Background Facial Weak Order Lattice and properties The facial weak order and its lattice of quotients Aram Dermenjian Joint work with: Christophe Hohlweg (LACIM) and Vincent Pilaud (CNRS & LIX) Université du Québec à Montréal 13 April 2019 On this day in 1909 Stan Ulam was born. “Knowing what is big and what is small is more important than being able to solve partial differential equations.” - Ulam. A. Dermenjian (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019 1/5?
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 � � s 1 , s 2 , s 3 | s 2 1 = s 2 2 = s 2 W B 3 = 4 Γ B 3 : s 1 s 2 s 3 A. Dermenjian (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019 2/10?
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 (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019 2/10?
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 (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019 2/10?
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 ) < ℓ ( ws ). A. Dermenjian (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019 ∼ π /10?
Background Coxeter Systems Facial Weak Order Motivation Lattice and properties History and Background Theorem (Björner [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 (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019 4/10?
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. 5/10 2 A. Dermenjian (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019
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? 5/10 2 A. Dermenjian (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019
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? 5/10 2 A. Dermenjian (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019
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? ∼ τ /10 2 A. Dermenjian (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019
Local Definition Background Global Definition Facial Weak Order Root Inversion Set Lattice and properties Equivalence Parabolic Subgroups Let I ⊆ S . W I = � I � is standard parabolic subgroup (long element: w ◦ , I ). W I : = { w ∈ W | ℓ ( w ) ≤ ℓ ( ws ), for all s ∈ I } is the set of min length coset representatives for W / W I . Unique factorization: w = w I · w I with w I ∈ W I , 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 7/10 2 A. Dermenjian (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019
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 . 8/10 2 A. Dermenjian (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019
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) t (2) (2) s (1) (1) W { t } W { s } e 9/10 2 A. Dermenjian (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019
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? 10/10 10 A. Dermenjian (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019
Local Definition Background Global Definition Facial Weak Order Root Inversion Set Lattice and properties Equivalence Facial Intervals Proposition (Björner, Las Vergas, Sturmfels, White, Ziegler ’93) Let ( W , S ) be a finite Coxeter system and xW I a standard parabolic coset. Then there exists a unique interval [ x , xw ◦ , I ] in the weak order such that xW I = [ x , xw ◦ , I ] . sts [ sts , sts ] tsW { t } stW { s } [ ts , sts ] [ st , sts ] sts [ ts , ts ] [ st , st ] ts st ts st tW { s } sW { t } [ t , ts ] [ e , sts ] [ s , st ] W s t e s t [ t , t ] [ s , s ] W { t } W { s } [ e , t ] [ e , s ] e [ e , e ] 11/10 10 A. Dermenjian (UQÀM) The facial weak order and its lattice of quotients 13 Apr 2019
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