The weak Bruhat order on the symmetric group is Sperner Yibo Gao Joint work with: Christian Gaetz Massachusetts Institute of Technology FPSAC 2019 Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 1 / 21
Overview The Sperner property of weak Bruhat order 1 The Sperner property of Posets An sl 2 -action on the weak Bruhat order of S n Open problems Further work related to the code weights 2 A determinant formula by Hamaker, Pechenik, Speyer and Weigandt Padded Schubert polynomials Weighted enumeration of chains in the (strong) Bruhat order Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 2 / 21
The Sperner property Let P be a ranked poset with rank decomposition P 0 ⊔ P 1 ⊔ · · · ⊔ P r . Definition P is called k - Sperner if no union of its k antichains is larger than the union of its largest k ranks. P is called Sperner if it is 1-Sperner. P is called strongly Sperner if it is k -Sperner for any k ∈ Z ≥ 1 . • • • • • � • � • • • � • � • • • • Figure: A Sperner poset (left) and a non-Sperner poset (right) Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 3 / 21
The Sperner property Further assume that P = P 0 ⊔ · · · ⊔ P r is rank symmetric: | P i | = | P r − i | for all i , rank unimodal: there exists m such that | P 0 | ≤ | P 1 | ≤ · · · ≤ | P m | ≥ · · · ≥ | P r − 1 | ≥ | P r | . Definition An order lowering operator is a linear map D : C P → C P such that � D · x = wt ( y , x ) · y , x ∈ P i . y ⋖ x • x 1 2 y • • z Dx = y + 2 z 2 1 • • 1 2 • Figure: An example of an order lowering operator. Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 4 / 21
The Sperner property (via linear isomorphism) Recall P = P 0 ⊔ · · · ⊔ P r is rank symmetric and rank unimodal. Lemma (Stanley 1980) If there exists an order lowering operator D such that D r − 2 i : C P r − i → C P i is an isomorphism for any 0 ≤ i ≤ ⌊ r / 2 ⌋ , then P is strongly Sperner. Together with the hard Lefschetz theorem in algebraic geometry, Stanley proved the following: Theorem (Stanley 1980) Let ( W , S ) be a Coxeter system for which W is a Weyl group. Then the (strong) Bruhat order on W or any parabolic quotient W J is rank symmetric, rank unimodal and strongly Sperner. Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 5 / 21
The Sperner property (via sl 2 representations) Definition An sl 2 representation on P consists of the following data: an order lowering operator D : C P i → C P i − 1 , ∀ i , a raising operator U : C P i → C P i +1 , ∀ i , ( U doesn’t need to respect the order) a modified rank function H : C P i → C P i , x �→ (2 i − r ) x , such that UD − DU = H . In fact, U , D , H make C P an sl 2 representation. Theorem (Proctor 1982) A ranked poset P admits an sl 2 representation if and only if P is rank symmetric, rank unimodal and strongly Sperner. Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 6 / 21
The weak and strong Bruhat orders (on S n ) For w ∈ S n , let ℓ ( w ) denote the usual Coxeter length. The (right) weak (Bruhat) order W n is generated by w ⋖ W ws i if ℓ ( ws i ) = ℓ ( w ) + 1 , where s i = ( i , i + 1) . The (strong) Bruhat order S n is generated by w ⋖ S wt ij if ℓ ( wt ij ) = ℓ ( w ) + 1 , where t ij = ( i , j ) . • 321 • 321 • • 312 • • 312 231 231 • • 132 • • 132 213 213 • 123 • 123 Figure: The weak and strong order on S 3 . Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 7 / 21
The weak and strong Bruhat orders (on S n ) Stanley (1980) showed that the strong Bruhat order (on any Weyl group) is strongly Sperner, and has a symmetric chain decomposition for types A n , B n , D n . Bj¨ orner (1984) conjectured that the weak Bruhat order is strongly Sperner. Stanley (2017) suggested an order lowering operator � D · w = i · ( ws i ) . ℓ ( ws i )= ℓ ( w ) − 1 Conjecture (Stanley 2017) For D defined as above, D ( n 2 ) − 2 i : C ( W n )( n 2 ) − i → C ( W n ) i has nonzero � n � determinant for 0 ≤ i ≤ / 2 . Thus, the weak Bruhat order W n is 2 strongly Sperner. Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 8 / 21
An sl 2 action on the weak Bruhat order W n Proposition (Gaetz and G. 2018) The following data give an sl 2 action on W n : the order lowering operator suggested by Stanley � D · w = i · ( ws i ) , ℓ ( ws i )= ℓ ( w ) − 1 a raising operator defined by � U · w = || code ( w ) − code ( u ) || L 1 · u , w ⋖ S u � � n �� H · w = 2 ℓ ( w ) − · w . 2 Recall code ( w ) i = { j > i : w ( j ) < w ( i ) } . Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 9 / 21
