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Diagrams of Affine Permutations and Their Labellings Taedong Yun Oracle June 23, 2014 Taedong Yun Permutation Diagrams and Labellings June 23, 2014 1 / 36 Contents Based on the joint work with Hwanchul Yoo (KIAS). Taedong Yun

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Diagrams of Affine Permutations and Their Labellings

Taedong Yun

Oracle

June 23, 2014

Taedong Yun Permutation Diagrams and Labellings June 23, 2014 1 / 36

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Contents

Based on the joint work with Hwanchul Yoo (KIAS).

Taedong Yun Permutation Diagrams and Labellings June 23, 2014 2 / 36

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Affine Balanced Labellings

Section 1 Affine Balanced Labellings

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Permutations

A permutation is a bijection w : {1, 2, . . . , n} → {1, 2, . . . , n}. Σn := the symmetric group, group of (finite) permutations of size n. Σn is generated by

1 the simple reflections s1, . . . , sn−1

(si is the permutation which interchanges the pair (i, i + 1))

2 and the following relations

s2

i = 1

for all i sisi+1si = si+1sisi+1 for all i sisj = sjsi for |i − j| ≥ 2.

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Affine Permutations

An affine permutation of period n is a bijection w : Z → Z such that w(i + n) = w(i) + n ∀i ∈ Z and w(1) + w(2) + · · · + w(n) = n(n + 1)/2.

  • Σn := the affine symmetric group, group of affine permutations of

period n.

  • Σn is generated by

1 the simple reflections s0, s1, . . . , sn−1

(si interchanges the pairs (i+rn, i + 1+rn) ∀r ∈ Z.)

2 and the following relations

s2

i = 1

for all i sisi+1si = si+1sisi+1 for all i sisj = sjsi for i, j not adjacent, i = j. where the indices are taken modulo n.

Taedong Yun Permutation Diagrams and Labellings June 23, 2014 5 / 36

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Affine Permutations

Example

  • Σ3 is generated by {s0, s1, s2}.

· · ·

  • 1

1 2 3 4 5 · · · id · · ·

  • 1

1 2 3 4 5 · · · s2 · · ·

  • 1

1 3 2 4 6 · · · s2s0 · · · 1

  • 1

3 4 2 6 · · · s2s0s1 · · ·

  • 4

1 3

  • 1

4 6 2 · · · s2s0s1s0 · · ·

  • 4

3 1

  • 1

6 4 2 · · · w = s2s0s1s0 ∈ Σ3 w = [1, −1, 6] : window notation

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Affine Permutations

Remark A permutation w = [w1, . . . , wn] can be viewed as an affine permutation [w1, . . . , wn] with period n, written in window notation. This corresponds to the natural embedding Σn ֒ − → Σn, si − → si.

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Reduced Words

A reduced decomposition of w is a decomposition w = si1 · · · siℓ where ℓ is the minimal number for which such a decomposition exists. That minimal ℓ = ℓ(w) is the length of w. The word i1i2 · · · iℓ is called a reduced word of w.

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Reduced Words

A reduced decomposition of w is a decomposition w = si1 · · · siℓ where ℓ is the minimal number for which such a decomposition exists. That minimal ℓ = ℓ(w) is the length of w. The word i1i2 · · · iℓ is called a reduced word of w. Fact The length of an affine permutation w is the number of inversions (i, j) with 1 ≤ i ≤ n. ℓ(w) = |{(i, j) | i < j, w(i) > w(j), 1 ≤ i ≤ n}|.

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Diagrams

The affine permutation diagram of w ∈ Σn is the set D(w) = {(i, w(j)) | i < j, w(i) > w(j)} ⊆ Z × Z. When w is a finite permutation, D(w) consists of infinite number of identical copies of the Rothe diagram of w diagonally.

