combinatorics of canonical bases revisited
TRANSCRIPT
![Page 1: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/1.jpg)
Combinatorics of canonical bases revisited
Bea Schumann (Volker Genz and Gleb Koshevoy)
Hannover05. Dezember, 2016
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 1/ 34
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Today I will talk about recent work with Volker Genz and GlebKoshevoy.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 2/ 34
![Page 3: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/3.jpg)
Today I will talk about recent work with Volker Genz and GlebKoshevoy.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 2/ 34
![Page 4: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/4.jpg)
Outline of the talk:
1 Crystal structures on PBW-type bases
2 Rhombic tilings
3 Crystal structures via Rhombic tilings
4 A duality between Lusztig’s and Kashiwara’s parametrizationof canonical bases
5 An interpretation in the setup of cluster varieties
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 3/ 34
![Page 5: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/5.jpg)
Outline of the talk:
1 Crystal structures on PBW-type bases
2 Rhombic tilings
3 Crystal structures via Rhombic tilings
4 A duality between Lusztig’s and Kashiwara’s parametrizationof canonical bases
5 An interpretation in the setup of cluster varieties
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 3/ 34
![Page 6: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/6.jpg)
Outline of the talk:
1 Crystal structures on PBW-type bases
2 Rhombic tilings
3 Crystal structures via Rhombic tilings
4 A duality between Lusztig’s and Kashiwara’s parametrizationof canonical bases
5 An interpretation in the setup of cluster varieties
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 3/ 34
![Page 7: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/7.jpg)
Outline of the talk:
1 Crystal structures on PBW-type bases
2 Rhombic tilings
3 Crystal structures via Rhombic tilings
4 A duality between Lusztig’s and Kashiwara’s parametrizationof canonical bases
5 An interpretation in the setup of cluster varieties
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 3/ 34
![Page 8: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/8.jpg)
Outline of the talk:
1 Crystal structures on PBW-type bases
2 Rhombic tilings
3 Crystal structures via Rhombic tilings
4 A duality between Lusztig’s and Kashiwara’s parametrizationof canonical bases
5 An interpretation in the setup of cluster varieties
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 3/ 34
![Page 9: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/9.jpg)
Outline of the talk:
1 Crystal structures on PBW-type bases
2 Rhombic tilings
3 Crystal structures via Rhombic tilings
4 A duality between Lusztig’s and Kashiwara’s parametrizationof canonical bases
5 An interpretation in the setup of cluster varieties
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 3/ 34
![Page 10: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/10.jpg)
Setup
g = sln := {A ∈ Mn(C) | traceA = 0}
h = {diagonal matrices} ⊂ sln Cartan subalgebra
Φ+ = {αk,`−1 = εk − ε` | k , ` ∈ {1, ..., n}; k < `} ⊂ h∗
positive roots of g, where εk (diag(h1, h2, . . . , hn)) = hk .
A total ordering ≤ on Φ+ is called convex if for eachβ1, β2, β3 ∈ Φ+ with β1 + β3 = β2
β1 ≤ β2 ≤ β3 or β3 ≤ β2 ≤ β1.
If we replace β1 ≤ β2 ≤ β3 by β3 ≤′ β2 ≤′ β1, we call theconvex ordering ≤′ a flip of ≤.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 4/ 34
![Page 11: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/11.jpg)
Setup
g = sln := {A ∈ Mn(C) | traceA = 0}
h = {diagonal matrices} ⊂ sln Cartan subalgebra
Φ+ = {αk,`−1 = εk − ε` | k , ` ∈ {1, ..., n}; k < `} ⊂ h∗
positive roots of g, where εk (diag(h1, h2, . . . , hn)) = hk .
A total ordering ≤ on Φ+ is called convex if for eachβ1, β2, β3 ∈ Φ+ with β1 + β3 = β2
β1 ≤ β2 ≤ β3 or β3 ≤ β2 ≤ β1.
If we replace β1 ≤ β2 ≤ β3 by β3 ≤′ β2 ≤′ β1, we call theconvex ordering ≤′ a flip of ≤.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 4/ 34
![Page 12: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/12.jpg)
Setup
g = sln := {A ∈ Mn(C) | traceA = 0}
h = {diagonal matrices} ⊂ sln Cartan subalgebra
Φ+ = {αk,`−1 = εk − ε` | k , ` ∈ {1, ..., n}; k < `} ⊂ h∗
positive roots of g, where εk (diag(h1, h2, . . . , hn)) = hk .
A total ordering ≤ on Φ+ is called convex if for eachβ1, β2, β3 ∈ Φ+ with β1 + β3 = β2
β1 ≤ β2 ≤ β3 or β3 ≤ β2 ≤ β1.
If we replace β1 ≤ β2 ≤ β3 by β3 ≤′ β2 ≤′ β1, we call theconvex ordering ≤′ a flip of ≤.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 4/ 34
![Page 13: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/13.jpg)
Setup
g = sln := {A ∈ Mn(C) | traceA = 0}
h = {diagonal matrices} ⊂ sln Cartan subalgebra
Φ+ = {αk,`−1 = εk − ε` | k , ` ∈ {1, ..., n}; k < `} ⊂ h∗
positive roots of g, where εk (diag(h1, h2, . . . , hn)) = hk .
A total ordering ≤ on Φ+ is called convex if for eachβ1, β2, β3 ∈ Φ+ with β1 + β3 = β2
β1 ≤ β2 ≤ β3 or β3 ≤ β2 ≤ β1.
If we replace β1 ≤ β2 ≤ β3 by β3 ≤′ β2 ≤′ β1, we call theconvex ordering ≤′ a flip of ≤.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 4/ 34
![Page 14: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/14.jpg)
Setup
g = sln := {A ∈ Mn(C) | traceA = 0}
h = {diagonal matrices} ⊂ sln Cartan subalgebra
Φ+ = {αk,`−1 = εk − ε` | k , ` ∈ {1, ..., n}; k < `} ⊂ h∗
positive roots of g, where εk (diag(h1, h2, . . . , hn)) = hk .
A total ordering ≤ on Φ+ is called convex if for eachβ1, β2, β3 ∈ Φ+ with β1 + β3 = β2
β1 ≤ β2 ≤ β3 or β3 ≤ β2 ≤ β1.
If we replace β1 ≤ β2 ≤ β3 by β3 ≤′ β2 ≤′ β1, we call theconvex ordering ≤′ a flip of ≤.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 4/ 34
![Page 15: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/15.jpg)
Setup
g = sln := {A ∈ Mn(C) | traceA = 0}
h = {diagonal matrices} ⊂ sln Cartan subalgebra
Φ+ = {αk,`−1 = εk − ε` | k , ` ∈ {1, ..., n}; k < `} ⊂ h∗
positive roots of g, where εk (diag(h1, h2, . . . , hn)) = hk .
A total ordering ≤ on Φ+ is called convex if for eachβ1, β2, β3 ∈ Φ+ with β1 + β3 = β2
β1 ≤ β2 ≤ β3 or β3 ≤ β2 ≤ β1.
