Hopf (pipe) dreams YORK Nantel Bergeron - Queen's Uwehlau/Herstmonceux/HerstTalks/... · 2016. 8....

Hopf (pipe) dreams

YORK

U N I V E R S I T E———————U N I V E R S I T Y

Nantel Bergeron

York Research Chair in Applied Algebra

www.math.yorku.ca/bergeron

E-Mail: [email protected]

with Cesar Ceballos and Vincent Pilaud

Herstmonceux, England Jul 2016

Outline

● Pipe Dreams

● Hopf Algebra of pipe dreams

● some Hopf subalgebra and enumarations

Jul 2016 outline

Pipe Dreams0 1 5 4 6 2 3

0

1

2

3

4

5

6

A pipe dream P ∈ Π6 where ωP = [1,5,4,6,2,3].

ω∶ ⊕n≥0

Πn Ð→⊕n≥0

Sn

Jul 2016 1/10 Hopf algebra of pipe dreams.

comultiplication of Pipe Dreams

∆n∶ Πn →n

⊕γ=0

Πγ ⊗Πn−γ

P ↦ ∑γ∈GD(ωP )∪{0,n}

∆γ,n−γ(P )



∆n∶ Πn →n

⊕γ=0

Πγ ⊗Πn−γ

P ↦ ∑γ∈GD(ωP )∪{0,n}

∆γ,n−γ(P )

∆4,2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

6 3 5 4 2 1

1

2

3

4

5

6

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

4 1 3 2

1

2

3

4

⊗

2 1

1

2



∆4

⎛⎜⎜⎜⎜⎜⎝

4 3 1 2

1

2

3

4

⎞⎟⎟⎟⎟⎟⎠

=

4 3 1 2

1

2

3

4

⊗ +

2 1

1

2

⊗

1 2

1

2

+

1

1

⊗

3 1 2

1

2

3

+ ⊗

4 3 1 2

1

2

3

4


comultiplication on permutation

∆γ,n−γ ∶ Πn → Πγ ⊗Πn−γ




↓ω ↓ω⊗ω

Sn → Sγ ⊗ Sn−γ




↓ω ↓ω⊗ω

Sn → Sγ ⊗ Sn−γ

∆4,2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

6 3 5 4 2 1

1

2

3

4

5

6

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

4 1 3 2

1

2

3

4

⊗

2 1

1

2

635421 ↦ 4132⊗ 21


multiplication of Pipe Dreams

µr,s∶ Πr ⊗Πs → Πr+s

P ⋅Q ↦ P ↶∆GD(ωP )+1(Q)

3 4 1 2

1

2

3

4

⋅

2 1

1

2

↦ ?


multiplication of Pipe Dreams

6 5 3 4 1 2

1

2

3

4

5

6

+

6 4 5 3 1 2

1

2

3

4

5

6

+

6 4 5 2 3 1

1

2

3

4

5

6

+

5 6 4 3 1 2

1

2

3

4

5

6

+

5 6 4 2 3 1

1

2

3

4

5

6

+

5 6 3 4 2 1

1

2

3

4

5

6


multiplication of permutations


↓ω⊗ω ↓ω

Sr ⊗ Ss → Sr+s

u ⋅ v ↦ u↶∆GD(u)+1(v)

3412 ⋅ 21↦ ?




↓ω⊗ω ↓ω

Sr ⊗ Ss → Sr+s

u ⋅ v ↦ u↶∆GD(u)+1(v)

3412 ⋅ 21↦ ?

3412 = 12 ● 12 21 = 1 ● 1




↓ω⊗ω ↓ω

Sr ⊗ Ss → Sr+s

3412 ⋅ 21 = (12 ● 12)⊔⊔●(1 ● 1)

= 12 ● 12 ● 1 ● 1 + 12 ● 1 ● 12 ● 1 + 1 ● 12 ● 12 ● 1

+ 12 ● 1 ● 1 ● 12 + 1 ● 12 ● 1 ● 12 + 1 ● 1 ● 12 ● 12

= 563421 + 564231 + 645231

+ 564312 + 645312 + 653412



3412 ⋅ 21 = 563421 + 564231 + 645231 + 564312 + 645312 + 6534126 5 3 4 1 2

1

2

3

4

5

6

+

6 4 5 3 1 2

1

2

3

4

5

6

+

6 4 5 2 3 1

1

2

3

4

5

6

+

5 6 4 3 1 2

1

2

3

4

5

6

+

5 6 4 2 3 1

1

2

3

4

5

6

+

5 6 3 4 2 1

1

2

3

4

5

6


Two Hopf Algebras

kΠ =⊕n≥0

kΠn kS =⊕n≥0

kSn

ω∶kΠ→ kS

kΠ: new

kS: new? What is relation with Malvenuto-Reutenauer


kΠ is free and cofree

Generators:

w1 ... wγ wγ+1... wn

1

⋮

n−γ

n−γ+1 +

⋮

n


kΠ is free and cofree

Generators:

w1 ... wγ wγ+1... wn

1

⋮

n−γ

n−γ+1 +

⋮

n

��


kΠS for a (fixed) set of atomics S

Given a set of atomics S [atomic=no global descent]

ΠS = {P ∈ Π ∶ Atomics(ωP ) ⊆ S}

Example S={1}: kΠ{1} ≅ Loday-Ronco Hopf Algebra

— dim deg n = Cn+1 (Catalan)

— number of generators deg n = Cn





Example S={12}: kΠ{12}

— number of generators deg n = C2n+1





Example S={213}: kΠ{213}






Example S={3214}: kΠ{3214}



kΠS for S = {1,12,123, . . .}Example S={1,12,123,. . . }: kΠS

— dim deg n = number of paths in positive quadran with steps

[Bousquet-Melou, Mishna]


kΠS for S = {1,12,123, . . .}


kΠS for S = {1,12,123, . . .}

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K A M S A H A E Y O

M E R C I

T H A N K S

G R A C I A S


Hopf (pipe) dreams YORK Nantel Bergeron - Queen's Uwehlau/Herstmonceux/HerstTalks/... · 2016. 8....

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Transcript of Hopf (pipe) dreams YORK Nantel Bergeron - Queen's Uwehlau/Herstmonceux/HerstTalks/... · 2016. 8....