Symmetric Group Tilings and Alternating Sign Matrices

symmetric group tilings The Waldspurger and Mienrenken Decompositions for Type A

James McKeown October 2016

University of Miami Outline

∙ Background, Waldspurger and Meinrenken Theorems ∙ Geometry ⇒ Permutations ∙ Permutations ⇒ Alternating Sign Matrices ∙ Alternating Sign Matrices ⇒ Geometry

1 Let G ⊂ O(n) be a ﬁnite group generated by reﬂections.

Reflection Groups

An element g ∈ O(n) is a reﬂection if it sends some nonzero vector α ∈ Rn to its negative and ﬁxes the hyperplane orthogonal to α pointwise.

2 Reflection Groups

An element g ∈ O(n) is a reﬂection if it sends some nonzero vector α ∈ Rn to its negative and ﬁxes the hyperplane orthogonal to α pointwise.

Let G ⊂ O(n) be a ﬁnite group generated by reﬂections.

2 Theorem (Coxeter): Let A be an arrangement in Rn of reﬂecting n hyperplanes for the reﬂection group G. Then G ↷ (R ∖ ∪H∈AH) freely and transitively on the chambers.

We pick one such chamber and call it the “weight cone” denoted CW. The

cone dual to CW we call the “root cone” and denote CR.

The Reflection Representation of Sn, and the Braid Arrangement

3 Figure: S3 ↷ R 2 3 Figure: S3 ↷ R = R /(1, 1, 1)

3 We pick one such chamber and call it the “weight cone” denoted CW. The

cone dual to CW we call the “root cone” and denote CR.

The Reflection Representation of Sn, and the Braid Arrangement

3 Figure: S3 ↷ R 2 3 Figure: S3 ↷ R = R /(1, 1, 1) Theorem (Coxeter): Let A be an arrangement in Rn of reﬂecting n hyperplanes for the reﬂection group G. Then G ↷ (R ∖ ∪H∈AH) freely and transitively on the chambers.

3 The Reflection Representation of Sn, and the Braid Arrangement

We pick one such chamber and call it the “weight cone” denoted CW. The

cone dual to CW we call the “root cone” and denote CR. 3 A first Example– Weights and Roots

α1

W2 α2

4 Weights and Roots

Fact (Coxeter): Cw is a simplicial cone.

Let w1, w2,..., wn be vectors generating the rays of CW [Jargon: called the “fundamental weights”] Then the dual cone is deﬁned as

n CR ∶= {x ∈ R ∶ (x, y) ≤ 0∀y ∈ CW}

Let α1, α2, . . . , αn be vectors which generate rays of CR [Jargon: called the “simple roots”]

5 The first example A2 (S3)

α1

W2 α2

2 −1 3w1 = −2α2 − 1α1 −α1 = 2w2−1w1 ( ) −1 2 3w2 = −1α2 − 2α1 −α2 = −1w2 + 2w1

6 Lengths of weights are then normalized so that

−1 ⎛ ∣ ∣ ∣ ∣ ⎞ ⎛ ∣ ∣ ∣ ∣ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ∣ ∣ ∣ ∣ ⎟ ⎜ ∣ ∣ ∣ ∣ ⎟ ⎜ ⎟ ⎜ ⎟ − ⎜ ⎟ = ⎜ ⎟ ⎜α1 α2 . . . αn⎟ ⎜w1 w2 ... wn⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ∣ ∣ ∣ ∣ ⎟ ⎜ ∣ ∣ ∣ ∣ ⎟ ⎝ ∣ ∣ ∣ ∣ ⎠ ⎝ ∣ ∣ ∣ ∣ ⎠

The matrix of α’s is called the Cartan Matrix. It gives the coordinates of the simple roots in the basis of fundamental weights.

Cartan Matrices: Roots in terms of Weights

In type A, it is conventional to let αi = ei − ei+1 so (αi, αi) = 2 ∀i

7 The matrix of α’s is called the Cartan Matrix. It gives the coordinates of the simple roots in the basis of fundamental weights.

Cartan Matrices: Roots in terms of Weights

In type A, it is conventional to let αi = ei − ei+1 so (αi, αi) = 2 ∀i

Lengths of weights are then normalized so that

7 Cartan Matrices: Roots in terms of Weights

In type A, it is conventional to let αi = ei − ei+1 so (αi, αi) = 2 ∀i

Lengths of weights are then normalized so that

The matrix of α’s is called the Cartan Matrix. It gives the coordinates of the simple roots in the basis of fundamental weights.