If we replace β1 ≤ β2 ≤ β3 by β3 ≤′ β2 ≤′ β1, we call theconvex ordering ≤′ a flip of ≤.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 4/ 34
![Page 16: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/16.jpg)
PBW-type bases
For each convex ordering β1 ≤ β2 ≤ . . . ≤ βN onΦ+ = {β1, β2, . . . , βN} Lusztig defined a PBW-type basis of U+
q
(=positive part of the quantized enveloping algebra of g)
B≤ = {F (xβ1)
β1F
(xβ2)
β2· · ·F (xβN )
βN| (xβ1 , xβ2 , . . . , xβN ) ∈ ZN
≥0}.
B≤ is in natural bijection with the canonical basis of U+q
we have a crystal structure (= a particular coloured graphstructure) on each B≤.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 5/ 34
![Page 17: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/17.jpg)
PBW-type bases
For each convex ordering β1 ≤ β2 ≤ . . . ≤ βN onΦ+ = {β1, β2, . . . , βN} Lusztig defined a PBW-type basis of U+
q
(=positive part of the quantized enveloping algebra of g)
B≤ = {F (xβ1)
β1F
(xβ2)
β2· · ·F (xβN )
βN| (xβ1 , xβ2 , . . . , xβN ) ∈ ZN
≥0}.
B≤ is in natural bijection with the canonical basis of U+q
we have a crystal structure (= a particular coloured graphstructure) on each B≤.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 5/ 34
![Page 18: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/18.jpg)
PBW-type bases
For each convex ordering β1 ≤ β2 ≤ . . . ≤ βN onΦ+ = {β1, β2, . . . , βN} Lusztig defined a PBW-type basis of U+
q
(=positive part of the quantized enveloping algebra of g)
B≤ = {F (xβ1)
β1F
(xβ2)
β2· · ·F (xβN )
βN| (xβ1 , xβ2 , . . . , xβN ) ∈ ZN
≥0}.
B≤ is in natural bijection with the canonical basis of U+q
we have a crystal structure (= a particular coloured graphstructure) on each B≤.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 5/ 34
![Page 19: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/19.jpg)
PBW-type bases
For each convex ordering β1 ≤ β2 ≤ . . . ≤ βN onΦ+ = {β1, β2, . . . , βN} Lusztig defined a PBW-type basis of U+
q
(=positive part of the quantized enveloping algebra of g)
B≤ = {F (xβ1)
β1F
(xβ2)
β2· · ·F (xβN )
βN| (xβ1 , xβ2 , . . . , xβN ) ∈ ZN
≥0}.
B≤ is in natural bijection with the canonical basis of U+q
we have a crystal structure (= a particular coloured graphstructure) on each B≤.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 5/ 34
![Page 20: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/20.jpg)
Crystal structure
{Vertices of the crystal} = { elements of B≤}
For each a ∈ {1, . . . , n − 1} we have an operatorfa : B≤ → B≤. For b, b′ ∈ B≤, there is an arrow in the crystal
graph ba−→ b′ ⇐⇒ fab = b′
εa(b) =length of string of consecutive arrows of colour aending at b.
fa is given explicitly if αa is the ≤-minimal positive root.Otherwise fa is given recursively via transition betweendifferent PBW-type bases obtained from each other by asequence of flips of the corresponding convex orderings(difficult to compute in general).
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 6/ 34
![Page 21: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/21.jpg)
Crystal structure
{Vertices of the crystal} = { elements of B≤}
For each a ∈ {1, . . . , n − 1} we have an operatorfa : B≤ → B≤.
For b, b′ ∈ B≤, there is an arrow in the crystal
graph ba−→ b′ ⇐⇒ fab = b′
εa(b) =length of string of consecutive arrows of colour aending at b.
fa is given explicitly if αa is the ≤-minimal positive root.Otherwise fa is given recursively via transition betweendifferent PBW-type bases obtained from each other by asequence of flips of the corresponding convex orderings(difficult to compute in general).
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 6/ 34
![Page 22: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/22.jpg)
Crystal structure
{Vertices of the crystal} = { elements of B≤}
For each a ∈ {1, . . . , n − 1} we have an operatorfa : B≤ → B≤. For b, b′ ∈ B≤, there is an arrow in the crystal
graph ba−→ b′ ⇐⇒ fab = b′
εa(b) =length of string of consecutive arrows of colour aending at b.
fa is given explicitly if αa is the ≤-minimal positive root.Otherwise fa is given recursively via transition betweendifferent PBW-type bases obtained from each other by asequence of flips of the corresponding convex orderings(difficult to compute in general).
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 6/ 34
![Page 23: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/23.jpg)
Crystal structure
{Vertices of the crystal} = { elements of B≤}
For each a ∈ {1, . . . , n − 1} we have an operatorfa : B≤ → B≤. For b, b′ ∈ B≤, there is an arrow in the crystal
graph ba−→ b′ ⇐⇒ fab = b′
εa(b) =length of string of consecutive arrows of colour aending at b.
fa is given explicitly if αa is the ≤-minimal positive root.Otherwise fa is given recursively via transition betweendifferent PBW-type bases obtained from each other by asequence of flips of the corresponding convex orderings(difficult to compute in general).
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 6/ 34
![Page 24: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/24.jpg)
Crystal structure
{Vertices of the crystal} = { elements of B≤}
For each a ∈ {1, . . . , n − 1} we have an operatorfa : B≤ → B≤. For b, b′ ∈ B≤, there is an arrow in the crystal
graph ba−→ b′ ⇐⇒ fab = b′
εa(b) =length of string of consecutive arrows of colour aending at b.
fa is given explicitly if αa is the ≤-minimal positive root.
Otherwise fa is given recursively via transition betweendifferent PBW-type bases obtained from each other by asequence of flips of the corresponding convex orderings(difficult to compute in general).
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 6/ 34
![Page 25: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/25.jpg)
Crystal structure
{Vertices of the crystal} = { elements of B≤}
For each a ∈ {1, . . . , n − 1} we have an operatorfa : B≤ → B≤. For b, b′ ∈ B≤, there is an arrow in the crystal
graph ba−→ b′ ⇐⇒ fab = b′
εa(b) =length of string of consecutive arrows of colour aending at b.
fa is given explicitly if αa is the ≤-minimal positive root.Otherwise fa is given recursively via transition betweendifferent PBW-type bases obtained from each other by asequence of flips of the corresponding convex orderings
(difficult to compute in general).
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 6/ 34
![Page 26: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/26.jpg)
Crystal structure
{Vertices of the crystal} = { elements of B≤}
For each a ∈ {1, . . . , n − 1} we have an operatorfa : B≤ → B≤. For b, b′ ∈ B≤, there is an arrow in the crystal
graph ba−→ b′ ⇐⇒ fab = b′
εa(b) =length of string of consecutive arrows of colour aending at b.
fa is given explicitly if αa is the ≤-minimal positive root.Otherwise fa is given recursively via transition betweendifferent PBW-type bases obtained from each other by asequence of flips of the corresponding convex orderings(difficult to compute in general).
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 6/ 34
![Page 27: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/27.jpg)
Aim:
Find nice description of fab for ≤ arbitrary!
(nice=withouth using piecewise-linear maps)
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 7/ 34
![Page 28: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/28.jpg)
Aim:
Find nice description of fab for ≤ arbitrary!