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Diagrams

1 1 6 6

Figure: diagram of w = [2, 5, 0, 7, 3, 4] ∈ Σ6

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Diagrams

1 1 6 6

Figure: diagram of w = [2, 5, 0, 7, 3, 4] ∈ Σ6

fundamental window [D(w)] := ([1, n] × Z) ∩ D(w). cell = {(i + rn, j + rn) | r ∈ Z} =: (i, j). ℓ(w) = #(cells) = |[D(w)]|

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Balanced Hook

Labelling of a diagram = labelling of the cells with positive integers.

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Balanced Hook

Labelling of a diagram = labelling of the cells with positive integers.

4 7 3 1 7 4 2 5

Hook Hij := ({i} × Z≥j) ∪ (Z≥i × {j})

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Balanced Hook

Labelling of a diagram = labelling of the cells with positive integers.

4 7 3 1 7 4 2 5

Hook Hij := ({i} × Z≥j) ∪ (Z≥i × {j}) Balanced hook: If one rearranges the labels in the hook so that they weakly increase from right to left and from top to bottom, then the corner label remains unchanged.

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Balanced Hook

Labelling of a diagram = labelling of the cells with positive integers.

4 7 3 1 7 4 2 5 4 4 5 7 7 3 2 1

Hook Hij := ({i} × Z≥j) ∪ (Z≥i × {j}) Balanced hook: If one rearranges the labels in the hook so that they weakly increase from right to left and from top to bottom, then the corner label remains unchanged.

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Balanced Labellings

Definition (Fomin-Greene-Reiner-Shimozono 1997) A labelling of a diagram D is called: balanced if every hook is balanced

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Balanced Labellings

Definition (Fomin-Greene-Reiner-Shimozono 1997) A labelling of a diagram D is called: balanced if every hook is balanced column-strict if no column contains two boxes with equal labels

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Affine Balanced Labellings Affine Permutations and Balanced Labellings

Balanced Labellings

Definition (Fomin-Greene-Reiner-Shimozono 1997) A labelling of a diagram D is called: balanced if every hook is balanced column-strict if no column contains two boxes with equal labels injective if each of the labels {1, 2, . . . , ℓ} appears exactly once in [D]. 1 1 2 2 3 3 4 4 3

Figure: injective balanced labelling

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Affine Balanced Labellings Reduced Words and Canonical Labellings

Canonical Labellings

Let a = a1a2 · · · aℓ be a reduced word of w ∈ Σn, i.e. w = sa1sa2 · · · saℓ. We define the canonical labelling Ta of a. e.g. w = [1, −1, 6] = s2s0s1s0.

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Affine Balanced Labellings Reduced Words and Canonical Labellings

Canonical Labellings

Let a = a1a2 · · · aℓ be a reduced word of w ∈ Σn, i.e. w = sa1sa2 · · · saℓ. We define the canonical labelling Ta of a. e.g. w = [1, −1, 6] = s2s0s1s0.

1 1 1

s2

1 1 1 2 2 2

s2s0

1 1 1 2 2 2 3 3

s2s0s1

1 1 2 2 3 3 4 4

s2s0s1s0

3

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Affine Balanced Labellings Reduced Words and Canonical Labellings

Canonical Labellings

Theorem (Yoo-Y. 2012, FGRS 1997 for finite case) For all w ∈ Σn, the map a − → Ta (the canonical labelling of a) is a bijection {reduced words of w} − → {injective balanced labellings of D(w)}. Proof. Find the inverse map.

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Symmetric Functions

Section 2 Symmetric Functions

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Symmetric Functions Affine Stanley Symmetric Functions

Stanley Symmetric Functions

v ∈ Σn is called decreasing if it has a decreasing reduced word. e.g. v = s4s2s1 ∈ Σ5. w = v1v2 · · · vr is a decreasing factorization of w if each vi ∈ Σn is decreasing and ℓ(w) = r

i=1 ℓ(vi).

(ℓ(v1), ℓ(v2), . . . , ℓ(vr)) is the type of the factorization. Definition (Stanley 1984) Let w ∈ Σn be a permutation. The Stanley symmetric function Fw(x)

  • f w is defined by

Fw(x) := Fw(x1, x2, · · · ) =

  • w=v1v2···vr

xℓ(v1)

1

xℓ(v2)

2

· · · xℓ(vr)

r

, where the sum is over all decreasing factorization of w.