7 Waldspurger Matrices

Consider the reﬂection representation of the symmetric group

ϕ ∶ Sn Ð→ GLn−1(R)

Let D be the matrix with columns the fundamental weights in basis of the simple roots (i.e. the n − 1 × n − 1 inverse of the Cartan matrix).

W(g) ∶= [ϕ(1) − ϕ(g)]D

expressed in the coordinates of simple roots we will call the Waldspurger Matrix of g.

8 Take away: Waldspurger matrices give a tiling of the root cone!

For G an afﬁne reﬂection group acting on a Euclidean vector space V, and

˚AW ⊂ V the interior of a fundamental domain for the action of G (sometimes called a fundamental alcove) one has the following decomposition:

V = ⊔(1 − g)˚AW g∈G

Take away: Convex hulls of columns of Waldspurger matrices (and the zero vector) give a tiling of the Rn!

Waldspurger and Mienrenken Theorems

Waldspurger’s Theorem (2005):

For G a ﬁnite reﬂection group acting on a Euclidean vector space V,

CR the (closed) cone over the positive roots, and

˚CW ⊂ V the interior of a fundamental domain for the action of G (sometimes called the weight cone), one has the following decomposition:

CR = ⊔(1 − g)˚CW g∈G

Meinrenken’s Theorem (2007):

9 For G an afﬁne reﬂection group acting on a Euclidean vector space V, and

˚AW ⊂ V the interior of a fundamental domain for the action of G (sometimes called a fundamental alcove) one has the following decomposition:

V = ⊔(1 − g)˚AW g∈G

Take away: Convex hulls of columns of Waldspurger matrices (and the zero vector) give a tiling of the Rn!

Waldspurger and Mienrenken Theorems

Waldspurger’s Theorem (2005):

For G a ﬁnite reﬂection group acting on a Euclidean vector space V,

CR the (closed) cone over the positive roots, and

˚CW ⊂ V the interior of a fundamental domain for the action of G (sometimes called the weight cone), one has the following decomposition:

CR = ⊔(1 − g)˚CW g∈G Take away: Waldspurger matrices give a tiling of the root cone!

Meinrenken’s Theorem (2007):

9 Take away: Waldspurger matrices give a tiling of the root cone!

For G an afﬁne reﬂection group acting on a Euclidean vector space V, and

˚AW ⊂ V the interior of a fundamental domain for the action of G (sometimes called a fundamental alcove) one has the following decomposition:

V = ⊔(1 − g)˚AW g∈G

Take away: Convex hulls of columns of Waldspurger matrices (and the zero vector) give a tiling of the Rn!

Waldspurger and Mienrenken Theorems

Waldspurger’s Theorem (2005):

Translation: Cones over columns of Waldspurger matrices give a tiling of the root cone!

Meinrenken’s Theorem (2007):

9 Take away: Waldspurger matrices give a tiling of the root cone!

Take away: Convex hulls of columns of Waldspurger matrices (and the zero vector) give a tiling of the Rn!

Waldspurger and Mienrenken Theorems

Waldspurger’s Theorem (2005):

Translation: Cones over columns of Waldspurger matrices give a tiling of the root cone!

Meinrenken’s Theorem (2007):

For G an afﬁne reﬂection group acting on a Euclidean vector space V, and

˚AW ⊂ V the interior of a fundamental domain for the action of G (sometimes called a fundamental alcove) one has the following decomposition:

V = ⊔(1 − g)˚AW g∈G

9 Take away: Waldspurger matrices give a tiling of the root cone!

Waldspurger and Mienrenken Theorems

Waldspurger’s Theorem (2005):

Translation: Cones over columns of Waldspurger matrices give a tiling of the root cone!

Meinrenken’s Theorem (2007):

For G an afﬁne reﬂection group acting on a Euclidean vector space V, and

˚AW ⊂ V the interior of a fundamental domain for the action of G (sometimes called a fundamental alcove) one has the following decomposition:

V = ⊔(1 − g)˚AW g∈G

Take away: Convex hulls of columns of Waldspurger matrices (and the zero vector) give a tiling of the Rn!