(nice=withouth using piecewise-linear maps)
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 7/ 34
![Page 29: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/29.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4 < α4 < α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 30: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/30.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4 < α4 < α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 31: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/31.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2=ε2−ε3
< α1,2 < α1 < α1,3 < α1,4 < α4 < α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 32: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/32.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2=ε2−ε3
< α1,2 < α1 < α1,3 < α1,4 < α4 < α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 33: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/33.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2=ε1−ε3
< α1 < α1,3 < α1,4 < α4 < α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 34: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/34.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2=ε1−ε3
< α1 < α1,3 < α1,4 < α4 < α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
3
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 35: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/35.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1=ε1−ε2
< α1,3 < α1,4 < α4 < α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 36: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/36.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1=ε1−ε2
< α1,3 < α1,4 < α4 < α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
1
2
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 37: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/37.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3=ε1−ε4
< α1,4 < α4 < α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
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Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3=ε1−ε4
< α1,4 < α4 < α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 39: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/39.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4=ε1−ε5
< α4 < α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 40: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/40.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4=ε1−ε5
< α4 < α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
41
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
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Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4 < α4=ε4−ε5
< α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 42: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/42.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4 < α4=ε4−ε5
< α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
4
5
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 43: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/43.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4 < α4 < α2,4=ε2−ε5
< α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 44: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/44.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4 < α4 < α2,4=ε2−ε5
< α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
4
5
2
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 45: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/45.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4 < α4 < α2,4 < α3,4=ε3−ε5
< α2,3 < α3
1
2
3
4
5
2
3
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 46: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/46.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4 < α4 < α2,4 < α3,4=ε3−ε5
< α2,3 < α3
1
2
3
4
5
2
3
1
4
5
3
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 47: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/47.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4 < α4 < α2,4 < α3,4 < α2,3=ε2−ε4
< α3
1
2
3
4
5
2
3
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 48: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/48.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4 < α4 < α2,4 < α3,4 < α2,3=ε2−ε4
< α3
1
2
3
4
5
2
3
1
4
5
2
4
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 49: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/49.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4 < α4 < α2,4 < α3,4 < α2,3 < α3=ε3−ε4
1
2
3
4
5
2
3
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 50: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/50.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4 < α4 < α2,4 < α3,4 < α2,3 < α3=ε3−ε4
1
2
3
4
5
2
3
1
4
5
3
4
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 51: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/51.jpg)
Rhombic tilings
Elnitsky associated to each convex ordering on Φ+ a decompositionof the regular 2n−gon into rhombi (Rhombic tilings).
Example (n = 5)
α2 < α1,2 < α1 < α1,3 < α1,4 < α4 < α2,4 < α3,4 < α2,3 < α3
1
2
3
4
5
2
3
1
4
5
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 8/ 34
![Page 52: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/52.jpg)
Lusztig data
For a convex ordering β1 ≤ β2 ≤ . . . ≤ βn, let T≤ be thecorresponding tiling.
We identify each positive root αk,`−1 = εk − ε` with the tile
k
〈k, `〉 :=
`
and an element
F(xβ1
)
β1F
(xβ2)
β2· · ·F (xβN )
βN∈ B≤
with the vector
x = (xβj )βj∈T≤ called ≤ −Lusztig-datum.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 9/ 34
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Lusztig data
For a convex ordering β1 ≤ β2 ≤ . . . ≤ βn, let T≤ be thecorresponding tiling.
We identify each positive root αk,`−1 = εk − ε` with the tile
k
〈k, `〉 :=
`
and an element
F(xβ1
)
β1F
(xβ2)
β2· · ·F (xβN )
βN∈ B≤
with the vector
x = (xβj )βj∈T≤ called ≤ −Lusztig-datum.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 9/ 34
![Page 54: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/54.jpg)
Lusztig data
For a convex ordering β1 ≤ β2 ≤ . . . ≤ βn, let T≤ be thecorresponding tiling.
We identify each positive root αk,`−1 = εk − ε` with the tile
k
〈k, `〉 :=
`
and an element
F(xβ1
)
β1F
(xβ2)
β2· · ·F (xβN )
βN∈ B≤
with the vector
x = (xβj )βj∈T≤ called ≤ −Lusztig-datum.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 9/ 34
![Page 55: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/55.jpg)
Lusztig data
For a convex ordering β1 ≤ β2 ≤ . . . ≤ βn, let T≤ be thecorresponding tiling.
We identify each positive root αk,`−1 = εk − ε` with the tile
k
〈k, `〉 :=
`
and an element
F(xβ1
)
β1F
(xβ2)
β2· · ·F (xβN )
βN∈ B≤
with the vector
x = (xβj )βj∈T≤
called ≤ −Lusztig-datum.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 9/ 34
![Page 56: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/56.jpg)
Lusztig data
For a convex ordering β1 ≤ β2 ≤ . . . ≤ βn, let T≤ be thecorresponding tiling.
We identify each positive root αk,`−1 = εk − ε` with the tile
k
〈k, `〉 :=
`
and an element
F(xβ1
)
β1F
(xβ2)
β2· · ·F (xβN )
βN∈ B≤
with the vector
x = (xβj )βj∈T≤ called ≤ −Lusztig-datum.
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 9/ 34
![Page 57: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/57.jpg)
Strips
For i ∈ {1, . . . , n} we define the i-strip Si to be the sequenceconsisting of all rhombi parallel to the edge with label i .
Example (2-strip in T )
2
3
S21
5
4
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 10/ 34
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Strips
For i ∈ {1, . . . , n} we define the i-strip Si to be the sequenceconsisting of all rhombi parallel to the edge with label i .
Example (2-strip in T )
2
3
S21
5
4
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 10/ 34
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Strips
For i ∈ {1, . . . , n} we define the i-strip Si to be the sequenceconsisting of all rhombi parallel to the edge with label i .
Example (2-strip in T )
2
3
S21
5
4
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 10/ 34
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Strips
For i ∈ {1, . . . , n} we define the i-strip Si to be the sequenceconsisting of all rhombi parallel to the edge with label i .
Example (2-strip in T )
2
3
S21
5
4
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 10/ 34
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Strips
For i ∈ {1, . . . , n} we define the i-strip Si to be the sequenceconsisting of all rhombi parallel to the edge with label i .
Example (2-strip in T )
2
3
S21
5
4
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 10/ 34
![Page 62: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/62.jpg)
Strips
For i ∈ {1, . . . , n} we define the i-strip Si to be the sequenceconsisting of all rhombi parallel to the edge with label i .
Example (2-strip in T )
2
3
S21
5
4
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 10/ 34
![Page 63: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/63.jpg)
Combs
For a ∈ {1, . . . , n − 1} we define the a-comb Wa to be the set ofrhombi lying in the region starting at the left boundary cut out bythe a-strip Sa and the (a + 1)-strip Sa+1 .
Example (W1 and W3 in T )
2
3
W11
5
4
2
3
W31
5
4
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 11/ 34
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Combs
For a ∈ {1, . . . , n − 1} we define the a-comb Wa to be the set ofrhombi lying in the region starting at the left boundary cut out bythe a-strip Sa and the (a + 1)-strip Sa+1 .
