unimodality of bruhat intervals
TRANSCRIPT
![Page 1: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/1.jpg)
Unimodality of Bruhat intervals
Gaston Burrull
The University of Sydney
December 7, 2021
65th Annual Meeting of the AustMS
![Page 2: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/2.jpg)
Plan
▶ Overview about unimodality and top-heaviness
▶ The Top-Heavy Conjecture
▶ The affine Weyl group
▶ Towards unimodality
![Page 3: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/3.jpg)
Plan
▶ Overview about unimodality and top-heaviness
▶ The Top-Heavy Conjecture
▶ The affine Weyl group
▶ Towards unimodality
![Page 4: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/4.jpg)
Plan
▶ Overview about unimodality and top-heaviness
▶ The Top-Heavy Conjecture
▶ The affine Weyl group
▶ Towards unimodality
![Page 5: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/5.jpg)
Plan
▶ Overview about unimodality and top-heaviness
▶ The Top-Heavy Conjecture
▶ The affine Weyl group
▶ Towards unimodality
![Page 6: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/6.jpg)
Introduction
Let us take look at the following graph.
By removing a vertex we obtain smaller subgraphs.
![Page 7: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/7.jpg)
Introduction
Let us take look at the following graph.
By removing a vertex we obtain smaller subgraphs.
![Page 8: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/8.jpg)
Introduction
Let us take look at the following graph.
By removing a vertex we obtain smaller subgraphs.
![Page 9: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/9.jpg)
Introduction
Let us take look at the following graph.
By removing a vertex we obtain smaller subgraphs.
![Page 10: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/10.jpg)
Introduction
Let us take look at the following graph.
By removing a vertex we obtain smaller subgraphs.
![Page 11: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/11.jpg)
Introduction
Let us take look at the following graph.
By removing a vertex we obtain smaller subgraphs.
![Page 12: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/12.jpg)
Introduction
Let us take look at the following graph.
By removing a vertex we obtain smaller subgraphs.
![Page 13: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/13.jpg)
Introduction
Let us take look at the following graph.
By removing a vertex we obtain smaller subgraphs.
![Page 14: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/14.jpg)
Introduction
We observe a “unimodal” and “top-heavy” behaviour
![Page 15: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/15.jpg)
Introduction
Unimodal and top-heavy behaviours also appear in partitions
![Page 16: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/16.jpg)
The Top-Heavy Conjecture
Top-Heavy Conjecture (Dowling and Wilson, 1974) (forvector spaces). Let V be a vector space and E a finite set ofvectors which span V .
For i ≤ dim(V ), let
bi = ♯
{i-dimensional subspaces of V
which are spanned by vectors of E
}.
Then
bi ≤ bn−i , for all i ≤ dim(V )
2.
![Page 17: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/17.jpg)
The Top-Heavy Conjecture
Top-Heavy Conjecture (Dowling and Wilson, 1974) (forvector spaces). Let V be a vector space and E a finite set ofvectors which span V . For i ≤ dim(V ), let
bi = ♯
{i-dimensional subspaces of V
which are spanned by vectors of E
}.
Then
bi ≤ bn−i , for all i ≤ dim(V )
2.
![Page 18: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/18.jpg)
The Top-Heavy Conjecture
Top-Heavy Conjecture (Dowling and Wilson, 1974) (forvector spaces). Let V be a vector space and E a finite set ofvectors which span V . For i ≤ dim(V ), let
bi = ♯
{i-dimensional subspaces of V
which are spanned by vectors of E
}.
Then
bi ≤ bn−i , for all i ≤ dim(V )
2.
![Page 19: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/19.jpg)
The Top-Heavy Conjecture: An example
Let V = R3 and
E =
100
,
000
,
001
,
111
.
Then we have b0 = 1, b1 = 4, b2 = 6, and b3 = 1.
Note1 = b0 ≤ b3 = 1,
and4 = b1 ≤ b2 = 6.
![Page 20: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/20.jpg)
The Top-Heavy Conjecture: An example
Let V = R3 and
E =
100
,
000
,
001
,
111
.
Then we have b0 = 1, b1 = 4, b2 = 6, and b3 = 1.