An sl 2 action on the weak Bruhat order W n • (2,1) • 321 1 1 1 2 • • (2,0) • • 312 (1,1) 231 1 1 3 2 1 • • (0,1) • • 132 (1,0) 213 1 1 1 2 • (0,0) • 123 Figure: The order lowering operator D and the raising operator U The (unique) raising operator U that corresponds to D doesn’t need to be supported on the strong order. It’s just nice combinatorics. Corollary (Gaetz and G. 2018) The weak order W n on the symmetric group is strongly Sperner. Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 10 / 21
Open Problems Conjecture The weak Bruhat order is strongly Sperner for any Coxeter group. Conjecture The weak Bruhat order of type A has a symmetric chain decomposition. • 321 • • 312 231 • • 132 213 • 123 Example (Leclerc 1994) The weak order of H 3 doesn’t have a symmetric chain decomposition, but is strongly Sperner. Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 11 / 21
Formulas by Hamaker, Pechenik, Speyer and Weigandt Hamaker, Pechenik, Speyer and Weigandt resolved the full determinant conjecture by Stanley. Theorem (Hamaker et al. 2018, conjectured by Stanley 2017) �� n � #( W n ) i � �� n � � k − 1 � − k − i det D ( n 2 ) − 2 k = ! #( W n ) k 2 − k 2 k − i i =0 Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 12 / 21
Formulas by Hamaker, Pechenik, Speyer and Weigandt Definition (Schubert Polynomials) The Schubert Polynomials S w , for w ∈ S n , can be defined as follows: S w 0 = x n − 1 x n − 2 · · · x n − 1 , 1 2 S w = ∂ i S ws i if ℓ ( w ) = ℓ ( ws i ) − 1, where ∂ i f = ( f − s i f ) / ( x i − x i +1 ) is the i th divided difference operator. Proposition (Hamaker et al. 2018) Let ∇ = � i ∂/∂ x i . Then � ∇ S w − 1 = i · S s i w − 1 . i : ℓ ( w )= ℓ ( ws i )+1 Corollary (Macdonald’s Identity) � n � � a 1 · · · a N = ! . 2 reduced s a 1 ··· s aN = w 0 Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 13 / 21
Padded Schubert Polynomials Recall that { S w } w ∈ S n form a basis of span C { x α : α ≤ ρ } where ρ = ( n − 1 , . . . , 1) is the staircase partition. Definition (Gaetz and G. 2018) The padded Schubert polynomial � S w is the image of S w under x α �→ x α y ρ − α . Define the following linear operators n − 1 n − 1 � � ∂ ∂ ∇ = y i , ∆ = x i . ∂ x i ∂ y i i =1 i =1 Proposition (Hamaker et al. 2018; Gaetz and G. 2018) S w − 1 = � 1 ∇ � i : ℓ ( w )= ℓ ( ws i )+1 i · � S s i w − 1 . S w − 1 = � 2 ∆ � u : u ≥ S w || code ( u ) − code ( w ) || L 1 · � S u − 1 . Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 14 / 21
Padded Schubert polynomials • x 2 • x 2 1 x 2 1 x 2 • • x 2 • • x 2 x 1 y 1 x 2 1 y 2 x 1 x 2 1 • • x 1 y 1 y 2 + y 2 • • x 1 + x 2 x 1 y 1 y 2 x 1 1 x 2 • y 2 • 1 1 y 2 Figure: Schubert polynomials and padded Schubert polynomials on S 3 We see that �� ∂ � ( x 1 y 1 y 2 + y 2 1 x 2 ) = 3 x 1 y 1 x 2 + x 2 x i 1 y 2 . ∂ y i Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 15 / 21
Weights on the strong Bruhat order w ( i ) w ( j ) u ( j ) u ( i ) • • A A • • i ⋖ • ∅ ∅ D B D B • • • j C C • Figure: Weights on the strong Bruhat order Let a w ⋖ u = { k < i : w ( i ) < w ( k ) < w ( j ) } and similarly define b w ⋖ u , c w ⋖ u and d w ⋖ u . For example, when w = 4127653, u = 4157623, a w ⋖ u = 1 , b w ⋖ u = 2 , c w ⋖ u = 1 , d w ⋖ u = 0 . Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 16 / 21
Weighted enumeration of maximal chains If wt : E → R is a weight function on covering relations, where R is a commutative ring, we can define, for x ≤ y , � � m wt ( x , y ) = wt ( e ) . C : x → y e ∈ C maximal chain Theorem (Gaetz and G. 2019) Let z A , z B , z C , z D be indeterminates and define a weight function on the covering relations on the strong Bruhat order of S n as follows: wt ( w ⋖ u ) = 1 + z A a w ⋖ u + z B b w ⋖ u + z C c w ⋖ u + z D d w ⋖ u . Then if { z A , z B , z C , z D } = { 0 , 0 , z , 2 − z } as multisets, � n � m wt ( id , w 0 ) = ! . 2 Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 17 / 21
Weighted enumeration of maximal chains Let wt ( w ⋖ u ) = 1 + z A a w ⋖ u + z B b w ⋖ u + z C c w ⋖ u + z D d w ⋖ u . • 321 1 1 • • 312 231 1 + z B 1 + z A 1 + z C 1 + z D • • 132 213 1 1 • 123 Figure: Weights on covering relations of S 3 Then m wt (123 , 321) = 4 + z A + z B + z C + z D , which is 6 = 3! if { z A , z B , z C , z D } = { 0 , 0 , z , 2 − z } . Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 18 / 21
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