23
Local Definition Background Global Definition Facial Weak Order Root Inversion Set Lattice and properties Equivalence Root System Let ( V , �· , ·� ) be a real Euclidean space. Let W be a group generated by a set of reflections S . W ֒ → O ( V ) gives representation as a finite reflection group. The reflection associated to α ∈ V \{ 0 } is − γ s t − α t − α s s α ( v ) = v − 2 � v , α � || α || 2 α ( v ∈ V ) α s α t γ = α s + α t A root system is Φ : = { α ∈ V | s α ∈ W , || α || = 1 } We have Φ = Φ + ⊔ Φ − decomposable into positive and negative roots. A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 11/23
Local Definition Background Global Definition Facial Weak Order Root Inversion Set Lattice and properties Equivalence Relationship between Root Systems and Coxeter Systems W A 2 = � s , t | s 2 = t 2 = ( st ) 3 = e � Γ A 2 : s t − γ s t − α t − α s α s α t γ = α s + α t A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 12/23
Local Definition Background Global Definition Facial Weak Order Root Inversion Set Lattice and properties Equivalence Relationship between Root Systems and Coxeter Systems W A 2 = � s , t | s 2 = t 2 = ( st ) 3 = e � Γ A 2 : s t Perm( W ) = { w ( x ) | w ∈ W } s t x A. Dermenjian (UQ` AM) Facial Weak Order 18 June 2017 12/23
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