[PPT] - The weak Bruhat order on the symmetric group is Sperner Yibo Gao PowerPoint Presentation

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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

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Overview

1

The Sperner property of weak Bruhat order The Sperner property of Posets An sl2-action on the weak Bruhat order of Sn Open problems

2

Further work related to the code weights 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

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The Sperner property

Let P be a ranked poset with rank decomposition P0 ⊔ P1 ⊔ · · · ⊔ Pr.

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

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The Sperner property

Further assume that P = P0 ⊔ · · · ⊔ Pr is rank symmetric: |Pi| = |Pr−i| for all i, rank unimodal: there exists m such that |P0| ≤ |P1| ≤ · · · ≤ |Pm| ≥ · · · ≥ |Pr−1| ≥ |Pr|.

Definition

An order lowering operator is a linear map D : CP → CP such that D · x =

y⋖x

wt(y, x) · y, x ∈ Pi.

x
y
z

1 2 1 2 1 2

Dx = y + 2z

Figure: An example of an order lowering operator.

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 4 / 21

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The Sperner property (via linear isomorphism)

Recall P = P0 ⊔ · · · ⊔ Pr is rank symmetric and rank unimodal.

Lemma (Stanley 1980)

If there exists an order lowering operator D such that Dr−2i : CPr−i → CPi 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

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The Sperner property (via sl2 representations)

Definition

An sl2 representation on P consists of the following data: an order lowering operator D : CPi → CPi−1, ∀i, a raising operator U : CPi → CPi+1, ∀i, (U doesn’t need to respect the order) a modified rank function H : CPi → CPi, x → (2i − r)x, such that UD − DU = H. In fact, U, D, H make CP an sl2 representation.

Theorem (Proctor 1982)

A ranked poset P admits an sl2 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

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The weak and strong Bruhat orders (on Sn)

For w ∈ Sn, let ℓ(w) denote the usual Coxeter length. The (right) weak (Bruhat) order Wn is generated by w ⋖W wsi if ℓ(wsi) = ℓ(w) + 1, where si = (i, i + 1). The (strong) Bruhat order Sn is generated by w ⋖S wtij if ℓ(wtij) = ℓ(w) + 1, where tij = (i, j).

123
213
231
321
132
312
123
213
231
321
132
312

Figure: The weak and strong order on S3.

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 7 / 21

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The weak and strong Bruhat orders (on Sn)

Stanley (1980) showed that the strong Bruhat order (on any Weyl group) is strongly Sperner, and has a symmetric chain decomposition for types An, Bn, Dn. Bj¨

rner (1984) conjectured that the weak Bruhat order is strongly Sperner.

Stanley (2017) suggested an order lowering operator D · w =

ℓ(wsi)=ℓ(w)−1

i · (wsi).

Conjecture (Stanley 2017)

For D defined as above, D(n

2)−2i : C(Wn)(n 2)−i → C(Wn)i has nonzero

determinant for 0 ≤ i ≤ n

2

/2. Thus, the weak Bruhat order Wn is

strongly Sperner.

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 8 / 21

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An sl2 action on the weak Bruhat order Wn

Proposition (Gaetz and G. 2018)

The following data give an sl2 action on Wn: the order lowering operator suggested by Stanley D · w =

ℓ(wsi)=ℓ(w)−1

i · (wsi), a raising operator defined by U · w =

w⋖Su

||code(w) − code(u)||L1 · u, H · w =

2ℓ(w) −

n

2

· w.

Recall code(w)i = {j > i : w(j) < w(i)}.

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 9 / 21

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An sl2 action on the weak Bruhat order Wn

123
213
231
321
132
312

1 2 1 2 1 2

(0,0)
(1,0)
(1,1)
(2,1)
(0,1)
(2,0)

1 1 1 1 3 1 1

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 Wn on the symmetric group is strongly Sperner.

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 10 / 21

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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.

123
213
231
321
132
312

Example (Leclerc 1994)

The weak order of H3 doesn’t have a symmetric chain decomposition, but is strongly Sperner.

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 11 / 21

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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)

det D(n

2)−2k =

n 2

− k
!#(Wn)k

k−1

i=0

n

2

− k − i

k − i #(Wn)i

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 12 / 21

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Formulas by Hamaker, Pechenik, Speyer and Weigandt

Definition (Schubert Polynomials)

The Schubert Polynomials Sw, for w ∈ Sn, can be defined as follows: Sw0 = xn−1

1

xn−2

2

· · · xn−1, Sw = ∂iSwsi if ℓ(w) = ℓ(wsi) − 1, where ∂if = (f − sif )/(xi − xi+1) is the ith divided difference operator.

Proposition (Hamaker et al. 2018)

Let ∇ =

i ∂/∂xi. Then

∇Sw−1 =

i: ℓ(w)=ℓ(wsi)+1

i · Ssiw−1.

