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Separable elements and splittings of Weyl groups Yibo Gao Joint - PowerPoint PPT Presentation

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Separable elements and splittings of Weyl groups Yibo Gao Joint work with: Christian Gaetz Massachusetts Institute of Technology MIT Combinatorics Seminar, Spring 2020 Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 1 / 33

  1. Root systems and Weyl groups Definition (Root system) Let E = R n . A root system Φ ⊂ E is a finite set of vectors, such that Φ spans E ; for α ∈ Φ, k α ∈ Φ iff k ∈ {± 1 } ; for α, β ∈ Φ, 2( α, β ) / ( α, α ) ∈ Z ; for α, β ∈ Φ, σ α ( β ) := β − 2( α, β ) ( α, α ) α ∈ Φ . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 8 / 33

  2. Root systems and Weyl groups Definition (Root system) Let E = R n . A root system Φ ⊂ E is a finite set of vectors, such that Φ spans E ; for α ∈ Φ, k α ∈ Φ iff k ∈ {± 1 } ; for α, β ∈ Φ, 2( α, β ) / ( α, α ) ∈ Z ; for α, β ∈ Φ, σ α ( β ) := β − 2( α, β ) ( α, α ) α ∈ Φ . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 8 / 33

  3. Root systems and Weyl groups α 1 2 α 1 + 3 α 2 α 2 Picking a generic hyperplane partitions Φ into positive roots Φ + and negative roots Φ − . This determines a unique set ∆ of simple roots such that ∆ = { α 1 , . . . , α n } is a basis for E ; every α ∈ Φ + is written as � n i =1 c i α i where c i ∈ Z ≥ 0 ∀ i . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 9 / 33

  4. Root systems and Weyl groups Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 10 / 33

  5. Root systems and Weyl groups We say Φ is irreducible if it cannot be partitioned into Φ ′ ⊔ Φ ′′ such that ( α, β ) = 0 for all α ∈ Φ ′ and β ∈ Φ ′′ . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 10 / 33

  6. Root systems and Weyl groups We say Φ is irreducible if it cannot be partitioned into Φ ′ ⊔ Φ ′′ such that ( α, β ) = 0 for all α ∈ Φ ′ and β ∈ Φ ′′ . Irreducible root systems can be classified using Dynkin diagrams. Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 10 / 33

  7. Root systems and Weyl groups We say Φ is irreducible if it cannot be partitioned into Φ ′ ⊔ Φ ′′ such that ( α, β ) = 0 for all α ∈ Φ ′ and β ∈ Φ ′′ . Irreducible root systems can be classified using Dynkin diagrams. Figure: Irreducible root systems (Wikipedia) Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 10 / 33

  8. Root systems and Weyl groups Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 11 / 33

  9. Root systems and Weyl groups The Weyl group W = W (Φ) that corresponds to Φ is a finite subgroup of GL ( E ) generated by all reflections across roots σ α , for α ∈ Φ, or equivalently, by s i := σ α i for α i ∈ ∆. Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 11 / 33

  10. Root systems and Weyl groups The Weyl group W = W (Φ) that corresponds to Φ is a finite subgroup of GL ( E ) generated by all reflections across roots σ α , for α ∈ Φ, or equivalently, by s i := σ α i for α i ∈ ∆. Fix ∆ ⊂ Φ + ⊂ Φ as above. For w ∈ W , its Coxeter length ℓ ( w ) is defined to be the smallest ℓ such that w can be written as s i 1 · · · s i ℓ . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 11 / 33

  11. Root systems and Weyl groups The Weyl group W = W (Φ) that corresponds to Φ is a finite subgroup of GL ( E ) generated by all reflections across roots σ α , for α ∈ Φ, or equivalently, by s i := σ α i for α i ∈ ∆. Fix ∆ ⊂ Φ + ⊂ Φ as above. For w ∈ W , its Coxeter length ℓ ( w ) is defined to be the smallest ℓ such that w can be written as s i 1 · · · s i ℓ . The left weak (Bruhat) order is generated by if ℓ ( s i w ) = ℓ ( w ) + 1 , where s i = σ α i , α i ∈ ∆ . w ⋖ L s i w Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 11 / 33

  12. Root systems and Weyl groups Example: root system of type A n − 1 E = R n / (1 , . . . , 1) . Φ = { e i − e j : i � = j } . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 12 / 33

  13. Root systems and Weyl groups Example: root system of type A n − 1 E = R n / (1 , . . . , 1) . Φ = { e i − e j : i � = j } . Φ + = { e i − e j : i < j } . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 12 / 33