Note1 = b0 ≤ b3 = 1,
and4 = b1 ≤ b2 = 6.
![Page 21: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/21.jpg)
The Top-Heavy Conjecture: An example
Let V = R3 and
E =
100
,
000
,
001
,
111
.
Then we have b0 = 1, b1 = 4, b2 = 6, and b3 = 1.
Note1 = b0 ≤ b3 = 1,
and4 = b1 ≤ b2 = 6.
![Page 22: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/22.jpg)
The Top-Heavy Conjecture
A matroid is a structure that generalizes the notion of linearindependent sets.
It borrows and abstracts notions from graphtheory, linear algebra, and number theory.
Top-Heavy Conjecture (Dowling and Wilson, 1974). Let Mbe matroid. For i ≤ rank(M), let bi be the number of elements ofM of rank i . Then
bi ≤ bn−i , for all i ≤ rank(V )
2.
This was recently proved (Braden, Huh, Matherine, Proudfood,and Wang, 2020).
![Page 23: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/23.jpg)
The Top-Heavy Conjecture
A matroid is a structure that generalizes the notion of linearindependent sets. It borrows and abstracts notions from graphtheory, linear algebra, and number theory.
Top-Heavy Conjecture (Dowling and Wilson, 1974). Let Mbe matroid. For i ≤ rank(M), let bi be the number of elements ofM of rank i . Then
bi ≤ bn−i , for all i ≤ rank(V )
2.
This was recently proved (Braden, Huh, Matherine, Proudfood,and Wang, 2020).
![Page 24: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/24.jpg)
The Top-Heavy Conjecture
A matroid is a structure that generalizes the notion of linearindependent sets. It borrows and abstracts notions from graphtheory, linear algebra, and number theory.
Top-Heavy Conjecture (Dowling and Wilson, 1974). Let Mbe matroid. For i ≤ rank(M), let bi be the number of elements ofM of rank i .
Then
bi ≤ bn−i , for all i ≤ rank(V )
2.
This was recently proved (Braden, Huh, Matherine, Proudfood,and Wang, 2020).
![Page 25: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/25.jpg)
The Top-Heavy Conjecture
A matroid is a structure that generalizes the notion of linearindependent sets. It borrows and abstracts notions from graphtheory, linear algebra, and number theory.
Top-Heavy Conjecture (Dowling and Wilson, 1974). Let Mbe matroid. For i ≤ rank(M), let bi be the number of elements ofM of rank i . Then
bi ≤ bn−i , for all i ≤ rank(V )
2.
This was recently proved (Braden, Huh, Matherine, Proudfood,and Wang, 2020).
![Page 26: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/26.jpg)
The Top-Heavy Conjecture
A matroid is a structure that generalizes the notion of linearindependent sets. It borrows and abstracts notions from graphtheory, linear algebra, and number theory.
Top-Heavy Conjecture (Dowling and Wilson, 1974). Let Mbe matroid. For i ≤ rank(M), let bi be the number of elements ofM of rank i . Then
bi ≤ bn−i , for all i ≤ rank(V )
2.
This was recently proved (Braden, Huh, Matherine, Proudfood,and Wang, 2020).
![Page 27: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/27.jpg)
The affine Weyl group
Let V be a m-dimensional Euclidean vector space with innerproduct (−,−).
A root system Φ is a spanning set of vectors of V with goodproperties, the two main ones are:
▶ The only scalar multiples of a root α ∈ Φ that belong to Φare α itself and −α.
▶ For every root α ∈ Φ, the set Φ is closed under reflection sαthrough the hyperplane perpendicular to α.
![Page 28: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/28.jpg)
The affine Weyl group
Let V be a m-dimensional Euclidean vector space with innerproduct (−,−).
A root system Φ is a spanning set of vectors of V with goodproperties, the two main ones are:
▶ The only scalar multiples of a root α ∈ Φ that belong to Φare α itself and −α.
▶ For every root α ∈ Φ, the set Φ is closed under reflection sαthrough the hyperplane perpendicular to α.
![Page 29: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/29.jpg)
The affine Weyl group
Let V be a m-dimensional Euclidean vector space with innerproduct (−,−).
