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 5 April 2016 A. Dermenjian (UQ` AM) Facial Weak Order 5 April 2016 1/24
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties History and Background The weak order was introduced on Coxeter groups by Bj¨ orner in 1984, it was shown to be a lattice. A. Dermenjian (UQ` AM) Facial Weak Order 5 April 2016 2/24
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties History and Background The weak order was introduced on Coxeter groups by Bj¨ orner in 1984, it was shown to be a lattice. 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 A. Dermenjian (UQ` AM) Facial Weak Order 5 April 2016 2/24
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties History and Background The weak order was introduced on Coxeter groups by Bj¨ orner in 1984, it was shown to be a lattice. 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 5 April 2016 2/24
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties History and Background The weak order was introduced on Coxeter groups by Bj¨ orner in 1984, it was shown to be a lattice. 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. Let the (right) weak order be the order on the Cayley graph where w ws and ℓ ( w ) < ℓ ( ws ). For finite Coxeter systems, there exists a longest element in the weak order, w ◦ . A. Dermenjian (UQ` AM) Facial Weak Order 5 April 2016 2/24
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties History and Background The weak order was introduced on Coxeter groups by Bj¨ orner in 1984, it was shown to be a lattice. Example Let Γ A 2 : s t . sts = w ◦ = tst ts st s t e A. Dermenjian (UQ` AM) Facial Weak Order 5 April 2016 2/24
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 5 April 2016 3/24
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 5 April 2016 3/24
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. Our motivation was to continue this work for all Coxeter groups. A. Dermenjian (UQ` AM) Facial Weak Order 5 April 2016 3/24
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. Our motivation was to continue this work for all Coxeter groups. A. Dermenjian (UQ` AM) Facial Weak Order 5 April 2016 3/24
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. Our motivation was to continue this work for all Coxeter groups. A. Dermenjian (UQ` AM) Facial Weak Order 5 April 2016 4/24
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 5 April 2016 5/24
Local Definition Background Global Definition Facial Weak Order Root Inversion Set Lattice and properties Equivalence Facial Weak Order Definition (Krob et.al. [2001], 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 5 April 2016 6/24
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 stW { s } tsW { t } (2) (2) st (1) (1) ts (2) (2) (2) (2) sW { t } tW { s } W (1) (1) (1) (1) s (2) (2) t W { s } W { t } (1) (1) e A. Dermenjian (UQ` AM) Facial Weak Order 5 April 2016 7/24
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. Our motivation was to continue this work for all Coxeter groups. A. Dermenjian (UQ` AM) Facial Weak Order 5 April 2016 8/24
Local Definition Background Global Definition Facial Weak Order Root Inversion Set Lattice and properties Equivalence Root System Let ( V , �· , ·� ) be a 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 5 April 2016 9/24
Local Definition Background Global Definition Facial Weak Order Root Inversion Set Lattice and properties Equivalence Inversion Sets Let ( W , S ) be a Coxeter system. Define (left) inversion sets as the set N ( w ) : = Φ + ∩ w (Φ − ). Example − γ − α t s t − α s Let Γ A 2 : s t , with Φ given by the roots N ( ts ) = Φ + ∩ ts (Φ − ) α s α t γ = α s + α t = Φ + ∩ { α t , γ, − α s } = { α t , γ } A. Dermenjian (UQ` AM) Facial Weak Order 5 April 2016 10/24
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