  14. Root systems and Weyl groups Example: root system of type A n − 1 E = R n / (1 , . . . , 1) . Φ = { e i − e j : i � = j } . Φ + = { e i − e j : i < j } . ∆ = { e i − e i +1 : i = 1 , . . . , n − 1 } . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 12 / 33

  15. Root systems and Weyl groups Example: root system of type A n − 1 E = R n / (1 , . . . , 1) . Φ = { e i − e j : i � = j } . Φ + = { e i − e j : i < j } . ∆ = { e i − e i +1 : i = 1 , . . . , n − 1 } . σ e i − e j : ( x 1 , . . . , x i , . . . , x j , . . . , x n ) �→ ( x 1 , . . . , x j , . . . , x i , . . . , x n ) . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 12 / 33

  16. Root systems and Weyl groups Example: root system of type A n − 1 E = R n / (1 , . . . , 1) . Φ = { e i − e j : i � = j } . Φ + = { e i − e j : i < j } . ∆ = { e i − e i +1 : i = 1 , . . . , n − 1 } . σ e i − e j : ( x 1 , . . . , x i , . . . , x j , . . . , x n ) �→ ( x 1 , . . . , x j , . . . , x i , . . . , x n ) . s i = σ e i − e i +1 = ( i i + 1), so W ∼ = S n . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 12 / 33

  17. Root systems and Weyl groups Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 13 / 33

  18. Root systems and Weyl groups Definition (Inversion set) For w ∈ W , I Φ ( w ) := { α ∈ Φ + : w α ∈ Φ − } . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 13 / 33

  19. Root systems and Weyl groups Definition (Inversion set) For w ∈ W , I Φ ( w ) := { α ∈ Φ + : w α ∈ Φ − } . The following proposition is well-known and useful. Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 13 / 33

  20. Root systems and Weyl groups Definition (Inversion set) For w ∈ W , I Φ ( w ) := { α ∈ Φ + : w α ∈ Φ − } . The following proposition is well-known and useful. Proposition I Φ ( w ) uniquely characterizes w ∈ W . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 13 / 33

  21. Root systems and Weyl groups Definition (Inversion set) For w ∈ W , I Φ ( w ) := { α ∈ Φ + : w α ∈ Φ − } . The following proposition is well-known and useful. Proposition I Φ ( w ) uniquely characterizes w ∈ W . S ⊂ Φ + is the inversion set of some w ∈ W iff S is biconvex : Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 13 / 33

  22. Root systems and Weyl groups Definition (Inversion set) For w ∈ W , I Φ ( w ) := { α ∈ Φ + : w α ∈ Φ − } . The following proposition is well-known and useful. Proposition I Φ ( w ) uniquely characterizes w ∈ W . S ⊂ Φ + is the inversion set of some w ∈ W iff S is biconvex : if α, β ∈ S and α + β ∈ Φ + , then α + β ∈ S; Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 13 / 33

  23. Root systems and Weyl groups Definition (Inversion set) For w ∈ W , I Φ ( w ) := { α ∈ Φ + : w α ∈ Φ − } . The following proposition is well-known and useful. Proposition I Φ ( w ) uniquely characterizes w ∈ W . S ⊂ Φ + is the inversion set of some w ∈ W iff S is biconvex : if α, β ∈ S and α + β ∈ Φ + , then α + β ∈ S; ∈ S and α + β ∈ Φ + , then α + β / ∈ S. if α, β / Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 13 / 33

  24. Root systems and Weyl groups Definition (Inversion set) For w ∈ W , I Φ ( w ) := { α ∈ Φ + : w α ∈ Φ − } . The following proposition is well-known and useful. Proposition I Φ ( w ) uniquely characterizes w ∈ W . S ⊂ Φ + is the inversion set of some w ∈ W iff S is biconvex : if α, β ∈ S and α + β ∈ Φ + , then α + β ∈ S; ∈ S and α + β ∈ Φ + , then α + β / ∈ S. if α, β / u ≤ L v in the (left) weak order iff I Φ ( u ) ⊂ I Φ ( v ) . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 13 / 33

  25. Root systems and Weyl groups Definition (Inversion set) For w ∈ W , I Φ ( w ) := { α ∈ Φ + : w α ∈ Φ − } . The following proposition is well-known and useful. Proposition I Φ ( w ) uniquely characterizes w ∈ W . S ⊂ Φ + is the inversion set of some w ∈ W iff S is biconvex : if α, β ∈ S and α + β ∈ Φ + , then α + β ∈ S; ∈ S and α + β ∈ Φ + , then α + β / ∈ S. if α, β / u ≤ L v in the (left) weak order iff I Φ ( u ) ⊂ I Φ ( v ) . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 13 / 33