A root system Φ is a spanning set of vectors of V with goodproperties, the two main ones are:
▶ The only scalar multiples of a root α ∈ Φ that belong to Φare α itself and −α.
▶ For every root α ∈ Φ, the set Φ is closed under reflection sαthrough the hyperplane perpendicular to α.
![Page 30: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/30.jpg)
The affine Weyl group
Let V be a m-dimensional Euclidean vector space with innerproduct (−,−).
A root system Φ is a spanning set of vectors of V with goodproperties, the two main ones are:
▶ The only scalar multiples of a root α ∈ Φ that belong to Φare α itself and −α.
▶ For every root α ∈ Φ, the set Φ is closed under reflection sαthrough the hyperplane perpendicular to α.
![Page 31: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/31.jpg)
Example: Type A2
An example of a root system of rank 2 is the following
![Page 32: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/32.jpg)
Example: Type A2
An example of a root system of rank 2 is the following
![Page 33: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/33.jpg)
The affine Weyl group
Let α ∈ Φ ⊂ V be a root.
We define the dual root α∨ ∈ V ∗ of αby
⟨α∨, λ⟩ = 2(α, λ)
(α, α),
for every λ ∈ V .
By following simple rules on the root system one can choose asubset Φ+ of positive roots of Φ.
We define the dominant chamber C+ of V by
C+ = {v ∈ V | ⟨α∨, v⟩ > 0 for every α ∈ Φ+}.
![Page 34: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/34.jpg)
The affine Weyl group
Let α ∈ Φ ⊂ V be a root. We define the dual root α∨ ∈ V ∗ of αby
⟨α∨, λ⟩ = 2(α, λ)
(α, α),
for every λ ∈ V .
By following simple rules on the root system one can choose asubset Φ+ of positive roots of Φ.
We define the dominant chamber C+ of V by
C+ = {v ∈ V | ⟨α∨, v⟩ > 0 for every α ∈ Φ+}.
![Page 35: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/35.jpg)
The affine Weyl group
Let α ∈ Φ ⊂ V be a root. We define the dual root α∨ ∈ V ∗ of αby
⟨α∨, λ⟩ = 2(α, λ)
(α, α),
for every λ ∈ V .
By following simple rules on the root system one can choose asubset Φ+ of positive roots of Φ.
We define the dominant chamber C+ of V by
C+ = {v ∈ V | ⟨α∨, v⟩ > 0 for every α ∈ Φ+}.
![Page 36: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/36.jpg)
The affine Weyl group
Let α ∈ Φ ⊂ V be a root. We define the dual root α∨ ∈ V ∗ of αby
⟨α∨, λ⟩ = 2(α, λ)
(α, α),
for every λ ∈ V .
By following simple rules on the root system one can choose asubset Φ+ of positive roots of Φ.
We define the dominant chamber C+ of V by
C+ = {v ∈ V | ⟨α∨, v⟩ > 0 for every α ∈ Φ+}.
![Page 37: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/37.jpg)
The affine Weyl group
Let α ∈ Φ ⊂ V be a root. We define the dual root α∨ ∈ V ∗ of αby
⟨α∨, λ⟩ = 2(α, λ)
(α, α),
for every λ ∈ V .
By following simple rules on the root system one can choose asubset Φ+ of positive roots of Φ.
We define the dominant chamber C+ of V by
C+ = {v ∈ V | ⟨α∨, v⟩ > 0 for every α ∈ Φ+}.
![Page 38: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/38.jpg)
The affine Weyl group
Let us denote by Aff(V ) the group of all rigid motions on V .
TheWeyl group Wf associated with Φ is the subgroup of Aff(V )generated by all the sα where α ∈ Φ.
The affine Weyl group W associated with Φ is the subgroup ofAff(V ) generated by all sα,d for α ∈ Φ and d ∈ Z, where
sα,d(µ) = µ− ⟨α∨, µ⟩α+ dα
is the reflection with respect to the affine hyperplane Hα,d given by
Hα∨,m = {v ∈ V | ⟨α∨, v⟩ = m}.