Example (W1 and W3 in T )
2
3
W11
5
4
2
3
W31
5
4
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 11/ 34
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Combs
For a ∈ {1, . . . , n − 1} we define the a-comb Wa to be the set ofrhombi lying in the region starting at the left boundary cut out bythe a-strip Sa and the (a + 1)-strip Sa+1 .
Example (W1 and W3 in T )
2
3
W11
5
4
2
3
W31
5
4
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 11/ 34
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Crossings
An a-crossing γ is a sequence of neighbour rhombi in Wa
connecting the first rhombus of Sa with the first rhombus of Sa+1
following strips oriented from Sa to Sa+1.
Example (A 3-crossing in T )
2
3
1
5
4
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Crossings
An a-crossing γ is a sequence of neighbour rhombi in Wa
connecting the first rhombus of Sa with the first rhombus of Sa+1
following strips oriented from Sa to Sa+1.
Example (A 3-crossing in T )
2
3
1
5
4
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 12/ 34
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Crossings
An a-crossing γ is a sequence of neighbour rhombi in Wa
connecting the first rhombus of Sa with the first rhombus of Sa+1
following strips oriented from Sa to Sa+1.
Example (A 3-crossing in T )
2
3
1
5
4
2
3
1
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 12/ 34
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Crossings
An a-crossing γ is a sequence of neighbour rhombi in Wa
connecting the first rhombus of Sa with the first rhombus of Sa+1
following strips oriented from Sa to Sa+1.
Example (A 3-crossing in T )
2
3
1
5
4
2
3
1
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 12/ 34
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Crossings
An a-crossing γ is a sequence of neighbour rhombi in Wa
connecting the first rhombus of Sa with the first rhombus of Sa+1
following strips oriented from Sa to Sa+1.
Example (A 3-crossing in T )
2
3
1
5
4
1
5
4
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 12/ 34
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Crossings
An a-crossing γ is a sequence of neighbour rhombi in Wa
connecting the first rhombus of Sa with the first rhombus of Sa+1
following strips oriented from Sa to Sa+1.
Example (A 3-crossing in T )
2
3
1
5
4
1
5
4
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 12/ 34
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Crossings
An a-crossing γ is a sequence of neighbour rhombi in Wa
connecting the first rhombus of Sa with the first rhombus of Sa+1
following strips oriented from Sa to Sa+1.
Example (A 3-crossing in T )
2
3
1
5
4
2
3
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 12/ 34
![Page 73: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/73.jpg)
Crossings
An a-crossing γ is a sequence of neighbour rhombi in Wa
connecting the first rhombus of Sa with the first rhombus of Sa+1
following strips oriented from Sa to Sa+1.
Example (A 3-crossing in T )
2
3
1
5
4
2
3
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 12/ 34
![Page 74: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/74.jpg)
Crossings
An a-crossing γ is a sequence of neighbour rhombi in Wa
connecting the first rhombus of Sa with the first rhombus of Sa+1
following strips oriented from Sa to Sa+1.
Example (A 3-crossing in T )
2
3
1
5
4
2
3
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 12/ 34
![Page 75: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/75.jpg)
Crossings
An a-crossing γ is a sequence of neighbour rhombi in Wa
connecting the first rhombus of Sa with the first rhombus of Sa+1
following strips oriented from Sa to Sa+1.
Example (A 3-crossing in T )
2
3
1
5
44
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 12/ 34
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Crossings
An a-crossing γ is a sequence of neighbour rhombi in Wa
connecting the first rhombus of Sa with the first rhombus of Sa+1
following strips oriented from Sa to Sa+1.
Example (A 3-crossing in T )
2
3
1
5
44
Combinatorics of canonical bases revisited Bea Schumann (Volker Genz and Gleb Koshevoy) 12/ 34
![Page 77: Combinatorics of canonical bases revisited](https://reader034.vdocuments.mx/reader034/viewer/2022052105/62872036c8cb242a7b7fc369/html5/thumbnails/77.jpg)
Crossings
An a-crossing γ is a sequence of neighbour rhombi in Wa
connecting the first rhombus of Sa with the first rhombus of Sa+1
following strips oriented from Sa to Sa+1.
Example (A 3-crossing in T )
2
3
1
5
44
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Crossings
An a-crossing γ is a sequence of neighbour rhombi in Wa
connecting the first rhombus of Sa with the first rhombus of Sa+1
following strips oriented from Sa to Sa+1.
Example (A 3-crossing in T )
2
3
1
5
4
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Example (All 3-crossings in T )
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Example (All 3-crossings in T )
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Example (All 3-crossings in T )
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Example (All 3-crossings in T )
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Example (All 3-crossings in T )
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Example (All 3-crossings in T )
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Example (All 3-crossings in T )
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Example (All 3-crossings in T )
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Example (All 3-crossings in T )
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Ordering on crossings
We order the set of all i-crossings by
γ1 � γ2 ⇔ γ1 is left of γ2
Example (Ordering of 3-crossings in T )
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Ordering on crossings
We order the set of all i-crossings by
γ1 � γ2 ⇔ γ1 is left of γ2
Example (Ordering of 3-crossings in T )
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Ordering on crossings
We order the set of all i-crossings by
γ1 � γ2 ⇔ γ1 is left of γ2
Example (Ordering of 3-crossings in T )
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Ordering on crossings
We order the set of all i-crossings by
γ1 � γ2 ⇔ γ1 is left of γ2
Example (Ordering of 3-crossings in T )
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Ordering on crossings
We order the set of all i-crossings by
γ1 � γ2 ⇔ γ1 is left of γ2
Example (Ordering of 3-crossings in T )
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Ordering on crossings
We order the set of all i-crossings by
γ1 � γ2 ⇔ γ1 is left of γ2
Example (Ordering of 3-crossings in T )
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Ordering on crossings
We order the set of all i-crossings by
γ1 � γ2 ⇔ γ1 is left of γ2
Example (Ordering of 3-crossings in T )
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Ordering on crossings
We order the set of all i-crossings by
γ1 � γ2 ⇔ γ1 is left of γ2
Example (Ordering of 3-crossings in T )
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Vectors associated to crossings
To an i-crossing γ we associate the vector [γ] ∈ ZΦ+given by
(1 ≤ k < ` ≤ n)
[γ]αk,`−1=
1 if γ turns from the k-strip to the `-strip at 〈k , `〉,0 if γ does not turn at 〈k , `〉,− 1 if γ turns from the `-strip to the k-strip at 〈k , `〉.
Example
2
3
1
5
4
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Vectors associated to crossings
To an i-crossing γ we associate the vector [γ] ∈ ZΦ+given by
(1 ≤ k < ` ≤ n)
[γ]αk,`−1=
1 if γ turns from the k-strip to the `-strip at 〈k , `〉,0 if γ does not turn at 〈k , `〉,− 1 if γ turns from the `-strip to the k-strip at 〈k , `〉.
Example
2
3
1
5
4
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Vectors associated to crossings
To an i-crossing γ we associate the vector [γ] ∈ ZΦ+given by
(1 ≤ k < ` ≤ n)
[γ]αk,`−1=
1 if γ turns from the k-strip to the `-strip at 〈k , `〉,0 if γ does not turn at 〈k , `〉,− 1 if γ turns from the `-strip to the k-strip at 〈k , `〉.