  26. Root systems and Weyl groups Definition (Inversion set) For w ∈ W , I Φ ( w ) := { α ∈ Φ + : w α ∈ Φ − } . The following proposition is well-known and useful. Proposition I Φ ( w ) uniquely characterizes w ∈ W . S ⊂ Φ + is the inversion set of some w ∈ W iff S is biconvex : if α, β ∈ S and α + β ∈ Φ + , then α + β ∈ S; ∈ S and α + β ∈ Φ + , then α + β / ∈ S. if α, β / u ≤ L v in the (left) weak order iff I Φ ( u ) ⊂ I Φ ( v ) . Definition (Root poset and support) For α, β ∈ Φ + , α ≤ β if β − α is written as a nonnegative linear combination of simple roots. For α ∈ Φ + , its support is defined as Supp ( α ) := { α i ∈ ∆ : α i ≤ α } . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 13 / 33

  27. Root systems and Weyl groups Example: root system of type A n − 1 E = R n / (1 , . . . , 1) . Φ = { e i − e j : i � = j } . Φ + = { e i − e j : i < j } . ∆ = { e i − e i +1 : i = 1 , . . . , n − 1 } . σ e i − e j : ( x 1 , . . . , x i , . . . , x j , . . . , x n ) �→ ( x 1 , . . . , x j , . . . , x i , . . . , x n ) . s i = σ e i − e i +1 = ( i i + 1), so W ∼ = S n . I Φ ( w ) = { e i − e j : i < j , and w i > w j } . e 1 − e 5 e 2 − e 5 e 3 − e 5 e 1 − e 2 e 2 − e 3 e 3 − e 4 e 4 − e 5 Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 14 / 33

  28. A restriction map (Billey-Postnikov 2005) Let E ′ ⊂ E be a subspace. Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 15 / 33

  29. A restriction map (Billey-Postnikov 2005) Let E ′ ⊂ E be a subspace. Φ ′ := E ′ ∩ Φ is a root system. (Φ ′ ) + := E ′ ∩ Φ + is a choice of positive roots. Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 15 / 33

  30. A restriction map (Billey-Postnikov 2005) Let E ′ ⊂ E be a subspace. Φ ′ := E ′ ∩ Φ is a root system. (Φ ′ ) + := E ′ ∩ Φ + is a choice of positive roots. For w ∈ W (Φ), I Φ ( w ) is biconvex. So I Φ ( w ) ∩ E ′ ⊂ (Φ ′ ) + is also biconvex. Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 15 / 33

  31. A restriction map (Billey-Postnikov 2005) Let E ′ ⊂ E be a subspace. Φ ′ := E ′ ∩ Φ is a root system. (Φ ′ ) + := E ′ ∩ Φ + is a choice of positive roots. For w ∈ W (Φ), I Φ ( w ) is biconvex. So I Φ ( w ) ∩ E ′ ⊂ (Φ ′ ) + is also biconvex. Define w | Φ ′ := w ′ to be the unique w ′ ∈ W (Φ ′ ) such that I Φ ′ ( w ′ ) = I Φ ( w ) ∩ E ′ . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 15 / 33

  32. A restriction map (Billey-Postnikov 2005) Let E ′ ⊂ E be a subspace. Φ ′ := E ′ ∩ Φ is a root system. (Φ ′ ) + := E ′ ∩ Φ + is a choice of positive roots. For w ∈ W (Φ), I Φ ( w ) is biconvex. So I Φ ( w ) ∩ E ′ ⊂ (Φ ′ ) + is also biconvex. Define w | Φ ′ := w ′ to be the unique w ′ ∈ W (Φ ′ ) such that I Φ ′ ( w ′ ) = I Φ ( w ) ∩ E ′ . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 15 / 33

  33. A restriction map (Billey-Postnikov 2005) Let E ′ ⊂ E be a subspace. Φ ′ := E ′ ∩ Φ is a root system. (Φ ′ ) + := E ′ ∩ Φ + is a choice of positive roots. For w ∈ W (Φ), I Φ ( w ) is biconvex. So I Φ ( w ) ∩ E ′ ⊂ (Φ ′ ) + is also biconvex. Define w | Φ ′ := w ′ to be the unique w ′ ∈ W (Φ ′ ) such that I Φ ′ ( w ′ ) = I Φ ( w ) ∩ E ′ . Example: restriction map in type A Let w = 6347215 ∈ W ( A 6 ). Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 15 / 33

