Staircase diagrams and the enumeration of smooth Schubert varieties - PowerPoint PPT Presentation

Staircase diagrams and the enumeration of smooth Schubert varieties Edward Richmond* and William Slofstra Oklahoma State University* University of Waterloo July 4, 2016 Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 1 / 16

Let Γ be a Dynkin diagram of finite type with vertex set S = { s 1 , . . . , s n } . Let D be a collection of subsets of S . Type A Dynkin diagram: s 1 s 2 s n − 1 s n Type A example: D = { [ s 1 , s 3 ] , [ s 2 , s 4 ] , [ s 3 , s 5 ] , [ s 6 ] , [ s 7 , s 9 ] , [ s 9 , s 10 ] , [ s 10 , s 11 ] } 3 4 5 10 11 2 3 4 6 9 10 1 2 3 7 8 9 For any s ∈ S , define D s := { B ∈ D | s ∈ B } . D s 3 = { [ s 1 , s 3 ] , [ s 2 , s 4 ] , [ s 3 , s 5 ] } Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 2 / 16

Definition: We say a partially ordered set ( D , ≺ ) is a staircase diagram over Γ if Each B ∈ D is connected. If B covers B ′ , then B ∪ B ′ is connected. For each s ∈ S , D s is a saturated chain. If s adj t , then D s ∪ D t is a chain. Each is B ∈ D is maximal (resp. minimal) in D s for some s ∈ S . Type A example: 3 4 5 10 11 2 3 4 6 9 10 1 2 3 7 8 9 { [ s 1 , s 3 ] ≺ [ s 2 , s 4 ] ≺ [ s 3 , s 5 ] ≻ [ s 6 ] ≻ [ s 7 , s 9 ] ≺ [ s 9 , s 10 ] ≺ [ s 10 , s 11 ] } Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 3 / 16

Type D examples: s 1 s 2 s 3 s 4 s 5 2 3 3 4 5 2 31 4 31 2 3 4 5 { [ s 1 , s 3 ] ≺ [ s 3 , s 5 ] ≺ [ s 2 , s 3 ] } { [ s 2 , s 5 ] ≺ ([ s 2 , s 4 ] ∪ { s 1 } ) } Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 4 / 16

Non-example 1: 2 3 4 0 1 2 3 4 5 Violates: If s adj t , then D s ∪ D t is a chain. Each is B ∈ D is maximal (resp. minimal) in D s for some s ∈ S . Non-example 2: 2 3 4 5 31 4 Violates: For each s ∈ S , D s is a saturated chain. Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 5 / 16

Let G be a finite-type Lie group with Weyl group ( W, S ) and Dynkin diagram Γ . For any J ⊆ S , let u J denote the maximal element in W J . Let D be a staircase diagram on Γ . For any B ∈ D , define J ( B ) := { s ∈ B | B � = min D s } Example: 2 3 4 5 6 1 2 3 6 7 J ([ s 2 , s 6 ]) = { s 2 , s 3 , s 6 } For each B ∈ D , define λ ( B ) := u B u J ( B ) ∈ W. Remark: λ ( B ) is the maximal element of W B ∩ W J ( B ) Remark: The map λ : D → W is called the maximal labelling of D . Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 6 / 16

Definition: Let ( B 1 < B 2 < · · · < B n ) be a linear extension of D . Define Λ( D ) := λ ( B n ) · λ ( B n − 1 ) · · · λ ( B 1 ) . If B, B ′ are incomparable, then they are disjoint and non-adjacent. Thus λ ( B ) , λ ( B ′ ) commute and hence Λ( D ) is well defined. Example: Let D = { [ s 1 , s 3 ] , [ s 5 , s 6 ] , [ s 2 , s 5 ] } . 2 3 4 5 1 2 3 5 6 Then λ ([ s 1 , s 3 ]) = s 1 s 2 s 3 s 1 s 2 s 1 , λ ([ s 5 , s 6 ]) = s 5 s 6 s 5 , λ ([ s 2 , s 5 ]) = ( s 3 s 2 s 4 s 3 s 5 s 4 s 5 s 2 s 3 s 2 )( s 2 s 3 s 2 s 5 ) = s 3 s 2 s 4 s 3 s 5 s 4 and Λ( D ) = ( s 3 s 2 s 4 s 3 s 5 s 4 )( s 5 s 6 s 5 )( s 1 s 2 s 3 s 1 s 2 s 1 ) . Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 7 / 16

Define D R ( D ) := { s ∈ S | s is not a “lower inner corner” of D} . Ex: 2 3 4 5 6 1 2 7 D R ( D ) = { s 1 , s 2 , s 4 , s 5 , s 7 } . Let flip( D ) denote the staircase diagram D with the reserve partial order. Ex: 2 3 4 5 1 2 3 6 1 2 3 6 2 3 4 5 Coxeter group properties of Λ( D ) : R-Slofstra (arXiv15) ℓ (Λ( D )) = ℓ ( λ ( B 1 )) + · · · + ℓ ( λ ( B n )) D R ( D ) is the right-decent set of Λ( D ) Λ( D ) − 1 = Λ(flip( D )) Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 8 / 16

Connection with geometry: Let G be a finite group with Weyl group W and let X ( w ) ⊆ G/B denote the Schubert variety indexed by w ∈ W . Let Γ denote the Dynkin diagram of G . Theorem: R-Slofstra (arXiv15) If G is a simply-laced, then the map D �→ X (Λ( D )) defines a bijection: � � � � ⇒ staircase diagrams over Γ smooth Schubert varieties in G/B If λ : D → W is a (rationally) smooth labelling, then define Λ( D , λ ) ∈ W accordingly. Theorem: R-Slofstra (arXiv15) If G is of finite type, then the map D �→ X (Λ( D , λ )) defines a bijection: � � � � staircase diagrams over Γ (rationally) smooth Schubert ⇒ with (rationally) smooth labellings varieties in G/B Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 9 / 16