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Symmetric Functions Affine Stanley Symmetric Functions

Affine Stanley Symmetric Functions

A word a1a2 · · · aℓ with letters in Z/nZ is cyclically decreasing if (1) each letter appears at most once, and (2) whenever i and i + 1 both appears in the word, i + 1 precedes i. v ∈ Σn is cyclically decreasing if it has a cyclically decreasing reduced

  • word. e.g. v = s2s0s4 ∈

Σ5. w = v1v2 · · · vr is a cyclically decreasing factorization of w if each vi ∈ Σn is cyclically decreasing and ℓ(w) = r

i=1 ℓ(vi).

Definition (Lam 2006) w ∈ Σn. The affine Stanley symmetric function Fw(x) of w is

  • Fw(x) :=

Fw(x1, x2, · · · ) =

  • w=v1v2···vr

xℓ(v1)

1

xℓ(v2)

2

· · · xℓ(vr)

r

, where the sum is over all cyclically decreasing factorization of w.

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Symmetric Functions Affine Stanley Symmetric Functions

Column-Strict Balanced Labellings

Theorem (Yoo-Y. 2012, FGRS 1997 for finite case) CB(D) = {column-strict balanced labellings of a diagram D}. (“column strict” := no column contains two boxes with equal labels.) For all w ∈ Σn,

  • Fw(x) =
  • T∈CB(D(w))

xT where xT denotes the monomial

(i,j)∈[D(w)] xT(i,j).

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Symmetric Functions Affine Stanley Symmetric Functions

Column-Strict Balanced Labellings

1 1 2 2 4 4 4 4 4

()()()(s0)

1 1 1 2 2 2 4 4

()()()(s1s0)

1 1 1 2 2 2

()()(id)(s1s0)

1 1 1 2 2 2

()(s0)(id)(s1s0)

1 1 1

(s2)(s0)(id)(s1s0) ⇒ w = v1v2v3v4 v1 = s2, v2 = s0, v3 = id, v4 = s1s0.

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Properties of Permutation Diagrams

Section 3 Properties of Permutation Diagrams

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Properties of Permutation Diagrams Affine Diagram

Affine Diagram

A collection D of unit square boxes on Z × Z is called an affine diagram (of period n) if there are finite number of boxes on each row and column, and (i, j) ∈ D ⇔ (i + n, j + n) ∈ D. Clearly, any affine permutation diagram of w ∈ Σn is an affine diagram of period n.

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Properties of Permutation Diagrams Affine Diagram

Question

When is an (affine) diagram an (affine) permutation diagram?

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Properties of Permutation Diagrams Affine Diagram

Theorem (Yoo-Y. 2012, FGRS 1997 for finite case) D affine permutation diagram of w, Reduced word of w ⇐ ⇒ Injective balanced labelling of D. Proof. Find the inverse map.

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Properties of Permutation Diagrams Affine Diagram

Question

Theorem (Yoo-Y. 2012, FGRS 1997 for finite case) D affine permutation diagram of w, Reduced word of w ⇐ ⇒ Injective balanced labelling of D. When is an affine diagram an affine permutation diagram? D an affine diagram

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Properties of Permutation Diagrams Affine Diagram

Question

Theorem (Yoo-Y. 2012, FGRS 1997 for finite case) D affine permutation diagram of w, Reduced word of w ⇐ ⇒ Injective balanced labelling of D. When is an affine diagram an affine permutation diagram? D an affine diagram = ⇒ Try to find an injective balanced labelling of D.

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Properties of Permutation Diagrams Affine Diagram

Question

Theorem (Yoo-Y. 2012, FGRS 1997 for finite case) D affine permutation diagram of w, Reduced word of w ⇐ ⇒ Injective balanced labelling of D. When is an affine diagram an affine permutation diagram? D an affine diagram = ⇒ Try to find an injective balanced labelling of D. = ⇒ Using the “inverse map”, find the reduced word a1a2 · · · aℓ corresponding to the labelling.