  34. A restriction map (Billey-Postnikov 2005) Let E ′ ⊂ E be a subspace. Φ ′ := E ′ ∩ Φ is a root system. (Φ ′ ) + := E ′ ∩ Φ + is a choice of positive roots. For w ∈ W (Φ), I Φ ( w ) is biconvex. So I Φ ( w ) ∩ E ′ ⊂ (Φ ′ ) + is also biconvex. Define w | Φ ′ := w ′ to be the unique w ′ ∈ W (Φ ′ ) such that I Φ ′ ( w ′ ) = I Φ ( w ) ∩ E ′ . Example: restriction map in type A Let w = 6347215 ∈ W ( A 6 ). Consider E ′ = span ( e 2 − e 4 , e 4 − e 5 ). Then Φ ′ is of type A 2 with the set of simple roots ∆ ′ = { e 2 − e 4 , e 4 − e 5 } = { e ′ 1 − e ′ 2 , e ′ 2 − e ′ 3 } . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 15 / 33

  35. A restriction map (Billey-Postnikov 2005) Let E ′ ⊂ E be a subspace. Φ ′ := E ′ ∩ Φ is a root system. (Φ ′ ) + := E ′ ∩ Φ + is a choice of positive roots. For w ∈ W (Φ), I Φ ( w ) is biconvex. So I Φ ( w ) ∩ E ′ ⊂ (Φ ′ ) + is also biconvex. Define w | Φ ′ := w ′ to be the unique w ′ ∈ W (Φ ′ ) such that I Φ ′ ( w ′ ) = I Φ ( w ) ∩ E ′ . Example: restriction map in type A Let w = 6347215 ∈ W ( A 6 ). Consider E ′ = span ( e 2 − e 4 , e 4 − e 5 ). Then Φ ′ is of type A 2 with the set of simple roots ∆ ′ = { e 2 − e 4 , e 4 − e 5 } = { e ′ 1 − e ′ 2 , e ′ 2 − e ′ 3 } . Then I Φ ( w ) ∩ E ′ = { e 4 − e 5 , e 2 − e 5 } = { e ′ 2 − e ′ 3 , e ′ 1 − e ′ 3 } since w (4) > w (5) and w (2) > w (5). Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 15 / 33

  36. A restriction map (Billey-Postnikov 2005) Let E ′ ⊂ E be a subspace. Φ ′ := E ′ ∩ Φ is a root system. (Φ ′ ) + := E ′ ∩ Φ + is a choice of positive roots. For w ∈ W (Φ), I Φ ( w ) is biconvex. So I Φ ( w ) ∩ E ′ ⊂ (Φ ′ ) + is also biconvex. Define w | Φ ′ := w ′ to be the unique w ′ ∈ W (Φ ′ ) such that I Φ ′ ( w ′ ) = I Φ ( w ) ∩ E ′ . Example: restriction map in type A Let w = 6347215 ∈ W ( A 6 ). Consider E ′ = span ( e 2 − e 4 , e 4 − e 5 ). Then Φ ′ is of type A 2 with the set of simple roots ∆ ′ = { e 2 − e 4 , e 4 − e 5 } = { e ′ 1 − e ′ 2 , e ′ 2 − e ′ 3 } . Then I Φ ( w ) ∩ E ′ = { e 4 − e 5 , e 2 − e 5 } = { e ′ 2 − e ′ 3 , e ′ 1 − e ′ 3 } since w (4) > w (5) and w (2) > w (5). So w | Φ ′ = 231 ∈ W ( A 2 ). Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 15 / 33

  37. Separable elements in Weyl groups Definition (Gaetz and G. 2019) Let w ∈ W (Φ). Then w is separable if one of the following holds: Φ is of type A 1 ; Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 16 / 33

  38. Separable elements in Weyl groups Definition (Gaetz and G. 2019) Let w ∈ W (Φ). Then w is separable if one of the following holds: Φ is of type A 1 ; Φ = � Φ i is reducible and w | Φ i is separable for all i ; Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 16 / 33

  39. Separable elements in Weyl groups Definition (Gaetz and G. 2019) Let w ∈ W (Φ). Then w is separable if one of the following holds: Φ is of type A 1 ; Φ = � Φ i is reducible and w | Φ i is separable for all i ; Φ is irreducible and there exists a pivot α i ∈ ∆ such that w | Φ ′ ∈ W (Φ ′ ) is separable, where Φ ′ is generated by ∆ \ { α i } and either { α ∈ Φ + : α ≥ α i } ⊂ I Φ ( w ) or { α ∈ Φ + : α ≥ α i } ∩ I Φ ( w ) = ∅ . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 16 / 33