Example
2
3
1
5
[γ]αk,`−1= 1
4
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Vectors associated to crossings
To an i-crossing γ we associate the vector [γ] ∈ ZΦ+given by
(1 ≤ k < ` ≤ n)
[γ]αk,`−1=
1 if γ turns from the k-strip to the `-strip at 〈k , `〉,0 if γ does not turn at 〈k , `〉,− 1 if γ turns from the `-strip to the k-strip at 〈k , `〉.
Example
1
2
3
1
5
1[γ]αk,`−1
= 1
4
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Vectors associated to crossings
To an i-crossing γ we associate the vector [γ] ∈ ZΦ+given by
(1 ≤ k < ` ≤ n)
[γ]αk,`−1=
1 if γ turns from the k-strip to the `-strip at 〈k , `〉,0 if γ does not turn at 〈k , `〉,− 1 if γ turns from the `-strip to the k-strip at 〈k , `〉.
Example
1
2
3
1
5
1[γ]αk,`−1
= −1
4
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Vectors associated to crossings
To an i-crossing γ we associate the vector [γ] ∈ ZΦ+given by
(1 ≤ k < ` ≤ n)
[γ]αk,`−1=
1 if γ turns from the k-strip to the `-strip at 〈k , `〉,0 if γ does not turn at 〈k , `〉,− 1 if γ turns from the `-strip to the k-strip at 〈k , `〉.
Example
1
2
3
1
−1
5
1[γ]αk,`−1
= −1
4
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Vectors associated to crossings
To an i-crossing γ we associate the vector [γ] ∈ ZΦ+given by
(1 ≤ k < ` ≤ n)
[γ]αk,`−1=
1 if γ turns from the k-strip to the `-strip at 〈k , `〉,0 if γ does not turn at 〈k , `〉,− 1 if γ turns from the `-strip to the k-strip at 〈k , `〉.
Example
01
2
03
0
1
−1
5
0
1[γ]αk,`−1
= 0
0
4
0
0
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The Crossing Formula
For a ≤-Lusztig datum x we define for 1 ≤ k < l ≤ n
Fa (x , γ) :=∑〈k,`〉∈γa∈[k,`]
xαk,`−1−
∑〈k,`〉∈γa/∈[k,`]
[γ]αk,`−1=0
xαk,`−1.
Example
Let γ be the following 3-crossing
2
3
1
5
4
For a Lusztig datum x ∈ NT , we have
F3 (x , γ) = −x[2,3] + x[2,5] + x[2,4] − x[4,5] + x[1,4].
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The Crossing Formula
For a ≤-Lusztig datum x we define for 1 ≤ k < l ≤ n
Fa (x , γ) :=∑〈k,`〉∈γa∈[k,`]
xαk,`−1−
∑〈k,`〉∈γa/∈[k,`]
[γ]αk,`−1=0
xαk,`−1.
Example
Let γ be the following 3-crossing
2
3
1
5
4
For a Lusztig datum x ∈ NT , we have
F3 (x , γ) = −x[2,3] + x[2,5] + x[2,4] − x[4,5] + x[1,4].
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The Crossing Formula
For a ≤-Lusztig datum x we define for 1 ≤ k < l ≤ n
Fa (x , γ) :=∑〈k,`〉∈γa∈[k,`]
xαk,`−1−
∑〈k,`〉∈γa/∈[k,`]
[γ]αk,`−1=0
xαk,`−1.
Example
Let γ be the following 3-crossing
2
3
1
5
4
For a Lusztig datum x ∈ NT , we have
F3 (x , γ) = −x[2,3] + x[2,5] + x[2,4] − x[4,5] + x[1,4].
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The Crossing Formula
For a ≤-Lusztig datum x we define for 1 ≤ k < l ≤ n
Fa (x , γ) :=∑〈k,`〉∈γa∈[k,`]
xαk,`−1−
∑〈k,`〉∈γa/∈[k,`]
[γ]αk,`−1=0
xαk,`−1.
Example
Let γ be the following 3-crossing
2
3
1
5
4
For a Lusztig datum x ∈ NT , we have
F3 (x , γ) = −x[2,3] + x[2,5] + x[2,4] − x[4,5] + x[1,4].
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Crossing Formula (Genz, Koshevoy, S.)
Let a ∈ {1, . . . , n − 1} and let x be a ≤-Lusztig datum. Then
εa(x) = maxγ
Fa(x , γ)
andfa(x)− x = [γx ],
where γx is the maximal a-crossing with Fa(x , γx) = εa(x).
Remark
The above theorem was inspired by a result by Reineke for convexorderings adapted to quivers.
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Crossing Formula (Genz, Koshevoy, S.)
Let a ∈ {1, . . . , n − 1} and let x be a ≤-Lusztig datum. Then
εa(x) = maxγ
Fa(x , γ)
andfa(x)− x = [γx ],
where γx is the maximal a-crossing with Fa(x , γx) = εa(x).
Remark
The above theorem was inspired by a result by Reineke for convexorderings adapted to quivers.
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Crossing Formula (Genz, Koshevoy, S.)
Let a ∈ {1, . . . , n − 1} and let x be a ≤-Lusztig datum. Then
εa(x) = maxγ
Fa(x , γ)
andfa(x)− x = [γx ],
where γx is the maximal a-crossing with Fa(x , γx) = εa(x).
Remark
The above theorem was inspired by a result by Reineke for convexorderings adapted to quivers.
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Crossing Formula (Genz, Koshevoy, S.)
Let a ∈ {1, . . . , n − 1} and let x be a ≤-Lusztig datum. Then
εa(x) = maxγ
Fa(x , γ)
andfa(x)− x = [γx ],
where γx is the maximal a-crossing with Fa(x , γx) = εa(x).
Remark
The above theorem was inspired by a result by Reineke for convexorderings adapted to quivers.
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The set of Reineke crossings
Question:
Do all vectors associated to crossings appear asactions of Kashiwara operators?
Definition
We say γ ∈ Γa is a Reineke a-crossing if it satisfies the followingcondition: For any γi = 〈s, t〉 ∈ γ such that γi−1, γi and γi+1 lie inthe same strip sequence Ls we have
s > t if t ≤ a
s < t if a + 1 ≤ t.
We denote the set of all Reineke a-crossings by Ra
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The set of Reineke crossings
Question:
Do all vectors associated to crossings appear asactions of Kashiwara operators?
Definition
We say γ ∈ Γa is a Reineke a-crossing if it satisfies the followingcondition: For any γi = 〈s, t〉 ∈ γ such that γi−1, γi and γi+1 lie inthe same strip sequence Ls we have
s > t if t ≤ a
s < t if a + 1 ≤ t.
We denote the set of all Reineke a-crossings by Ra
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The set of Reineke crossings
Question:
Do all vectors associated to crossings appear asactions of Kashiwara operators?
Definition
We say γ ∈ Γa is a Reineke a-crossing if it satisfies the followingcondition: For any γi = 〈s, t〉 ∈ γ such that γi−1, γi and γi+1 lie inthe same strip sequence Ls we have
s > t if t ≤ a
s < t if a + 1 ≤ t.
We denote the set of all Reineke a-crossings by Ra
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The set of Reineke crossings
Question:
Do all vectors associated to crossings appear asactions of Kashiwara operators?