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Properties of Permutation Diagrams Affine Diagram

Question

Theorem (Yoo-Y. 2012, FGRS 1997 for finite case) D affine permutation diagram of w, Reduced word of w ⇐ ⇒ Injective balanced labelling of D. When is an affine diagram an affine permutation diagram? D an affine diagram = ⇒ Try to find an injective balanced labelling of D. = ⇒ Using the “inverse map”, find the reduced word a1a2 · · · aℓ corresponding to the labelling. = ⇒ Check if w = sa1sa2 · · · saℓ gives you the diagram D.

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Properties of Permutation Diagrams Classification Theorem

Classification Theorem

Theorem (Yoo-Y. 2012) D an affine diagram. D is an affine permutation diagram if and only if it is North-West and admits a content map.

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Properties of Permutation Diagrams Classification Theorem

Classification Theorem

Theorem (Yoo-Y. 2012) D an affine diagram. D is an affine permutation diagram if and only if it is North-West and admits a content map. An affine diagram is North-West if whenever there is a box at (i, j) and at (k, ℓ) with i < k and j > ℓ, there must be a box at (i, ℓ).

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Properties of Permutation Diagrams Content Map

Content Map

Definition (Yoo-Y. 2012) D an affine diagram with period n. C : D → Z is a content map if it satisfies the following four conditions. (C1) If boxes b1 and b2 are in the same row (respectively, column), b2 being to the east (resp., south) to b1, and there are no boxes between b1 and b2, then C(b2) − C(b1) = 1. (C2) If b2 is strictly to the southeast of b1, then C(b2) − C(b1) ≥ 2. (C3) For each row (resp., column), the content of the leftmost (resp., topmost) box is equal to the row (resp., column) index. (C4) If b1 = (i, j) and b2 = (i + n, j + n) coordinate-wise, then C(b2) − C(b1) = n.

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Properties of Permutation Diagrams Content Map

Content Map

5 6 7 4 4 3 2 1

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Properties of Permutation Diagrams Content Map

Content Map

5 6 7 4 4 3 2 1

(C1) If boxes b1 and b2 are in the same row, b2 being to the east to b1, and there are no boxes between b1 and b2, then C(b2) − C(b1) = 1.

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Properties of Permutation Diagrams Content Map

Content Map

5 6 7 4 4 3 2 1

(C1) If boxes b1 and b2 are in the same row, b2 being to the east to b1, and there are no boxes between b1 and b2, then C(b2) − C(b1) = 1. (C2) If b2 is strictly to the southeast

  • f b1, then C(b2) − C(b1) ≥ 2.

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Properties of Permutation Diagrams Content Map

Content Map

5 6 7 4 4 3 2 1

(C1) If boxes b1 and b2 are in the same row, b2 being to the east to b1, and there are no boxes between b1 and b2, then C(b2) − C(b1) = 1. (C2) If b2 is strictly to the southeast

  • f b1, then C(b2) − C(b1) ≥ 2.

(C3) For each row, the content of the leftmost box is equal to the row index.

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Properties of Permutation Diagrams Content Map

Content Map

5 6 7 4 4 3 2 1

(C1) If boxes b1 and b2 are in the same row, b2 being to the east to b1, and there are no boxes between b1 and b2, then C(b2) − C(b1) = 1. (C2) If b2 is strictly to the southeast

  • f b1, then C(b2) − C(b1) ≥ 2.

(C3) For each row, the content of the leftmost box is equal to the row index. (C4) If b1 = (i, j) and b2 = (i + n, j + n) coordinate-wise, then C(b2) − C(b1) = n.

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Properties of Permutation Diagrams Proof of Classification Theorem

Theorem (Yoo-Y. 2012) D an (affine) diagram. D is an (affine) permutation diagram if and only if it is North-West and admits a content map. Proof. We re-construct the affine permutation from a North-West diagram with a content map.