  40. Separable elements in Weyl groups Definition (Gaetz and G. 2019) Let w ∈ W (Φ). Then w is separable if one of the following holds: Φ is of type A 1 ; Φ = � Φ i is reducible and w | Φ i is separable for all i ; Φ is irreducible and there exists a pivot α i ∈ ∆ such that w | Φ ′ ∈ W (Φ ′ ) is separable, where Φ ′ is generated by ∆ \ { α i } and either { α ∈ Φ + : α ≥ α i } ⊂ I Φ ( w ) or { α ∈ Φ + : α ≥ α i } ∩ I Φ ( w ) = ∅ . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 16 / 33

  41. Separable elements in Weyl groups Definition (Gaetz and G. 2019) Let w ∈ W (Φ). Then w is separable if one of the following holds: Φ is of type A 1 ; Φ = � Φ i is reducible and w | Φ i is separable for all i ; Φ is irreducible and there exists a pivot α i ∈ ∆ such that w | Φ ′ ∈ W (Φ ′ ) is separable, where Φ ′ is generated by ∆ \ { α i } and either { α ∈ Φ + : α ≥ α i } ⊂ I Φ ( w ) or { α ∈ Φ + : α ≥ α i } ∩ I Φ ( w ) = ∅ . Compare the following equivalent definition of separable permutations. Definition Let w ∈ S n . Then w is separable if one of the following holds: n ≤ 2; there exists 1 < m < n such that either w 1 · · · w m is a separable permutation on { 1 , . . . , m } and w m +1 · · · w n is a separable permutation on { m + 1 , . . . , n } ; or w 1 · · · w m is a separable permutation on { n − m +1 , . . . , n } and w m +1 · · · w n is a separable permutation on { 1 , . . . , n − m } . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 16 / 33

  42. Separable elements in Weyl groups Example (separable elements in W ( B 2 )) Φ + = { α 1 , α 2 , α 1 + α 2 , α 1 + 2 α 2 } . ∆ = { α 1 , α 2 } . Dynkin diagram • • . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 17 / 33

  43. Separable elements in Weyl groups Example (separable elements in W ( B 2 )) Φ + = { α 1 , α 2 , α 1 + α 2 , α 1 + 2 α 2 } . ∆ = { α 1 , α 2 } . Dynkin diagram • • . • Φ + � { α 1 , α 1 + α 2 , α 1 + 2 α 2 } � • � • { α 2 , α 1 + α 2 , α 1 + 2 α 2 } { α 1 , α 1 + α 2 } • • { α 2 , α 1 + 2 α 2 } { α 1 } � • � • { α 2 } � • ∅ Figure: Weak order of type B 2 labeled by inversion sets, where separable elements are circled. Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 17 / 33

  44. Properties of separable elements The definition of separable elements, together with the following theorem, answers an open problem of Fan Wei. Theorem (Gaetz and G. 2019) Let w ∈ W = W (Φ) be separable. Then the upper order ideal V w and the lower order ideal Λ w in weak order are both rank-symmetric and rank-unimodal, and V w ( q )Λ w ( q ) = W ( q ) . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 18 / 33

  45. Properties of separable elements The definition of separable elements, together with the following theorem, answers an open problem of Fan Wei. Theorem (Gaetz and G. 2019) Let w ∈ W = W (Φ) be separable. Then the upper order ideal V w and the lower order ideal Λ w in weak order are both rank-symmetric and rank-unimodal, and V w ( q )Λ w ( q ) = W ( q ) . 0 in the parabolic quotient W J is separable. In The longest element w J this case we recover the well-known that that W J ( q ) W J ( q ) = W ( q ) , where W J is the parabolic subgroup . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 18 / 33

  46. Classification via pattern avoidance Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 19 / 33

  47. Classification via pattern avoidance Definition (Pattern avoidance) We say that w ∈ W (Φ) avoids pattern w ′ ∈ W (Φ ′ ) if there does not exists a subspace E ′ ⊂ E such that Φ ′ ≃ E ′ ∩ Φ and w | Φ ′ = w ′ . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 19 / 33

  48. Classification via pattern avoidance Definition (Pattern avoidance) We say that w ∈ W (Φ) avoids pattern w ′ ∈ W (Φ ′ ) if there does not exists a subspace E ′ ⊂ E such that Φ ′ ≃ E ′ ∩ Φ and w | Φ ′ = w ′ . Theorem (Gaetz and G. 2019) An element w ∈ W (Φ) is separable if (and only if) it avoids: 2413 and 3142 in W ( A 3 ) , two patterns of length 2 in W ( B 2 ) , and six patterns of length 2,3,4 in W ( G 2 ) . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 19 / 33