![Page 39: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/39.jpg)
The affine Weyl group
Let us denote by Aff(V ) the group of all rigid motions on V . TheWeyl group Wf associated with Φ is the subgroup of Aff(V )generated by all the sα where α ∈ Φ.
The affine Weyl group W associated with Φ is the subgroup ofAff(V ) generated by all sα,d for α ∈ Φ and d ∈ Z, where
sα,d(µ) = µ− ⟨α∨, µ⟩α+ dα
is the reflection with respect to the affine hyperplane Hα,d given by
Hα∨,m = {v ∈ V | ⟨α∨, v⟩ = m}.
![Page 40: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/40.jpg)
The affine Weyl group
Let us denote by Aff(V ) the group of all rigid motions on V . TheWeyl group Wf associated with Φ is the subgroup of Aff(V )generated by all the sα where α ∈ Φ.
The affine Weyl group W associated with Φ is the subgroup ofAff(V ) generated by all sα,d for α ∈ Φ and d ∈ Z
, where
sα,d(µ) = µ− ⟨α∨, µ⟩α+ dα
is the reflection with respect to the affine hyperplane Hα,d given by
Hα∨,m = {v ∈ V | ⟨α∨, v⟩ = m}.
![Page 41: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/41.jpg)
The affine Weyl group
Let us denote by Aff(V ) the group of all rigid motions on V . TheWeyl group Wf associated with Φ is the subgroup of Aff(V )generated by all the sα where α ∈ Φ.
The affine Weyl group W associated with Φ is the subgroup ofAff(V ) generated by all sα,d for α ∈ Φ and d ∈ Z, where
sα,d(µ) = µ− ⟨α∨, µ⟩α+ dα
is the reflection with respect to the affine hyperplane Hα,d given by
Hα∨,m = {v ∈ V | ⟨α∨, v⟩ = m}.
![Page 42: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/42.jpg)
The affine Weyl group
Let us denote by Aff(V ) the group of all rigid motions on V . TheWeyl group Wf associated with Φ is the subgroup of Aff(V )generated by all the sα where α ∈ Φ.
The affine Weyl group W associated with Φ is the subgroup ofAff(V ) generated by all sα,d for α ∈ Φ and d ∈ Z, where
sα,d(µ) = µ− ⟨α∨, µ⟩α+ dα
is the reflection with respect to the affine hyperplane Hα,d given by
Hα∨,m = {v ∈ V | ⟨α∨, v⟩ = m}.
![Page 43: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/43.jpg)
The affine Weyl group
Let us denote by Aff(V ) the group of all rigid motions on V . TheWeyl group Wf associated with Φ is the subgroup of Aff(V )generated by all the sα where α ∈ Φ.
The affine Weyl group W associated with Φ is the subgroup ofAff(V ) generated by all sα,d for α ∈ Φ and d ∈ Z, where
sα,d(µ) = µ− ⟨α∨, µ⟩α+ dα
is the reflection with respect to the affine hyperplane Hα,d given by
Hα∨,m = {v ∈ V | ⟨α∨, v⟩ = m}.
![Page 44: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/44.jpg)
The affine Weyl group
Equivalently, the affine Weyl group W associated with Φ is thesemidirect product Wf ⋉ ZΦ.
The elements of W will berepresented by pairs (λ,w) where λ ∈ ZΦ and w ∈ Wf .The groupmultiplication is given by
(λ,w) · (λ′,w ′) = (λ+ wλ′,w · w ′).
The group ZΦ is a lattice and its called the root lattice.
![Page 45: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/45.jpg)
The affine Weyl group
Equivalently, the affine Weyl group W associated with Φ is thesemidirect product Wf ⋉ ZΦ. The elements of W will berepresented by pairs (λ,w) where λ ∈ ZΦ and w ∈ Wf .
The groupmultiplication is given by
(λ,w) · (λ′,w ′) = (λ+ wλ′,w · w ′).
The group ZΦ is a lattice and its called the root lattice.
![Page 46: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/46.jpg)
The affine Weyl group
Equivalently, the affine Weyl group W associated with Φ is thesemidirect product Wf ⋉ ZΦ. The elements of W will berepresented by pairs (λ,w) where λ ∈ ZΦ and w ∈ Wf .The groupmultiplication is given by
(λ,w) · (λ′,w ′) = (λ+ wλ′,w · w ′).
