separable elements and splittings of weyl groups
play

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

Aug 16, 2023 •7 likes •1.3k views

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

Recommend

Separable elements in Weyl groups Yibo Gao Joint work with: Christian Gaetz Massachusetts

Separable elements in Weyl groups Yibo Gao Joint work with: Christian Gaetz Massachusetts

Separable elements in Weyl groups Yibo Gao Joint work with: Christian Gaetz Massachusetts Institute of Technology Summer Combo in Vermont, 2019 Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 1 / 23 Overview Background

1.18k views • 89 slides
Splittings and cross-sections in topological groups Madrid, December 3-4, 2015 Hugo J. Bello,

Splittings and cross-sections in topological groups Madrid, December 3-4, 2015 Hugo J. Bello,

Preliminaries Extensions of topological abelian groups Free abelian topological groups Main Result Splittings and cross-sections in topological groups Madrid, December 3-4, 2015 Hugo J. Bello, University of Navarra (Based on a joint work with

303 views • 26 slides
Models and refined models for involutory reflection groups and classical Weyl groups FABRIZIO

Models and refined models for involutory reflection groups and classical Weyl groups FABRIZIO

Models and refined models for involutory reflection groups and classical Weyl groups Models and refined models for involutory reflection groups and classical Weyl groups FABRIZIO CASELLI AND ROBERTA FULCI FPSAC10 - San Francisco Models and

1.25k views • 91 slides
GAGTA7 Conference Dynamics for the splittings of free-by-cyclic groups Ilya Kapovich University

GAGTA7 Conference Dynamics for the splittings of free-by-cyclic groups Ilya Kapovich University

GAGTA7 Conference Dynamics for the splittings of free-by-cyclic groups Ilya Kapovich University of Illinois at Urbana-Champaign Based on joint work with Spencer Dowdall and Chris Leininger arXiv:1301.7739 GAGTA7 Conference, New York; May,

1.53k views • 135 slides
Permutations and Weyl Groups: Seeing Irreducibility in Cycle Structures Noah Hughes Appalachian

Permutations and Weyl Groups: Seeing Irreducibility in Cycle Structures Noah Hughes Appalachian

Permutations and Weyl Groups: Seeing Irreducibility in Cycle Structures Noah Hughes Appalachian State University hughesna@appstate.edu Saturday, October 5, 2013 Outline: * A brief introduction to finite simple Lie Algebras * Root Systems,

200 views • 18 slides
Weyl algebra S p g q7 S p g q Deformation of Weyl algebra Problems with Weyl algebra

Weyl algebra S p g q7 S p g q Deformation of Weyl algebra Problems with Weyl algebra

C OMPLETED H OPF ALGEBROIDS D OCTORAL PRESENTATION Martina Stoji c Faculty of Science Department of Mathematics October 20, 2017 I NTRODUCTION T HE CATEGORY indproVect H OPF ALGEBROID AND SCALAR EXTENSION H EISENBERG DOUBLES E XAMPLES 1. I

432 views • 39 slides
Adopting Sample Ballot Design Elements Orange County adopted many of the main design elements

Adopting Sample Ballot Design Elements Orange County adopted many of the main design elements

Adopting Sample Ballot Design Elements Orange County adopted many of the main design elements Before After Make it Work for Your Jurisdiction Input from community groups (advisory groups, language communities, etc) Allow groups

263 views • 3 slides
Chapter 15 - Chapter 15 - The Elements: he Elements: The Last Four Main Groups The Last Four

Chapter 15 - Chapter 15 - The Elements: he Elements: The Last Four Main Groups The Last Four

Chapter 15 - Chapter 15 - The Elements: he Elements: The Last Four Main Groups The Last Four Main Groups Group 15/V: The Nitrogen Family Group 16/VI: The Oxygen Family Group 17/VII: The Halogens Group 18/VIII: The Nobel gases

941 views • 58 slides
Fully Commutative Elements in Coxeter Groups Philippe Nadeau (CNRS &amp; Universit e Lyon 1)

Fully Commutative Elements in Coxeter Groups Philippe Nadeau (CNRS &amp; Universit e Lyon 1)

Fully Commutative Elements in Coxeter Groups Philippe Nadeau (CNRS &amp; Universit e Lyon 1) Jiao Tong University, Shanghai, October 29th Organization I. Coxeter groups II. Fully commutative elements and Heaps III. FC elements in type A

803 views • 65 slides
Sparse Separable Nonnegative Matrix Factorization Extending Separable NMF with 0 sparsity

Sparse Separable Nonnegative Matrix Factorization Extending Separable NMF with 0 sparsity

Sparse Separable Nonnegative Matrix Factorization Extending Separable NMF with 0 sparsity constraints Nicolas Nadisic, Arnaud Vandaele, Jeremy Cohen, Nicolas Gillis 9 October 2020 GdR MIA Thematic Day Universit e de Mons, Belgium

1.03k views • 44 slides
Generalized Weyl algebras and their global dimension V. V. Bavula 1 Generalized Weyl algebras

Generalized Weyl algebras and their global dimension V. V. Bavula 1 Generalized Weyl algebras

Generalized Weyl algebras and their global dimension V. V. Bavula 1 Generalized Weyl algebras Definition of the generalized Weyl algebras . Let D be a ring, = ( 1 , ..., n ) a set of commuting automorphisms of D , ( i j = j i