  49. Classification via pattern avoidance Definition (Pattern avoidance) We say that w ∈ W (Φ) avoids pattern w ′ ∈ W (Φ ′ ) if there does not exists a subspace E ′ ⊂ E such that Φ ′ ≃ E ′ ∩ Φ and w | Φ ′ = w ′ . Theorem (Gaetz and G. 2019) An element w ∈ W (Φ) is separable if (and only if) it avoids: 2413 and 3142 in W ( A 3 ) , two patterns of length 2 in W ( B 2 ) , and six patterns of length 2,3,4 in W ( G 2 ) . Our proof is fairly technical, type-dependent and computer-assisted. Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 19 / 33

  50. Classification via pattern avoidance Definition (Pattern avoidance) We say that w ∈ W (Φ) avoids pattern w ′ ∈ W (Φ ′ ) if there does not exists a subspace E ′ ⊂ E such that Φ ′ ≃ E ′ ∩ Φ and w | Φ ′ = w ′ . Theorem (Gaetz and G. 2019) An element w ∈ W (Φ) is separable if (and only if) it avoids: 2413 and 3142 in W ( A 3 ) , two patterns of length 2 in W ( B 2 ) , and six patterns of length 2,3,4 in W ( G 2 ) . Our proof is fairly technical, type-dependent and computer-assisted. Remark | W ( E 8 ) | = 696 , 729 , 600. Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 19 / 33

  51. Faces of graph associahedra Let Γ be a simple graph. The graph associahedron A (Γ) is a polytope which can be defined as the Minkowski sum of coordinate simplices corresponding to the connected subgraphs of Γ. Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 20 / 33

  52. Faces of graph associahedra Let Γ be a simple graph. The graph associahedron A (Γ) is a polytope which can be defined as the Minkowski sum of coordinate simplices corresponding to the connected subgraphs of Γ. Definition (Postnikov 2009) A collection N of subsets of Γ is a nested set if for all J ∈ N , the induced subgraph Γ | J is connected, for any I , J ∈ N , either I ⊂ J , J ⊂ I or I ∩ J = ∅ , for any collection of k ≥ 2 disjoint J 1 , . . . , J k ⊂ N , then subgraph Γ | J 1 ∪···∪ J k is not connected. Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 20 / 33

  53. Faces of graph associahedra Let Γ be a simple graph. The graph associahedron A (Γ) is a polytope which can be defined as the Minkowski sum of coordinate simplices corresponding to the connected subgraphs of Γ. Definition (Postnikov 2009) A collection N of subsets of Γ is a nested set if for all J ∈ N , the induced subgraph Γ | J is connected, for any I , J ∈ N , either I ⊂ J , J ⊂ I or I ∩ J = ∅ , for any collection of k ≥ 2 disjoint J 1 , . . . , J k ⊂ N , then subgraph Γ | J 1 ∪···∪ J k is not connected. Proposition (Postnikov 2009) The poset of faces of A (Γ) is isomorphic to the poset of nested sets on Γ which contain all connected components of Γ , ordered by reverse containment. Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 20 / 33

  54. Faces of graph associahedra Theorem (Gaetz and G. 2019) Let W be a finite Weyl group whose Dynkin diagram Γ contains r connected components. Then 1 the nested sets on Γ are in bijection with separable elements of W : � N �→ w 0 ( J ) =: w ( N ) , J ∈N where the product is taken in the order of any linear extension In particular, separable elements of W are in bijection with 2 r copies of faces of A (Γ). Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 21 / 33

  55. Faces of graph associahedra Theorem (Gaetz and G. 2019) Let W be a finite Weyl group whose Dynkin diagram Γ contains r connected components. Then 1 the nested sets on Γ are in bijection with separable elements of W : � N �→ w 0 ( J ) =: w ( N ) , J ∈N where the product is taken in the order of any linear extension 2 the rank generating function of the intervals [ e , w ( N )] is � J ∈N even W J ( q ) Λ L w ( N ) ( q ) = J ∈N odd W J ( q ) . � In particular, separable elements of W are in bijection with 2 r copies of faces of A (Γ). Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 21 / 33

  56. Faces of graph associahedra Example: bijection from nested sets to separable elements Let W be a Weyl group of type A 4 with simple roots α 1 , α 2 , α 3 , α 4 . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 22 / 33

  57. Faces of graph associahedra Example: bijection from nested sets to separable elements Let W be a Weyl group of type A 4 with simple roots α 1 , α 2 , α 3 , α 4 . The rank generating function for W is W ( q ) = [5]! q . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 22 / 33

  58. Faces of graph associahedra Example: bijection from nested sets to separable elements Let W be a Weyl group of type A 4 with simple roots α 1 , α 2 , α 3 , α 4 . The rank generating function for W is W ( q ) = [5]! q . N = {{ α 1 , α 2 , α 3 , α 4 } , { α 1 , α 2 } , { α 2 } , { α 4 }} . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 22 / 33

