Post on 29-Jul-2021
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: bergeron@yorku.ca
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 )
Jul 2016 2/10 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 )
∆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
Jul 2016 2/10 Hopf algebra of pipe dreams.
comultiplication of Pipe Dreams
∆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
Jul 2016 2/10 Hopf algebra of pipe dreams.
comultiplication on permutation
∆γ,n−γ ∶ Πn → Πγ ⊗Πn−γ
Jul 2016 3/10 Hopf algebra of pipe dreams.
comultiplication on permutation
∆γ,n−γ ∶ Πn → Πγ ⊗Πn−γ
↓ω ↓ω⊗ω
Sn → Sγ ⊗ Sn−γ
Jul 2016 3/10 Hopf algebra of pipe dreams.
comultiplication on permutation
∆γ,n−γ ∶ Πn → Πγ ⊗Πn−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
Jul 2016 3/10 Hopf algebra of pipe dreams.
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
↦ ?
Jul 2016 4/10 Hopf algebra of pipe dreams.
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
Jul 2016 4/10 Hopf algebra of pipe dreams.
multiplication of permutations
µr,s∶ Πr ⊗Πs → Πr+s
↓ω⊗ω ↓ω
Sr ⊗ Ss → Sr+s
u ⋅ v ↦ u↶∆GD(u)+1(v)
3412 ⋅ 21↦ ?
Jul 2016 5/10 Hopf algebra of pipe dreams.
multiplication of permutations
µr,s∶ Πr ⊗Πs → Πr+s
↓ω⊗ω ↓ω
Sr ⊗ Ss → Sr+s
u ⋅ v ↦ u↶∆GD(u)+1(v)
3412 ⋅ 21↦ ?
3412 = 12 ● 12 21 = 1 ● 1
Jul 2016 5/10 Hopf algebra of pipe dreams.
multiplication of permutations
µr,s∶ Πr ⊗Πs → Πr+s
↓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
Jul 2016 5/10 Hopf algebra of pipe dreams.
multiplication of permutations
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
Jul 2016 5/10 Hopf algebra of pipe dreams.
Two Hopf Algebras
kΠ =⊕n≥0
kΠn kS =⊕n≥0
kSn
ω∶kΠ→ kS
kΠ: new
kS: new? What is relation with Malvenuto-Reutenauer
Jul 2016 6/10 Hopf algebra of pipe dreams.
kΠ is free and cofree
Generators:
w1 ... wγ wγ+1... wn
1
⋮
n−γ
n−γ+1 +
⋮
n
Jul 2016 7/10 Hopf algebra of pipe dreams.
kΠ is free and cofree
Generators:
w1 ... wγ wγ+1... wn
1
⋮
n−γ
n−γ+1 +
⋮
n
����
Jul 2016 7/10 Hopf algebra of pipe dreams.
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
Jul 2016 8/10 Hopf algebra of pipe dreams.
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={12}: kΠ{12}
— number of generators deg n = C2n+1
Jul 2016 8/10 Hopf algebra of pipe dreams.
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={213}: kΠ{213}
— number of generators deg n = C3n+2
Jul 2016 8/10 Hopf algebra of pipe dreams.
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={3214}: kΠ{3214}
— number of generators deg n = C4n+3
Jul 2016 8/10 Hopf algebra of pipe dreams.
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]
Jul 2016 9/10 Hopf algebra of pipe dreams.
kΠS for S = {1,12,123, . . .}
Jul 2016 10/10 Hopf algebra of pipe dreams.
kΠS for S = {1,12,123, . . .}
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Jul 2016 10/10 Hopf algebra of pipe dreams.
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
Jul 2016 1000000/10 Hopf algebra of pipe dreams.