295 views • 11 slides
Derangements and p -elements in permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk

Derangements and p -elements in permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk

Derangements and p -elements in permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk Groups and their ApplicationsManchester, 14 February 2007 uous closed curve in S which cannot be lifted In the beginning . . . A derangement is a

231 views • 5 slides
The nilpotency of elements of the stable homotopy groups of spheres Richard Wong eCHT Kan

The nilpotency of elements of the stable homotopy groups of spheres Richard Wong eCHT Kan

Introduction Nilpotence for elements of order p . Nilpotence for all elements. The nilpotency of elements of the stable homotopy groups of spheres Richard Wong eCHT Kan seminar Fall 2019 Richard Wong University of Texas at Austin The

601 views • 42 slides
Asymptotic density of test elements in free groups and surface groups Ilir Snopche Universidade

Asymptotic density of test elements in free groups and surface groups Ilir Snopche Universidade

Asymptotic density of test elements in free groups and surface groups Ilir Snopche Universidade Federal do Rio de Janeiro Joint work with S. Tanushevski Groups St Andrews 2017 in Birmingham August, 2017 Ilir Snopche Asymptotic density of

829 views • 44 slides
Derangements and p -elements in permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk

Derangements and p -elements in permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk

Derangements and p -elements in permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk Groups and their Applications Manchester, 14 February 2007 In the beginning . . . A derangement is a permutation with no fixed points. In the

1.14k views • 77 slides
Simpson correspondence in characteristic p &gt; 0 and splittings of the algebra of PD-differential

Simpson correspondence in characteristic p &gt; 0 and splittings of the algebra of PD-differential

Simpson correspondence in characteristic p &gt; 0 and splittings of the algebra of PD-differential operators Michel Gros (CNRS &amp;University of Rennes I) Simons Symposium, Schloss Elmau, 7-13 May 2017 S a flat

558 views • 30 slides
Three-body approach to d + scattering and bound state using realistic forces in a separable or

Three-body approach to d + scattering and bound state using realistic forces in a separable or

6 Li bound state and d + scattering Motivation EST separable expansion Faddeev-AGS equations Summary and outlook Three-body approach to d + scattering and bound state using realistic forces in a separable or non-separable representation

618 views • 34 slides
Parallel computations of Grbner bases in the Weyl algebra Something to run on a machine with

Parallel computations of Grbner bases in the Weyl algebra Something to run on a machine with

Weyl algebra Grbner basis Faugres F 4 algorithm Parallel computations of Grbner bases in the Weyl algebra Something to run on a machine with 128 cores Anton Leykin Institute for Mathematics and its Applications, Minneapolis MSRI,

813 views • 62 slides
40 years of the Weyl anomaly M. J. Duff Physics Department Imperial College London CFT Beyond

40 years of the Weyl anomaly M. J. Duff Physics Department Imperial College London CFT Beyond

40 years of the Weyl anomaly M. J. Duff Physics Department Imperial College London CFT Beyond Two Dimensions Texas A&amp;M March 2012 1 / 50 Abstract Classically, Weyl invariance S ( g , ) = S ( g , ) under = ( x )

881 views • 50 slides
The Superconformal Index and the Weyl Anomaly Great Lakes Strings 2018, University of Chicago

The Superconformal Index and the Weyl Anomaly Great Lakes Strings 2018, University of Chicago

The Superconformal Index and the Weyl Anomaly Great Lakes Strings 2018, University of Chicago Brian McPeak Leinweber Center for Theoretical Physics University of Michigan Based on work presented in 1804.04155 Conformal Symmetry and Weyl

387 views • 10 slides
Random elements of large groups Continuous case Viktor Kiss Etvs Lornd University

Random elements of large groups Continuous case Viktor Kiss Etvs Lornd University

Random elements of large groups Continuous case Viktor Kiss Etvs Lornd University Toposym, July 25, 2016 Joint work with Udayan B. Darji, Mrton Elekes, Kende Kalina, Zoltn Vidnynszky Viktor Kiss Random elements of large

714 views • 35 slides
On Stein-Rosenberg type theorems for nonnegative splittings Dimitrios Noutsos

On Stein-Rosenberg type theorems for nonnegative splittings Dimitrios Noutsos

On Stein-Rosenberg type theorems for nonnegative splittings Dimitrios Noutsos dnoutsos@cc.uoi.gr Department of Mathematics, University of Ioannina, GREECE Computational methods with Applications, Harrachov-2007 Czech Republic, August 1925,

1.34k views • 24 slides
On Reeb graphs derived from Heegaard splittings with distance 2 g (

On Reeb graphs derived from Heegaard splittings with distance 2 g (

On Reeb graphs derived from Heegaard splittings with distance 2 g ( D1) 20 / 12 / 2010 @ 1. Preliminaries M : a closed orientable 3-manifold Def. A P B : a (genus g ) Heegaard

380 views • 17 slides
Diffeomorphisms and Heegaard splittings of 3-manifolds Hyamfest Melbourne, July 2011

Diffeomorphisms and Heegaard splittings of 3-manifolds Hyamfest Melbourne, July 2011

Diffeomorphisms and Heegaard splittings of 3-manifolds Hyamfest Melbourne, July 2011 Some philosophy Adding geometric structure tends to restrict automor- phisms.

691 views • 20 slides

More recommend