9 The Waldspurger Decomposition for A2 (S3)

(12)

˚ CR = ⊔(1 − g)CW g∈G

(13) (123) α1

id (132) W1

W2 α2

(23)

10 The Waldspurger Decomposition for A2 (S3)

(12)

˚ CR = ⊔(1 − g)CW g∈G

(13) (123) α1

id (132) W1

W2 α2

(23)

10 The Waldspurger Decomposition for A2 (S3)

(12)

˚ CR = ⊔(1 − g)CW g∈G

(13) (123) α1

id (132) W1

W2 α2

(23)

10 The Waldspurger Decomposition for A2 (S3)

(12)

˚ CR = ⊔(1 − g)CW g∈G

(13) (123) α1

id (132) W1

W2 α2

(23)

10 The Waldspurger Decomposition for A2 (S3)

(12)

˚ CR = ⊔(1 − g)CW g∈G

(13) (123) α1

id (132) W1

W2 α2

(23)

10 The Waldspurger Decomposition for A2 (S3)

(12)

˚ CR = ⊔(1 − g)CW g∈G

(13) (123) α1

id (132) W1

W2 α2

(23)

10 The Mienrenken Decomposition for A2 (S3)

˚ V = ⊔(1 − g)AW g∈G

α1

W2 α2

11 The Mienrenken Decomposition for A2 (S3)

˚ V = ⊔(1 − g)AW g∈G

α1

W2 α2

11 The Mienrenken Decomposition for A2 (S3)

˚ V = ⊔(1 − g)AW g∈G

α1

W2 α2

11 The Waldspurger Decomposition for A3 (S4)

000 010

110 011

111

100 001

12 Part 2: Geometry ⇒ Permutations

Meinrenken Tile for A3 = S4

13 Meinrenken Tile for A3 = S4

Part 2: Geometry ⇒ Permutations

13 ⎧ R ⎫ ⎪ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 0 ⎞R ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟R ⎪ ⎪ ⎜ 1 ⎟ ⎜ 2 ⎟ ⎜ 2 ⎟ ⎜ 1 ⎟R ⎪ that is, 43512 ↦ ⎨a ⎜ ⎟ + b ⎜ ⎟ + c ⎜ ⎟ + d ⎜ ⎟Ra, b, c, d ∈ R≥ ⎬ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟R 0⎪ ⎪ 1 1 2 1 R ⎪ ⎪ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠R ⎪ ⎩ 0 0 1 1 R ⎭

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

14 ⎧ R ⎫ ⎪ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 0 ⎞R ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟R ⎪ ⎪ ⎜ 1 ⎟ ⎜ 2 ⎟ ⎜ 2 ⎟ ⎜ 1 ⎟R ⎪ that is, 43512 ↦ ⎨a ⎜ ⎟ + b ⎜ ⎟ + c ⎜ ⎟ + d ⎜ ⎟Ra, b, c, d ∈ R≥ ⎬ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟R 0⎪ ⎪ 1 1 2 1 R ⎪ ⎪ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠R ⎪ ⎩ 0 0 1 1 R ⎭

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

14 The Fundamental Transformation

Consider 43512 ∈ S5

1 1 1 0

1 2 2 1

1 1 2 1

0 0 1 1

⎧ R ⎫ ⎪ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 0 ⎞R ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟R ⎪ ⎪ ⎜ 1 ⎟ ⎜ 2 ⎟ ⎜ 2 ⎟ ⎜ 1 ⎟R ⎪ that is, 43512 ↦ ⎨a ⎜ ⎟ + b ⎜ ⎟ + c ⎜ ⎟ + d ⎜ ⎟Ra, b, c, d ∈ R≥ ⎬ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟R 0⎪ ⎪ 1 1 2 1 R ⎪ ⎪ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠R ⎪ ⎩ 0 0 1 1 R ⎭ 14 ∙ four copies of

S3 picture

∙ Codimension ↕ # cycles (c(n, k))

∙ NOT A CW- complex!

Classical Permutation Statistics, something weird...

(23)

(132) (234) (1342)

(13) *(13)(24)* (24) (1324) (1423) (134) (142)

(1234) (14) (1432) (123) (243)

(124) (143) (1243)

(12) (12)(34) (34)

15 ∙ Codimension ↕ # cycles (c(n, k))

∙ NOT A CW- complex!

Classical Permutation Statistics, something weird...

(23)

∙ four copies of

(132) (234) S3 picture (1342)

(13) *(13)(24)* (24) (1324) (1423) (134) (142)

(1234) (14) (1432) (123) (243)

(124) (143) (1243)

(12) (12)(34) (34)

15 ∙ NOT A CW- complex!

Classical Permutation Statistics, something weird...