Definition
We say γ ∈ Γa is a Reineke a-crossing if it satisfies the followingcondition: For any γi = 〈s, t〉 ∈ γ such that γi−1, γi and γi+1 lie inthe same strip sequence Ls we have
s > t if t ≤ a
s < t if a + 1 ≤ t.
We denote the set of all Reineke a-crossings by Ra
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The set of Reineke crossings
Question:
Do all vectors associated to crossings appear asactions of Kashiwara operators?
Definition
We say γ ∈ Γa is a Reineke a-crossing if it satisfies the followingcondition: For any γi = 〈s, t〉 ∈ γ such that γi−1, γi and γi+1 lie inthe same strip sequence Ls we have
s > t if t ≤ a
s < t if a + 1 ≤ t.
We denote the set of all Reineke a-crossings by Ra
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In our running example all 3-crossings are Reineke 3-crossingsexcept:
2
3
1
5
4
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In our running example all 3-crossings are Reineke 3-crossingsexcept:
2
3
1
5
4
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In our running example all 3-crossings are Reineke 3-crossingsexcept:
2
3
1
5
4
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In our running example all 3-crossings are Reineke 3-crossingsexcept:
2
3
1
5
4
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The Set of Reineke vectors
Theorem (Genz, Koshevoy, S.)
We have{fax − x | x ∈ NT } = {[γ] | γ ∈ Ra}.
Definition
The subset Ra(T ) := {fax − x | x ∈ NT } ⊂ ZT is called the set ofa-Reineke vectors.
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The Set of Reineke vectors
Theorem (Genz, Koshevoy, S.)
We have{fax − x | x ∈ NT } = {[γ] | γ ∈ Ra}.
Definition
The subset Ra(T ) := {fax − x | x ∈ NT } ⊂ ZT is called the set ofa-Reineke vectors.
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Kashiwara’s parametrization
Each convex ordering ≤ on Φ+ yields another parametrizationS≤ ⊂ ZN of the canonical basis by counting the lengths of certaincanonical paths from the unique source to each element b in thecrystal graph.
By Berenstein-Zelevinsky and Littelmann the set S≤ is a polyhedralcone in ZN .
The cone S≤ is called the string cone (corresponding to ≤).
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Kashiwara’s parametrization
Each convex ordering ≤ on Φ+ yields another parametrizationS≤ ⊂ ZN of the canonical basis by counting the lengths of certaincanonical paths from the unique source to each element b in thecrystal graph.
By Berenstein-Zelevinsky and Littelmann the set S≤ is a polyhedralcone in ZN .
The cone S≤ is called the string cone (corresponding to ≤).
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Kashiwara’s parametrization
Each convex ordering ≤ on Φ+ yields another parametrizationS≤ ⊂ ZN of the canonical basis by counting the lengths of certaincanonical paths from the unique source to each element b in thecrystal graph.
By Berenstein-Zelevinsky and Littelmann the set S≤ is a polyhedralcone in ZN .
The cone S≤ is called the string cone (corresponding to ≤).
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Opposite orderings
For a convex ordering
β1 ≤ β2 ≤ . . . ≤ βN
on Φ+
we define another convex ordering ≤op by
βN ≤op βN−1 ≤op . . . ≤op β1.
Recall that for a tiling T
Ra(T ) := {fax − x | x ∈ NT } ⊂ ZT .
We have the following duality between Lusztig’s and Kashiwara’sparametrization:
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Opposite orderings
For a convex ordering
β1 ≤ β2 ≤ . . . ≤ βN
on Φ+ we define another convex ordering ≤op by
βN ≤op βN−1 ≤op . . . ≤op β1.
Recall that for a tiling T
Ra(T ) := {fax − x | x ∈ NT } ⊂ ZT .
We have the following duality between Lusztig’s and Kashiwara’sparametrization:
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Opposite orderings
For a convex ordering
β1 ≤ β2 ≤ . . . ≤ βN
on Φ+ we define another convex ordering ≤op by
βN ≤op βN−1 ≤op . . . ≤op β1.
Recall that for a tiling T
Ra(T ) := {fax − x | x ∈ NT } ⊂ ZT .
We have the following duality between Lusztig’s and Kashiwara’sparametrization:
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Opposite orderings
For a convex ordering
β1 ≤ β2 ≤ . . . ≤ βN
on Φ+ we define another convex ordering ≤op by
βN ≤op βN−1 ≤op . . . ≤op β1.
Recall that for a tiling T
Ra(T ) := {fax − x | x ∈ NT } ⊂ ZT .
We have the following duality between Lusztig’s and Kashiwara’sparametrization:
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The duality
Theorem (Genz, Koshevoy, S.)
Let ≤ be a convex ordering on Φ+ and let T be the tilingassociated to the convex ordering ≤op.
We have
S≤ = {v ∈ ZN | (v , x) ≥ 0 ∀x ∈ Ra(T ), ∀a}.
Here (, ) is the standard scalar product.
Remark
The defining inequalities obtained for S≤ by the theoremabove were already found by Gleizer-Postnikov.
In the special case of convex orderings ≤ adapted to quiversof type A, the duality between Lusztig’s and Kashiwara’sparametrization above was discovered by Zelikson using workof Reineke.
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The duality
Theorem (Genz, Koshevoy, S.)
Let ≤ be a convex ordering on Φ+ and let T be the tilingassociated to the convex ordering ≤op. We have
S≤ = {v ∈ ZN | (v , x) ≥ 0 ∀x ∈ Ra(T ), ∀a}.
Here (, ) is the standard scalar product.
Remark
The defining inequalities obtained for S≤ by the theoremabove were already found by Gleizer-Postnikov.
In the special case of convex orderings ≤ adapted to quiversof type A, the duality between Lusztig’s and Kashiwara’sparametrization above was discovered by Zelikson using workof Reineke.
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The duality
Theorem (Genz, Koshevoy, S.)
Let ≤ be a convex ordering on Φ+ and let T be the tilingassociated to the convex ordering ≤op. We have
S≤ = {v ∈ ZN | (v , x) ≥ 0 ∀x ∈ Ra(T ), ∀a}.
Here (, ) is the standard scalar product.
Remark
The defining inequalities obtained for S≤ by the theoremabove were already found by Gleizer-Postnikov.
In the special case of convex orderings ≤ adapted to quiversof type A, the duality between Lusztig’s and Kashiwara’sparametrization above was discovered by Zelikson using workof Reineke.
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The duality
Theorem (Genz, Koshevoy, S.)
Let ≤ be a convex ordering on Φ+ and let T be the tilingassociated to the convex ordering ≤op. We have
S≤ = {v ∈ ZN | (v , x) ≥ 0 ∀x ∈ Ra(T ), ∀a}.
Here (, ) is the standard scalar product.
Remark
The defining inequalities obtained for S≤ by the theoremabove were already found by Gleizer-Postnikov.
In the special case of convex orderings ≤ adapted to quiversof type A, the duality between Lusztig’s and Kashiwara’sparametrization above was discovered by Zelikson using workof Reineke.
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The duality
Theorem (Genz, Koshevoy, S.)