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Properties of Permutation Diagrams Proof of Classification Theorem

Proof of Classification Theorem (Wiring Diagrams)

5 6 7 4 4 3 2 1

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Properties of Permutation Diagrams Proof of Classification Theorem

Proof of Classification Theorem (Wiring Diagrams)

7 1 2 3 4 5 4 6 1 2 3 4 5 6 w1 w2 w3 w4 w5 7 8 (9) w6 w7 w8 (w9) 10 11

1-st row 1-st column Taedong Yun Permutation Diagrams and Labellings June 23, 2014 29 / 36

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Properties of Permutation Diagrams Proof of Classification Theorem

Proof of Classification Theorem (Wiring Diagrams)

7 1 2 3 4 5 4 6 1 2 3 4 5 6 w1 w2 w3 w4 w5 7 8 (9) w6 w7 w8 (w9) 10 11

1-st row 1-st column

1 2 3 4 5 6 w1 w2 w3 w4 w5 7 8 (9) w6 w7 w8 (w9)

Taedong Yun Permutation Diagrams and Labellings June 23, 2014 29 / 36

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Properties of Permutation Diagrams Proof of Classification Theorem

Proof of Classification Theorem (Wiring Diagrams)

= ⇒ w = [2, 6, 1, 4, 3, 7, 8, 5, 9].

7 1 2 3 4 5 4 6 1 2 3 4 5 6 w1 w2 w3 w4 w5 7 8 (9) w6 w7 w8 (w9) 10 11

1-st row 1-st column

1 2 3 4 5 6 w1 w2 w3 w4 w5 7 8 (9) w6 w7 w8 (w9)

Taedong Yun Permutation Diagrams and Labellings June 23, 2014 29 / 36

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SLIDE 49

Set-Valued Balanced Labellings

Section 4 Set-Valued Balanced Labellings

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SLIDE 50

Set-Valued Balanced Labellings

Set-Valued Labellings 125 3 4 6 u3u3u2u3u2u1

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Set-Valued Balanced Labellings

Related Objects

labelling set-valued (s-v) labelling balanced labelling s-v balanced labelling reduced words nilHecke words Stanley symm. func. stable Grothendieck poly. (Lascoux-Sch¨ utzenberger) Schubert poly. Sw Grothendieck poly. Gw (Lascoux-Sch¨ utzenberger) affine symm. group affine nilHecke algebra Un (Lam) affine Stanley symm. func. Fw affine stable Grothendieck poly. Gw (Lam)

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SLIDE 52

Set-Valued Balanced Labellings

125 3 4 6 u3u3u2u3u2u1

Theorem (Yoo-Y. 2013) w ∈ Σn, Un affine nilHecke algebra. There is a bijection from {nilHecke words a in Un with S(a) = w} to {s-v injective balanced labellings of D(w)}.

Taedong Yun Permutation Diagrams and Labellings June 23, 2014 33 / 36

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Set-Valued Balanced Labellings

Theorem (Yoo-Y. 2013) w an affine permutation, Gw affine stable Grothendieck polynomial.

  • Gw(x) =

T(−1)|T|−ℓ(w)xT,

  • ver all s-v column-strict balanced labellings T of D(w),

xT :=

b∈[D(w)]

  • k∈T(b) xk.

Theorem (Buch 2002) λ/µ a skew Young diagram (equiv., a diagram of 321-avoiding finite permutation). Gλ/µ(x) =

T(−1)|T|−|λ/µ|xT,

  • ver all set-valued tableaux T of shape λ/µ, xT :=

b∈λ/µ

  • k∈T(b) xk.

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Set-Valued Balanced Labellings

Theorem (Yoo-Y. 2013) w a finite permutation. Gw Grothendieck polynomial. Gw(x) =

  • T

(−1)|T|−ℓ(w)xT,

  • ver all column-strict s-v balanced labellings T of D(w) with flag

conditions: ∀t ∈ T(i, j), t ≤ i. Yoo-Yun, Diagrams of affine permutations, balanced labellings, and symmetric functions, arXiv:1305.0129.

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Set-Valued Balanced Labellings (math.tedyun.com, arXiv:1305.0129)

Happy Birthday, Richard!

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