The group ZΦ is a lattice and its called the root lattice.
![Page 47: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/47.jpg)
The affine Weyl group
Equivalently, the affine Weyl group W associated with Φ is thesemidirect product Wf ⋉ ZΦ. The elements of W will berepresented by pairs (λ,w) where λ ∈ ZΦ and w ∈ Wf .The groupmultiplication is given by
(λ,w) · (λ′,w ′) = (λ+ wλ′,w · w ′).
The group ZΦ is a lattice and its called the root lattice.
![Page 48: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/48.jpg)
Example: Type A2
Let us see the root system from before
In this case the finite Weyl group
Wf = ⟨sα, sβ : s2α = s2β = id, (sαsβ)3 = id⟩
is isomorphic to the symmetric group S3 = Sym({1, 2, 3})consisting of 6 elements.
![Page 49: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/49.jpg)
Example: Type A2
Let us see the root system from before
In this case the finite Weyl group
Wf = ⟨sα, sβ : s2α = s2β = id, (sαsβ)3 = id⟩
is isomorphic to the symmetric group S3 = Sym({1, 2, 3})consisting of 6 elements.
![Page 50: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/50.jpg)
Example: Type A2
Let us see the root system from before
In this case the finite Weyl group
Wf = ⟨sα, sβ : s2α = s2β = id, (sαsβ)3 = id⟩
is isomorphic to the symmetric group S3 = Sym({1, 2, 3})consisting of 6 elements.
![Page 51: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/51.jpg)
Example: Type A2
Let us add all the affine hyperplanes corresponding to sα,d forα ∈ Φ and d ∈ Z.
![Page 52: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/52.jpg)
Example: Type A2
Let us add all the affine hyperplanes corresponding to sα,d forα ∈ Φ and d ∈ Z.
The points in the root lattice ZΦ are the orange circles in thepicture.
![Page 53: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/53.jpg)
Example: Type A2
Let us add all the affine hyperplanes corresponding to sα,d forα ∈ Φ and d ∈ Z.
The points in the root lattice ZΦ are the orange circles in thepicture.
![Page 54: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/54.jpg)
The affine Weyl group
Consider A the set of alcoves of V with respect to W
, which is theset of connected components of
V∖⋃
{Hα∨ , d : α ∈ Φ and d ∈ Z} .
Let A+ ∈ A be the fundamental alcove: The unique alcovecontained in C+ whose closure contains the origin. There is abijection
W → Aw 7→ wA+.
![Page 55: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/55.jpg)
The affine Weyl group
Consider A the set of alcoves of V with respect to W , which is theset of connected components of
V∖⋃
{Hα∨ , d : α ∈ Φ and d ∈ Z} .
Let A+ ∈ A be the fundamental alcove: The unique alcovecontained in C+ whose closure contains the origin. There is abijection
W → Aw 7→ wA+.
![Page 56: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/56.jpg)
The affine Weyl group
Consider A the set of alcoves of V with respect to W , which is theset of connected components of
V∖⋃
{Hα∨ , d : α ∈ Φ and d ∈ Z} .
Let A+ ∈ A be the fundamental alcove:
The unique alcovecontained in C+ whose closure contains the origin. There is abijection
W → Aw 7→ wA+.
![Page 57: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/57.jpg)
The affine Weyl group
Consider A the set of alcoves of V with respect to W , which is theset of connected components of
V∖⋃
{Hα∨ , d : α ∈ Φ and d ∈ Z} .
Let A+ ∈ A be the fundamental alcove: The unique alcovecontained in C+ whose closure contains the origin.
There is abijection
W → Aw 7→ wA+.
![Page 58: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/58.jpg)
The affine Weyl group
Consider A the set of alcoves of V with respect to W , which is theset of connected components of
V∖⋃
{Hα∨ , d : α ∈ Φ and d ∈ Z} .
Let A+ ∈ A be the fundamental alcove: The unique alcovecontained in C+ whose closure contains the origin. There is abijection
W → Aw 7→ wA+.