  59. Faces of graph associahedra Example: bijection from nested sets to separable elements Let W be a Weyl group of type A 4 with simple roots α 1 , α 2 , α 3 , α 4 . The rank generating function for W is W ( q ) = [5]! q . N = {{ α 1 , α 2 , α 3 , α 4 } , { α 1 , α 2 } , { α 2 } , { α 4 }} . w ( N ) = 54321 · 32145 · 13245 · 12354 = 35412 . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 22 / 33

  60. Faces of graph associahedra Example: bijection from nested sets to separable elements Let W be a Weyl group of type A 4 with simple roots α 1 , α 2 , α 3 , α 4 . The rank generating function for W is W ( q ) = [5]! q . N = {{ α 1 , α 2 , α 3 , α 4 } , { α 1 , α 2 } , { α 2 } , { α 4 }} . w ( N ) = 54321 · 32145 · 13245 · 12354 = 35412 . We see that 354 | 12 has a pivot at α 3 . And Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 22 / 33

  61. Faces of graph associahedra Example: bijection from nested sets to separable elements Let W be a Weyl group of type A 4 with simple roots α 1 , α 2 , α 3 , α 4 . The rank generating function for W is W ( q ) = [5]! q . N = {{ α 1 , α 2 , α 3 , α 4 } , { α 1 , α 2 } , { α 2 } , { α 4 }} . w ( N ) = 54321 · 32145 · 13245 · 12354 = 35412 . We see that 354 | 12 has a pivot at α 3 . And Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 22 / 33

  62. Faces of graph associahedra Example: bijection from nested sets to separable elements Let W be a Weyl group of type A 4 with simple roots α 1 , α 2 , α 3 , α 4 . The rank generating function for W is W ( q ) = [5]! q . N = {{ α 1 , α 2 , α 3 , α 4 } , { α 1 , α 2 } , { α 2 } , { α 4 }} . w ( N ) = 54321 · 32145 · 13245 · 12354 = 35412 . We see that 354 | 12 has a pivot at α 3 . And w ( N ) ( q ) = [5]! q [2]! q = q 7 + 2 q 6 + 3 q 5 + 4 q 4 + 4 q 3 + 3 q 2 + 2 q + 1 . Λ L [3]! q [2]! q N ′ = {{ α 1 , α 2 } , { α 2 } , { α 4 }} . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 22 / 33

  63. Faces of graph associahedra Example: bijection from nested sets to separable elements Let W be a Weyl group of type A 4 with simple roots α 1 , α 2 , α 3 , α 4 . The rank generating function for W is W ( q ) = [5]! q . N = {{ α 1 , α 2 , α 3 , α 4 } , { α 1 , α 2 } , { α 2 } , { α 4 }} . w ( N ) = 54321 · 32145 · 13245 · 12354 = 35412 . We see that 354 | 12 has a pivot at α 3 . And w ( N ) ( q ) = [5]! q [2]! q = q 7 + 2 q 6 + 3 q 5 + 4 q 4 + 4 q 3 + 3 q 2 + 2 q + 1 . Λ L [3]! q [2]! q N ′ = {{ α 1 , α 2 } , { α 2 } , { α 4 }} . w ( N ′ ) = 32145 · 13245 · 12354 = 31254 . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 22 / 33

  64. Faces of graph associahedra Example: bijection from nested sets to separable elements Let W be a Weyl group of type A 4 with simple roots α 1 , α 2 , α 3 , α 4 . The rank generating function for W is W ( q ) = [5]! q . N = {{ α 1 , α 2 , α 3 , α 4 } , { α 1 , α 2 } , { α 2 } , { α 4 }} . w ( N ) = 54321 · 32145 · 13245 · 12354 = 35412 . We see that 354 | 12 has a pivot at α 3 . And w ( N ) ( q ) = [5]! q [2]! q = q 7 + 2 q 6 + 3 q 5 + 4 q 4 + 4 q 3 + 3 q 2 + 2 q + 1 . Λ L [3]! q [2]! q N ′ = {{ α 1 , α 2 } , { α 2 } , { α 4 }} . w ( N ′ ) = 32145 · 13245 · 12354 = 31254 . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 22 / 33