Corollary (Macdonald’s Identity)

reduced sa1···saN =w0

a1 · · · aN = n 2

!.

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 13 / 21

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Padded Schubert Polynomials

Recall that {Sw}w∈Sn form a basis of spanC{xα : α ≤ ρ} where ρ = (n − 1, . . . , 1) is the staircase partition.

Definition (Gaetz and G. 2018)

The padded Schubert polynomial Sw is the image of Sw under xα → xαyρ−α. Define the following linear operators ∇ =

n−1

i=1

∂ ∂xi yi, ∆ =

n−1

i=1

∂ ∂yi xi.

Proposition (Hamaker et al. 2018; Gaetz and G. 2018)

1 ∇

Sw−1 =

i: ℓ(w)=ℓ(wsi)+1 i ·

Ssiw−1.

2 ∆

Sw−1 =

u: u≥Sw ||code(u) − code(w)||L1 ·

Su−1.

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 14 / 21

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Padded Schubert polynomials

1
x1
x1x2
x2

1x2

x1 + x2
x2

1

y 2

1 y2

x1y1y2
x1y1x2
x2

1x2

x1y1y2 + y 2

1 x2

x2

1y2

Figure: Schubert polynomials and padded Schubert polynomials on S3

We see that ∂ ∂yi xi

(x1y1y2 + y2

1 x2) = 3x1y1x2 + x2 1y2.

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 15 / 21

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Weights on the strong Bruhat order

⋖

i

j w(i) w(j) u(j) u(i)

∅ ∅

A B C D A B C D

Figure: Weights on the strong Bruhat order

Let aw⋖u = {k < i : w(i) < w(k) < w(j)} and similarly define bw⋖u, cw⋖u and dw⋖u. For example, when w = 4127653, u = 4157623, aw⋖u = 1, bw⋖u = 2, cw⋖u = 1, dw⋖u = 0.

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 16 / 21

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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, mwt(x, y) =

C:x→y

maximal chain

e∈C

wt(e).

Theorem (Gaetz and G. 2019)

Let zA, zB, zC, zD be indeterminates and define a weight function on the covering relations on the strong Bruhat order of Sn as follows: wt(w ⋖ u) = 1 + zAaw⋖u + zBbw⋖u + zCcw⋖u + zDdw⋖u. Then if {zA, zB, zC, zD} = {0, 0, z, 2 − z} as multisets, mwt(id, w0) = n 2

!.

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 17 / 21

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Weighted enumeration of maximal chains

Let wt(w ⋖ u) = 1 + zAaw⋖u + zBbw⋖u + zCcw⋖u + zDdw⋖u.

123
213
231
321
132
312

1 1 1 1 1 + zA 1 + zC 1 + zD 1 + zB

Figure: Weights on covering relations of S3

Then mwt(123, 321) = 4 + zA + zB + zC + zD, which is 6 = 3! if {zA, zB, zC, zD} = {0, 0, z, 2 − z}.

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 18 / 21

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Weighted enumeration of maximal chains

Theorem (Gaetz and G. 2019)

Let zA, zB, zC, zD be indeterminates and define a weight function on the covering relations on the strong Bruhat order of Sn as follows: wt(w ⋖ u) = 1 + zAaw⋖u + zBbw⋖u + zCcw⋖u + zDdw⋖u. Then if {zA, zB, zC, zD} = {0, 0, z, 2 − z} as multisets, mwt(id, w0) = n 2

!.

Special cases:

1 (zA, zB, zC, zD) = (0, 1, 0, 1), wt(w ⋖ wtij) = j − i, 2 (zA, zB, zC, zD) = (0, 0, 2, 0), wt(w ⋖ u) = ||code(w) − code(u)||L1.

The “j − i” weight is commonly known as the Chevalley weight, which is investigated by Stembridge (2002) and further by Postnikov and Stanley (2009). It is still open to find a combinatorial proof.

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 19 / 21

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References

Zachary Hamaker, Oliver Pechenik, David E Speyer, and Anna Weigandt. Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley. arXiv:1812.00321 [math.CO]. Richard P. Stanley. Some Schubert shenanigans. arXiv:1704.00851 [math.CO]. Christian Gaetz and Yibo Gao. A combinatorial sl2-action and the Sperner property for the weak order. arXiv:1811.05501 [math.CO]. Christian Gaetz and Yibo Gao. A combinatorial duality between the weak and strong Bruhat orders. arXiv:1812.05126 [math.CO]. Christian Gaetz and Yibo Gao. Padded Schubert polynomials and weighted enumeration of Bruhat chains. arXiv:1905.00047 [math.CO].

Yibo Gao (MIT) The weak order is Sperner FPSAC 2019 20 / 21

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Thanks

Thanks: Alex Postnikov and Richard Stanley. Thank you for listening!

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