![Page 59: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/59.jpg)
Towards unimodality
Let us define a length function ℓ : W → N. Given by:
ℓ(w) = ♯
{Hyperplanes separating the alcove wA+
from the alcove A+
}.
An alcove A is called dominant if it is contained in C+
Let us consider fW = Wf \W the set of minimal length rightWf -coset representatives of W . The set of dominant alcoves is inbijection with fW .
There is also a partial order in W called the Bruhat order, it isgenerated by the relation
tx < x
where t is a reflection with respect to an hyperplane in W , and thealcoves txA+ and A+ lie in the same side of such hyperplane.
![Page 60: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/60.jpg)
Towards unimodality
Let us define a length function ℓ : W → N. Given by:
ℓ(w) = ♯
{Hyperplanes separating the alcove wA+
from the alcove A+
}.
An alcove A is called dominant if it is contained in C+
Let us consider fW = Wf \W the set of minimal length rightWf -coset representatives of W . The set of dominant alcoves is inbijection with fW .
There is also a partial order in W called the Bruhat order, it isgenerated by the relation
tx < x
where t is a reflection with respect to an hyperplane in W , and thealcoves txA+ and A+ lie in the same side of such hyperplane.
![Page 61: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/61.jpg)
Towards unimodality
Let us define a length function ℓ : W → N. Given by:
ℓ(w) = ♯
{Hyperplanes separating the alcove wA+
from the alcove A+
}.
An alcove A is called dominant if it is contained in C+
Let us consider fW = Wf \W the set of minimal length rightWf -coset representatives of W . The set of dominant alcoves is inbijection with fW .
There is also a partial order in W called the Bruhat order, it isgenerated by the relation
tx < x
where t is a reflection with respect to an hyperplane in W , and thealcoves txA+ and A+ lie in the same side of such hyperplane.
![Page 62: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/62.jpg)
Towards unimodality
Let us define a length function ℓ : W → N. Given by:
ℓ(w) = ♯
{Hyperplanes separating the alcove wA+
from the alcove A+
}.
An alcove A is called dominant if it is contained in C+
Let us consider fW = Wf \W the set of minimal length rightWf -coset representatives of W .
The set of dominant alcoves is inbijection with fW .
There is also a partial order in W called the Bruhat order, it isgenerated by the relation
tx < x
where t is a reflection with respect to an hyperplane in W , and thealcoves txA+ and A+ lie in the same side of such hyperplane.
![Page 63: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/63.jpg)
Towards unimodality
Let us define a length function ℓ : W → N. Given by:
ℓ(w) = ♯
{Hyperplanes separating the alcove wA+
from the alcove A+
}.
An alcove A is called dominant if it is contained in C+
Let us consider fW = Wf \W the set of minimal length rightWf -coset representatives of W . The set of dominant alcoves is inbijection with fW .
There is also a partial order in W called the Bruhat order, it isgenerated by the relation
tx < x
where t is a reflection with respect to an hyperplane in W , and thealcoves txA+ and A+ lie in the same side of such hyperplane.
![Page 64: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/64.jpg)
Towards unimodality
Let us define a length function ℓ : W → N. Given by:
ℓ(w) = ♯
{Hyperplanes separating the alcove wA+
from the alcove A+
}.
An alcove A is called dominant if it is contained in C+
Let us consider fW = Wf \W the set of minimal length rightWf -coset representatives of W . The set of dominant alcoves is inbijection with fW .
There is also a partial order in W called the Bruhat order,
it isgenerated by the relation
tx < x
where t is a reflection with respect to an hyperplane in W , and thealcoves txA+ and A+ lie in the same side of such hyperplane.
![Page 65: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/65.jpg)
Towards unimodality
Let us define a length function ℓ : W → N. Given by:
ℓ(w) = ♯
{Hyperplanes separating the alcove wA+
from the alcove A+
}.
An alcove A is called dominant if it is contained in C+
Let us consider fW = Wf \W the set of minimal length rightWf -coset representatives of W . The set of dominant alcoves is inbijection with fW .
There is also a partial order in W called the Bruhat order, it isgenerated by the relation
tx < x
where t is a reflection with respect to an hyperplane in W , and thealcoves txA+ and A+ lie in the same side of such hyperplane.