Theorem: Ryan (87), Wolper (89), R-Slofstra (arVix14) (Rationally) smooth Schubert varieties are iterated fiber bundles of (rationally) smooth “Grassmannian Schubert varieties”. (Rationally) smooth Grassmannian Schubert varieties are classified. Let P ⊆ G and consider the fibration P/B ֒ → G/B ։ G/P. The labelling map Λ( D ) = λ ( B n ) · Λ( D \ { B n } ) corresponds to a fibration of Schubert varieties → X (Λ( D )) ։ X P ( λ ( B n )) X (Λ( D \ { B n } )) ֒ Where the parabolic P is defined by the support of D \ { B n } in S . Example: 2 3 4 5 1 2 3 5 6 The support of D \ { B 3 } is { s 1 , s 2 , s 3 } ⊔ { s 5 , s 6 } . Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 10 / 16

Application to enumeration: Define generating series ∞ ∞ ∞ � � � a n t n , b n t n , c n t n , A ( t ) := B ( t ) := C ( t ) := n =0 n =0 n =0 ∞ ∞ � � d n t n , bc n t n , D ( t ) := BC ( t ) := n =3 n =0 where the coefficients a n , b n , c n , d n denote the number of smooth Schubert varieties of types A n , B n , C n , D n respectively, and bc n denotes the number of rationally smooth Schubert varieties of type B n or C n . Theorem: Haiman (90s), Bona (98), R-Slofstra (arXiv15) Let W ( t ) := � w n t n where W = A , B , C , D , or BC . Then P W ( t ) + Q W ( t ) √ 1 − 4 t W ( t ) = (1 − t ) 2 (1 − 6 t + 8 t 2 − 4 t 3 ) for some polynomials P W ( t ) and Q W ( t ) . Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 11 / 16

Theorem: Haiman (90s), Bona (98), R-Slofstra (arXiv15) P W ( t ) Q W ( t ) Type (1 − 4 t )(1 − t ) 3 t (1 − t ) 2 A (1 − 5 t + 5 t 2 )(1 − t ) 3 (2 t − t 2 )(1 − t ) 3 B 1 − 7 t + 15 t 2 − 11 t 3 − 2 t 4 + 5 t 5 t − t 2 − t 3 + 3 t 4 − t 5 C ( − 4 t + 19 t 2 + 8 t 3 − 30 t 4 + 16 t 5 )(1 − t ) 2 (4 t − 15 t 2 + 11 t 3 − 2 t 5 )(1 − t ) D 1 − 8 t + 23 t 2 − 29 t 3 + 14 t 4 2 t − 6 t 2 + 7 t 3 − 2 t 4 BC a n b n c n d n bc n n = 1 2 2 2 2 n = 2 6 7 7 8 n = 3 22 28 28 22 34 n = 4 88 116 114 108 142 n = 5 366 490 472 490 596 n = 6 1552 2094 1988 2164 2530 n = 7 6652 9014 8480 9474 10842 n = 8 28696 38988 36474 41374 46766 Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 12 / 16

Asymptotics: The smallest singularity of W ( t ) is the root √ √ α := 1 � � � � 3 3 4 − 17 + 3 33 + − 17 + 3 33 ≈ 0 . 228155 6 of the polynomial 1 − 6 t + 8 t 2 − 4 t 3 appearing in the denominator. Corollary: R-Slofstra (arXiv15) Let W ( t ) = � w n t n , where W = A , B , C , D , or BC . Then w n ∼ W α α n +1 , where W α := lim t → α ( α − t ) W ( t ) . In particular, w n +1 = α − 1 ≈ 4 . 382985 lim w n n →∞ A B C D BC W α ≈ 0 . 045352 0 . 062022 0 . 057301 0 . 067269 0 . 073972 Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 13 / 16

Proof of enumeration: We say a staircase diagram D is elementary if: The support of D is connected. If |D s | = 1 , then s is a leaf of the support. Examples: 6 7 8 5 6 7 3 4 2 3 4 5 4 5 31 4 5 5 6 Step 1: Decompose a staircase diagram into elementary diagrams. 5 6 7 5 6 6 7 1 2 1 2 4 5 7 8 → 4 5 7 8 2 3 2 3 4 8 9 3 4 8 9 Step 2: Count elementary diagrams. Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 14 / 16

Key observation: Elementary diagrams “grow” recursively at the rate of Catalan numbers! 2 3 1 2 3 4 2 3 4 2 3 1 2 3 1 2 4 5 3 4 5 4 5 3 4 5 2 3 4 5 3 4 2 3 4 2 3 4 2 3 4 1 2 3 4 2 3 1 2 3 1 2 3 1 2 1 2 1 � 2 n � Step 3: Use the generating series for Catalan numbers. Let c n := and n +1 n c n t n = 1 − √ 1 − 4 t ∞ � Cat ( t ) := . 2 t n =0 Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 15 / 16

Further directions: Analogous enumerative results hold for affine type A (R-Slofstra, in progress). Example: 0 5 6 7 0 0 1 2 7 0 1 2 3 4 5 What about other affine classical Lie types? Kac-Moody types? Find a generating series for the number of staircase diagrams over the Dynkin diagrams of E n . Thanks! Richmond-Slofstra (OSU-Waterloo) Staircase diagrams July 4, 2016 16 / 16