Let ≤ be a convex ordering on Φ+ and let T be the tilingassociated to the convex ordering ≤op. We have
S≤ = {v ∈ ZN | (v , x) ≥ 0 ∀x ∈ Ra(T ), ∀a}.
Here (, ) is the standard scalar product.
Remark
The defining inequalities obtained for S≤ by the theoremabove were already found by Gleizer-Postnikov.
In the special case of convex orderings ≤ adapted to quiversof type A, the duality between Lusztig’s and Kashiwara’sparametrization above was discovered by Zelikson using workof Reineke.
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Reduced double Bruhat cells
Let N be the set of upper triangular matrices in SLn(C) withdiagonal entries all equal to one.
For
S = {x1, x2, . . . , xk} ⊂ {1, 2, . . . , n},
let ∆S ∈ C[N] be the function associating to a matrix A the(chamber) minor corresponding to the columns x1, x2, . . . , xk andthe first k rows. The variety
Le,w0 = {A ∈ N | ∆{n−k,n−k+1,...,n}A 6= 0}
is called the reduced double Bruhat cell (associated to e,w0).
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Reduced double Bruhat cells
Let N be the set of upper triangular matrices in SLn(C) withdiagonal entries all equal to one. For
S = {x1, x2, . . . , xk} ⊂ {1, 2, . . . , n},
let ∆S ∈ C[N] be the function associating to a matrix A the(chamber) minor corresponding to the columns x1, x2, . . . , xk andthe first k rows. The variety
Le,w0 = {A ∈ N | ∆{n−k,n−k+1,...,n}A 6= 0}
is called the reduced double Bruhat cell (associated to e,w0).
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Reduced double Bruhat cells
Let N be the set of upper triangular matrices in SLn(C) withdiagonal entries all equal to one. For
S = {x1, x2, . . . , xk} ⊂ {1, 2, . . . , n},
let ∆S ∈ C[N] be the function associating to a matrix A the(chamber) minor corresponding to the columns x1, x2, . . . , xk andthe first k rows.
The variety
Le,w0 = {A ∈ N | ∆{n−k,n−k+1,...,n}A 6= 0}
is called the reduced double Bruhat cell (associated to e,w0).
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Reduced double Bruhat cells
Let N be the set of upper triangular matrices in SLn(C) withdiagonal entries all equal to one. For
S = {x1, x2, . . . , xk} ⊂ {1, 2, . . . , n},
let ∆S ∈ C[N] be the function associating to a matrix A the(chamber) minor corresponding to the columns x1, x2, . . . , xk andthe first k rows. The variety
Le,w0 = {A ∈ N | ∆{n−k,n−k+1,...,n}A 6= 0}
is called the reduced double Bruhat cell (associated to e,w0).
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Vertices of a tiling as chamber weights
We identify the vertices of the tiling with a subset of the chamberminors by collecting the indices of the strips running below eachvertex.
Example
2
3
1
5
4
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Vertices of a tiling as chamber weights
We identify the vertices of the tiling with a subset of the chamberminors by collecting the indices of the strips running below eachvertex.
Example
2
3
1
5
4
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Vertices of a tiling as chamber weights
We identify the vertices of the tiling with a subset of the chamberminors by collecting the indices of the strips running below eachvertex.
Example
2
3
1
5
x4
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Vertices of a tiling as chamber weights
We identify the vertices of the tiling with a subset of the chamberminors by collecting the indices of the strips running below eachvertex.
Example
2
2
3
1
3
5
x4
4
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Vertices of a tiling as chamber weights
We identify the vertices of the tiling with a subset of the chamberminors by collecting the indices of the strips running below eachvertex.
Example
2
3
1
5
∆{2,3,4}4
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The chamber minors corresponding to the vertices on the leftboundary of any tiling T equal the principal minors.
Thus ∆S = 1for all S on the left boundary of T .
Example
2
1 = ∆{1,2}
3
1 = ∆{1,2,3}
1
1 = ∆{1}
51 = ∆{1,2,3,4,5}
4
1 = ∆{1,2,3,4}
1 = ∆∅
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The chamber minors corresponding to the vertices on the leftboundary of any tiling T equal the principal minors. Thus ∆S = 1for all S on the left boundary of T .
Example
2
1 = ∆{1,2}
3
1 = ∆{1,2,3}
1
1 = ∆{1}
51 = ∆{1,2,3,4,5}
4
1 = ∆{1,2,3,4}
1 = ∆∅
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The chamber minors corresponding to the vertices on the leftboundary of any tiling T equal the principal minors. Thus ∆S = 1for all S on the left boundary of T .
Example
2
1 = ∆{1,2}
3
1 = ∆{1,2,3}
1
1 = ∆{1}
51 = ∆{1,2,3,4,5}
4
1 = ∆{1,2,3,4}
1 = ∆∅
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Cluster associated to tilings
The coordinate ring C[Le,w0 ] has the structure of a cluster algebraby Fomin-Zelevinsky.
This means that we have distinguishedoverlapping subsets of generators of C[Le,w0 ] (clusters).
Let V (T ) be the set of vertices of T not lying on the leftboundary. For each tiling T the set V (T ) of chamber minors formsa cluster in C[Le,w0 ].
To get from the cluster V (T≤) to the cluster V (T≤′) we have toapply a sequence of maps (mutations) corresponding to a sequenceof flips transforming ≤ to ≤′.
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Cluster associated to tilings
The coordinate ring C[Le,w0 ] has the structure of a cluster algebraby Fomin-Zelevinsky. This means that we have distinguishedoverlapping subsets of generators of C[Le,w0 ] (clusters).
Let V (T ) be the set of vertices of T not lying on the leftboundary. For each tiling T the set V (T ) of chamber minors formsa cluster in C[Le,w0 ].
To get from the cluster V (T≤) to the cluster V (T≤′) we have toapply a sequence of maps (mutations) corresponding to a sequenceof flips transforming ≤ to ≤′.
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Cluster associated to tilings
The coordinate ring C[Le,w0 ] has the structure of a cluster algebraby Fomin-Zelevinsky. This means that we have distinguishedoverlapping subsets of generators of C[Le,w0 ] (clusters).
Let V (T ) be the set of vertices of T not lying on the leftboundary.
For each tiling T the set V (T ) of chamber minors formsa cluster in C[Le,w0 ].
To get from the cluster V (T≤) to the cluster V (T≤′) we have toapply a sequence of maps (mutations) corresponding to a sequenceof flips transforming ≤ to ≤′.
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Cluster associated to tilings
The coordinate ring C[Le,w0 ] has the structure of a cluster algebraby Fomin-Zelevinsky. This means that we have distinguishedoverlapping subsets of generators of C[Le,w0 ] (clusters).
Let V (T ) be the set of vertices of T not lying on the leftboundary. For each tiling T the set V (T ) of chamber minors formsa cluster in C[Le,w0 ].
To get from the cluster V (T≤) to the cluster V (T≤′) we have toapply a sequence of maps (mutations) corresponding to a sequenceof flips transforming ≤ to ≤′.
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Cluster associated to tilings
The coordinate ring C[Le,w0 ] has the structure of a cluster algebraby Fomin-Zelevinsky. This means that we have distinguishedoverlapping subsets of generators of C[Le,w0 ] (clusters).