![Page 66: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/66.jpg)
Towards unimodality
Let us define a length function ℓ : W → N. Given by:
ℓ(w) = ♯
{Hyperplanes separating the alcove wA+
from the alcove A+
}.
An alcove A is called dominant if it is contained in C+
Let us consider fW = Wf \W the set of minimal length rightWf -coset representatives of W . The set of dominant alcoves is inbijection with fW .
There is also a partial order in W called the Bruhat order, it isgenerated by the relation
tx < x
where t is a reflection with respect to an hyperplane in W , and thealcoves txA+ and A+ lie in the same side of such hyperplane.
![Page 67: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/67.jpg)
Example: Type A2
![Page 68: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/68.jpg)
Towards unimodality
The Poincare polynomial of an interval [id,w ] ⊂ fW is given by
P([id,w ]) =∑y≤w
qℓ(y) =∑
fiqi .
An open question is. Are the Poincare polynomials for theintervals [id,w ] ⊂ fW unimodal?
A similar question for the whole group W is false in type A. Acounterexample comes from an element in the associated Schubertvariety X = Gr4(C12), where the corresponding Betti numbersb2i = fi (which count the number of cells of dimension 2i in X ) are
1,1,2,3,5,6,9,11,15,17,21,23,27,28,31,30,31,27,24,18,14,8,5,2,1.
![Page 69: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/69.jpg)
Towards unimodality
The Poincare polynomial of an interval [id,w ] ⊂ fW is given by
P([id,w ]) =∑y≤w
qℓ(y) =∑
fiqi .
An open question is. Are the Poincare polynomials for theintervals [id,w ] ⊂ fW unimodal?
A similar question for the whole group W is false in type A. Acounterexample comes from an element in the associated Schubertvariety X = Gr4(C12), where the corresponding Betti numbersb2i = fi (which count the number of cells of dimension 2i in X ) are
1,1,2,3,5,6,9,11,15,17,21,23,27,28,31,30,31,27,24,18,14,8,5,2,1.
![Page 70: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/70.jpg)
Towards unimodality
The Poincare polynomial of an interval [id,w ] ⊂ fW is given by
P([id,w ]) =∑y≤w
qℓ(y) =∑
fiqi .
An open question is. Are the Poincare polynomials for theintervals [id,w ] ⊂ fW unimodal?
A similar question for the whole group W is false in type A. Acounterexample comes from an element in the associated Schubertvariety X = Gr4(C12), where the corresponding Betti numbersb2i = fi (which count the number of cells of dimension 2i in X ) are
1,1,2,3,5,6,9,11,15,17,21,23,27,28,31,30,31,27,24,18,14,8,5,2,1.
![Page 71: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/71.jpg)
Towards unimodality
The Poincare polynomial of an interval [id,w ] ⊂ fW is given by
P([id,w ]) =∑y≤w
qℓ(y) =∑
fiqi .
An open question is. Are the Poincare polynomials for theintervals [id,w ] ⊂ fW unimodal?
A similar question for the whole group W is false in type A. Acounterexample comes from an element in the associated Schubertvariety X = Gr4(C12)
, where the corresponding Betti numbersb2i = fi (which count the number of cells of dimension 2i in X ) are
1,1,2,3,5,6,9,11,15,17,21,23,27,28,31,30,31,27,24,18,14,8,5,2,1.
![Page 72: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/72.jpg)
Towards unimodality
The Poincare polynomial of an interval [id,w ] ⊂ fW is given by
P([id,w ]) =∑y≤w
qℓ(y) =∑
fiqi .
An open question is. Are the Poincare polynomials for theintervals [id,w ] ⊂ fW unimodal?
A similar question for the whole group W is false in type A. Acounterexample comes from an element in the associated Schubertvariety X = Gr4(C12), where the corresponding Betti numbersb2i = fi (which count the number of cells of dimension 2i in X ) are
1,1,2,3,5,6,9,11,15,17,21,23,27,28,31,30,31,27,24,18,14,8,5,2,1.
![Page 73: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/73.jpg)
Towards unimodality
For [id,w ] ⊂ fW
P([id,w ]) =∑y≤w
qℓ(y) =∑
fiqi .