  65. Faces of graph associahedra Example: bijection from nested sets to separable elements Let W be a Weyl group of type A 4 with simple roots α 1 , α 2 , α 3 , α 4 . The rank generating function for W is W ( q ) = [5]! q . N = {{ α 1 , α 2 , α 3 , α 4 } , { α 1 , α 2 } , { α 2 } , { α 4 }} . w ( N ) = 54321 · 32145 · 13245 · 12354 = 35412 . We see that 354 | 12 has a pivot at α 3 . And w ( N ) ( q ) = [5]! q [2]! q = q 7 + 2 q 6 + 3 q 5 + 4 q 4 + 4 q 3 + 3 q 2 + 2 q + 1 . Λ L [3]! q [2]! q N ′ = {{ α 1 , α 2 } , { α 2 } , { α 4 }} . w ( N ′ ) = 32145 · 13245 · 12354 = 31254 . And w ( N ′ ) ( q ) = [3]! q [2]! q = q 3 + 2 q 2 + 2 q + 1 . Λ L [2]! q Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 22 / 33

  66. Generalized quotients and splittings of Weyl groups Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 23 / 33

  67. Generalized quotients and splittings of Weyl groups Definition (Bj¨ orner and Wachs 1988) Given a subset U of a Weyl group W , the generalized quotient is W / U := { w ∈ W | ℓ ( wu ) = ℓ ( w ) + ℓ ( u ) , ∀ u ∈ U } . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 23 / 33

  68. Generalized quotients and splittings of Weyl groups Definition (Bj¨ orner and Wachs 1988) Given a subset U of a Weyl group W , the generalized quotient is W / U := { w ∈ W | ℓ ( wu ) = ℓ ( w ) + ℓ ( u ) , ∀ u ∈ U } . It generalizes parabolic quotients, since W J = W / W J . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 23 / 33

  69. Generalized quotients and splittings of Weyl groups Definition (Bj¨ orner and Wachs 1988) Given a subset U of a Weyl group W , the generalized quotient is W / U := { w ∈ W | ℓ ( wu ) = ℓ ( w ) + ℓ ( u ) , ∀ u ∈ U } . It generalizes parabolic quotients, since W J = W / W J . Proposition (Bj¨ orner and Wachs 1988) Let u 0 = � R u ∈ U u. Then W / U = [ e , w 0 u − 1 0 ] L . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 23 / 33

  70. Generalized quotients and splittings of Weyl groups Definition (Bj¨ orner and Wachs 1988) Given a subset U of a Weyl group W , the generalized quotient is W / U := { w ∈ W | ℓ ( wu ) = ℓ ( w ) + ℓ ( u ) , ∀ u ∈ U } . It generalizes parabolic quotients, since W J = W / W J . Proposition (Bj¨ orner and Wachs 1988) Let u 0 = � R u ∈ U u. Then W / U = [ e , w 0 u − 1 0 ] L . In finite Weyl groups, generalized quotients are just intervals in the left weak order. Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 23 / 33

  71. Generalized quotients and splittings of Weyl groups Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 24 / 33

  72. Generalized quotients and splittings of Weyl groups Definition (Bj¨ orner and Wachs 1988) A pair ( X , Y ) of arbitrary subsets X , Y ⊂ W such that the multiplication map X × Y → W sending ( x , y ) �→ xy is length-additive and bijective is called a splitting of W . Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 24 / 33

  73. Generalized quotients and splittings of Weyl groups Definition (Bj¨ orner and Wachs 1988) A pair ( X , Y ) of arbitrary subsets X , Y ⊂ W such that the multiplication map X × Y → W sending ( x , y ) �→ xy is length-additive and bijective is called a splitting of W . For example, ( W J , W J ) is a splitting of W , for any J ⊂ ∆. Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 24 / 33

  74. Generalized quotients and splittings of Weyl groups Definition (Bj¨ orner and Wachs 1988) A pair ( X , Y ) of arbitrary subsets X , Y ⊂ W such that the multiplication map X × Y → W sending ( x , y ) �→ xy is length-additive and bijective is called a splitting of W . For example, ( W J , W J ) is a splitting of W , for any J ⊂ ∆. Problem (Bj¨ orner and Wachs 1988) In the case W = S n , for which U ⊂ W is the multiplication map W / U × U → W a splitting of W ? Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 24 / 33

  75. Generalized quotients and splittings of Weyl groups Definition (Bj¨ orner and Wachs 1988) A pair ( X , Y ) of arbitrary subsets X , Y ⊂ W such that the multiplication map X × Y → W sending ( x , y ) �→ xy is length-additive and bijective is called a splitting of W . For example, ( W J , W J ) is a splitting of W , for any J ⊂ ∆. Problem (Bj¨ orner and Wachs 1988) In the case W = S n , for which U ⊂ W is the multiplication map W / U × U → W a splitting of W ? This map is length-additive by definition. So the problem is asking for which U is this map a bijection? Yibo Gao (MIT) Separable elements in Weyl groups Feb 21, 2020 24 / 33

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