Let V (T ) be the set of vertices of T not lying on the leftboundary. For each tiling T the set V (T ) of chamber minors formsa cluster in C[Le,w0 ].
To get from the cluster V (T≤) to the cluster V (T≤′) we have toapply a sequence of maps (mutations) corresponding to a sequenceof flips transforming ≤ to ≤′.
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Flips and mutations
A flip of a convex ordering ≤ corresponds to a flip of a subhexagonof T≤.
Example
1
2
3
4
1
2
3
4
1
2
3
∆{3}
4
1
2
3
∆{2,4}
4
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Flips and mutations
A flip of a convex ordering ≤ corresponds to a flip of a subhexagonof T≤.
Example
1
2
3
4
1
2
3
4
1
2
3
∆{3}
4
1
2
3
∆{2,4}
4
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Parametrizations of canonical bases for cluster algebras
Recall that N can be obtained from Le,w0 by allowing the chamberminors to vanish which correspond to the vertices on the rightboundary of any tiling T .
Gross-Hacking-Keel-Kontsevich construct a canonical basis B forC[N].For each cluster X we have a parametrization of B by a polyhedralcone CX in ZX defined via a tropical function W :
CX := {v ∈ ZX |W |ZX (v) ≥ 0}.
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Parametrizations of canonical bases for cluster algebras
Recall that N can be obtained from Le,w0 by allowing the chamberminors to vanish which correspond to the vertices on the rightboundary of any tiling T .Gross-Hacking-Keel-Kontsevich construct a canonical basis B forC[N].
For each cluster X we have a parametrization of B by a polyhedralcone CX in ZX defined via a tropical function W :
CX := {v ∈ ZX |W |ZX (v) ≥ 0}.
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Parametrizations of canonical bases for cluster algebras
Recall that N can be obtained from Le,w0 by allowing the chamberminors to vanish which correspond to the vertices on the rightboundary of any tiling T .Gross-Hacking-Keel-Kontsevich construct a canonical basis B forC[N].For each cluster X we have a parametrization of B by a polyhedralcone CX in ZX defined via a tropical function W :
CX := {v ∈ ZX |W |ZX (v) ≥ 0}.
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Parametrizations of canonical bases for cluster algebras
Recall that N can be obtained from Le,w0 by allowing the chamberminors to vanish which correspond to the vertices on the rightboundary of any tiling T .Gross-Hacking-Keel-Kontsevich construct a canonical basis B forC[N].For each cluster X we have a parametrization of B by a polyhedralcone CX in ZX defined via a tropical function W :
CX := {v ∈ ZX |W |ZX (v) ≥ 0}.
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Here W =∑
a∈{1,2,...,n−1}Wa and for a tiling Ta such that the tile〈a, a + 1〉 intersects the right boundary with two edges
Wa|ZV (Ta)(v) := (v , e∆{n−a,n−a+1,...,n}).
For two clusters X , Y , there is a natural piecewise-lineartransformation between ZX and ZY which can be used to computeW |ZY from W |ZX .
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Here W =∑
a∈{1,2,...,n−1}Wa and for a tiling Ta such that the tile〈a, a + 1〉 intersects the right boundary with two edges
Wa|ZV (Ta)(v) := (v , e∆{n−a,n−a+1,...,n}).
For two clusters X , Y , there is a natural piecewise-lineartransformation between ZX and ZY which can be used to computeW |ZY from W |ZX .
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Here W =∑
a∈{1,2,...,n−1}Wa and for a tiling Ta such that the tile〈a, a + 1〉 intersects the right boundary with two edges
Wa|ZV (Ta)(v) := (v , e∆{n−a,n−a+1,...,n}).
For two clusters X , Y , there is a natural piecewise-lineartransformation between ZX and ZY which can be used to computeW |ZY from W |ZX .
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The tropical Chamber Ansatz
To relate the tropical function W (potential function) to thecrystal operations on NT , we define a bijection
CAT : ZV (T ) → ZT
v = (vS)S∈V (T ) 7→ (CAT (v)T )T∈T ,
given by the tropicalisation of Berenstein-Fomin-Zelevinsky’sChamber Ansatz.I.e. for a tile T ∈ T , we have
vu(T )
v`(T ) T
vo(T )
vr(T ), CAT (v)T = v`(T ) + vr(T ) − vo(T ) − vu(T ).
.
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The tropical Chamber Ansatz
To relate the tropical function W (potential function) to thecrystal operations on NT , we define a bijection
CAT : ZV (T ) → ZT
v = (vS)S∈V (T ) 7→ (CAT (v)T )T∈T ,
given by the tropicalisation of Berenstein-Fomin-Zelevinsky’sChamber Ansatz.
I.e. for a tile T ∈ T , we have
vu(T )
v`(T ) T
vo(T )
vr(T ), CAT (v)T = v`(T ) + vr(T ) − vo(T ) − vu(T ).
.
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The tropical Chamber Ansatz
To relate the tropical function W (potential function) to thecrystal operations on NT , we define a bijection
CAT : ZV (T ) → ZT
v = (vS)S∈V (T ) 7→ (CAT (v)T )T∈T ,
given by the tropicalisation of Berenstein-Fomin-Zelevinsky’sChamber Ansatz.I.e. for a tile T ∈ T , we have
vu(T )
v`(T ) T
vo(T )
vr(T ), CAT (v)T = v`(T ) + vr(T ) − vo(T ) − vu(T ).
.
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Crystal operations and potential functions
Recall that for a tiling T
Ra(T ) := {fax − x | x ∈ NT } ⊂ ZT .
We obtain the following relation of the crystal operations onLusztig’s parametrizations and the parametrizations of theGHKK-canonical basis for C[N].
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Crystal operations and potential functions
Recall that for a tiling T
Ra(T ) := {fax − x | x ∈ NT } ⊂ ZT .
We obtain the following relation of the crystal operations onLusztig’s parametrizations and the parametrizations of theGHKK-canonical basis for C[N].
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Theorem (Genz,Koshevoy,S.)
We have
Wa|ZT≤ (v) = min{(CA−1T≤op
(x), v) | x ∈ Ra(T≤op)}.
We obtain as a direct consequence .
Corollary
CV (T≤) = CA∗T≤(ST≤
).
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Theorem (Genz,Koshevoy,S.)
We have
Wa|ZT≤ (v) = min{(CA−1T≤op
(x), v) | x ∈ Ra(T≤op)}.
We obtain as a direct consequence
.
Corollary
CV (T≤) = CA∗T≤(ST≤
).
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Theorem (Genz,Koshevoy,S.)
We have
Wa|ZT≤ (v) = min{(CA−1T≤op
(x), v) | x ∈ Ra(T≤op)}.
We obtain as a direct consequence .
Corollary
CV (T≤) = CA∗T≤(ST≤
).
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Remark
A unimodular transformation from the weighted string cone forflag and Schubert varieties to the respective cone arising from thepotential functions of the appropriate cluster varieties was recentlyobtained by Bossinger-Fourier.
THANK YOU!!
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Remark
A unimodular transformation from the weighted string cone forflag and Schubert varieties to the respective cone arising from thepotential functions of the appropriate cluster varieties was recentlyobtained by Bossinger-Fourier.
THANK YOU!!
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