We know a partial results for W , and fW : The Poincarepolynomial is top-heavy, and it is unimodal in the first half.
Theorem (Bjorner and Ekhedal, 2009). The Betti numbersb2i for a “good stratified” variety X satisfy:
b2ℓ(w)−i ≤ b2ℓ(w)+i , for i ≤ n,
bi ≤ bi+2j , for 0 ≤ j ≤ n − i .
![Page 74: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/74.jpg)
Towards unimodality
For [id,w ] ⊂ fW
P([id,w ]) =∑y≤w
qℓ(y) =∑
fiqi .
We know a partial results for W , and fW : The Poincarepolynomial is top-heavy, and it is unimodal in the first half.
Theorem (Bjorner and Ekhedal, 2009). The Betti numbersb2i for a “good stratified” variety X satisfy:
b2ℓ(w)−i ≤ b2ℓ(w)+i , for i ≤ n,
bi ≤ bi+2j , for 0 ≤ j ≤ n − i .
![Page 75: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/75.jpg)
Towards unimodality
For [id,w ] ⊂ fW
P([id,w ]) =∑y≤w
qℓ(y) =∑
fiqi .
We know a partial results for W , and fW : The Poincarepolynomial is top-heavy, and it is unimodal in the first half.
Theorem (Bjorner and Ekhedal, 2009). The Betti numbersb2i for a “good stratified” variety X satisfy:
b2ℓ(w)−i ≤ b2ℓ(w)+i , for i ≤ n,
bi ≤ bi+2j , for 0 ≤ j ≤ n − i .
![Page 76: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/76.jpg)
Towards unimodality
For [id,w ] ⊂ fW
P([id,w ]) =∑y≤w
qℓ(y) =∑
fiqi .
A direct corollary of the previous theorem (by taking fi = b2i ) is:
fi ≤ fj , for i ≤ j ≤ ℓ(w)/2,
fi ≤ fℓ(w)−i , for i < ℓ(w)/2.
![Page 77: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/77.jpg)
Towards unimodality
For [id,w ] ⊂ fW
P([id,w ]) =∑y≤w
qℓ(y) =∑
fiqi .
A direct corollary of the previous theorem (by taking fi = b2i ) is:
fi ≤ fj , for i ≤ j ≤ ℓ(w)/2,
fi ≤ fℓ(w)−i , for i < ℓ(w)/2.
![Page 78: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/78.jpg)
Example: Type A2
The Poincare polynomial for [id,w ] in the picture
is1 + q + 2q2 + 2q3 + 3q4 + 3q5 + 4q6 + 2q7 + q8
![Page 79: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/79.jpg)
Example: Type A2
The Poincare polynomial for [id,w ] in the picture
is1 + q + 2q2 + 2q3 + 3q4 + 3q5 + 4q6 + 2q7 + q8
![Page 80: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/80.jpg)
Example: Type A2
The Poincare polynomial for [id,w ] in the picture
is1 + q + 2q2 + 2q3 + 3q4 + 3q5 + 4q6 + 2q7 + q8
![Page 81: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/81.jpg)
Example: Type A2
In bar graphics.
1 + q + 2q2 + 2q3 + 3q4 + 3q5 + 4q6 + 2q7 + q8
![Page 82: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/82.jpg)
Example: Type A2
In bar graphics.
1 + q + 2q2 + 2q3 + 3q4 + 3q5 + 4q6 + 2q7 + q8
![Page 83: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/83.jpg)
Example: Type F4
The Poincare polynomial for the dominant lattice interval [0, 2ρ].
![Page 84: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/84.jpg)
Example: Type F4
The Poincare polynomial for the dominant lattice interval [0, 2ρ].
![Page 85: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/85.jpg)
Example: Type F4
The Poincare polynomial for the interval [id, (2ρ, id)] ⊂ fW .
![Page 86: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/86.jpg)
Example: Type F4
The Poincare polynomial for the interval [id, (2ρ, id)] ⊂ fW .
![Page 87: Unimodality of Bruhat intervals](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fff049cbe0406b5363334/html5/thumbnails/87.jpg)
The End
Thank you!