(23)

∙ four copies of

(132) (234) S3 picture (1342)

∙ Codimension (13) *(13)(24)* (24) ↕ (1324) (1423) (134) (142) # cycles (c(n, k))

(1234) (14) (1432) (123) (243)

(124) (143) (1243)

(12) (12)(34) (34)

15 Classical Permutation Statistics, something weird...

(23)

∙ four copies of

(132) (234) S3 picture (1342)

∙ Codimension (13) *(13)(24)* (24) ↕ (1324) (1423) (134) (142) # cycles (c(n, k))

(1234) (14) (1432) (123) (243) ∙ NOT A CW- complex! (124) (143) (1243)

(12) (12)(34) (34)

15 ∙ Unimodal Motzkin Paths. ∙ Elements of the root lattice inside a certain polytope ∙ Abelian ideals in the nilradical of sln ∙ Young diagrams with hooklength less than n.

Interesting Bijections:

UM Vectors

Theorem: n There are exactly 2 possible columns of Waldspurger matrices of type An. We will call them UM vectors.

16 ∙ Unimodal Motzkin Paths. ∙ Elements of the root lattice inside a certain polytope ∙ Abelian ideals in the nilradical of sln ∙ Young diagrams with hooklength less than n.

UM Vectors

Theorem: n There are exactly 2 possible columns of Waldspurger matrices of type An. We will call them UM vectors.

Interesting Bijections:

16 ∙ Elements of the root lattice inside a certain polytope ∙ Abelian ideals in the nilradical of sln ∙ Young diagrams with hooklength less than n.

UM Vectors

Theorem: n There are exactly 2 possible columns of Waldspurger matrices of type An. We will call them UM vectors.

Interesting Bijections: ∙ Unimodal Motzkin Paths.

16 ∙ Abelian ideals in the nilradical of sln ∙ Young diagrams with hooklength less than n.

UM Vectors

Theorem: n There are exactly 2 possible columns of Waldspurger matrices of type An. We will call them UM vectors.

Interesting Bijections: ∙ Unimodal Motzkin Paths. ∙ Elements of the root lattice inside a certain polytope

16 ∙ Young diagrams with hooklength less than n.

UM Vectors

Theorem: n There are exactly 2 possible columns of Waldspurger matrices of type An. We will call them UM vectors.

Interesting Bijections: ∙ Unimodal Motzkin Paths. ∙ Elements of the root lattice inside a certain polytope ∙ Abelian ideals in the nilradical of sln

16 UM Vectors

Theorem: n There are exactly 2 possible columns of Waldspurger matrices of type An. We will call them UM vectors.

Interesting Bijections: ∙ Unimodal Motzkin Paths. ∙ Elements of the root lattice inside a certain polytope ∙ Abelian ideals in the nilradical of sln ∙ Young diagrams with hooklength less than n.

16 Warning: Note that WT (M) may be “over-determined” on the diagonal. In the case where M is a permutation matrix, but in general this need not be the case.

If an n × n matrix M has this property, we will say it is sum-symmetric

M ∈ SSn The map is linear and surjective, with kernel the diagonal matrices.

WT ∶ SSn ↠ Matn−1

The Waldspurger Transform applied to other Matrices

From an n × n matrix M, deﬁne the n − 1 × n − 1 matrix, WT (M) where ⎧ ⎪∑ M i ≤ j ⎪ a,b ⎪a≤i ⎪ > WT (M) ∶= ⎨b j . i,j ⎪ ≥ ⎪∑ Ma,b i j ⎪ ⎪a>i ⎩b≤j

17 If an n × n matrix M has this property, we will say it is sum-symmetric

M ∈ SSn The map is linear and surjective, with kernel the diagonal matrices.

WT ∶ SSn ↠ Matn−1

The Waldspurger Transform applied to other Matrices

From an n × n matrix M, deﬁne the n − 1 × n − 1 matrix, WT (M) where ⎧ ⎪∑ M i ≤ j ⎪ a,b ⎪a≤i ⎪ > WT (M) ∶= ⎨b j . i,j ⎪ ≥ ⎪∑ Ma,b i j ⎪ ⎪a>i ⎩b≤j Warning: Note that WT (M) may be “over-determined” on the diagonal. In the case where M is a permutation matrix, but in general this need not be the case.

17 If an n × n matrix M has this property, we will say it is sum-symmetric

M ∈ SSn The map is linear and surjective, with kernel the diagonal matrices.

WT ∶ SSn ↠ Matn−1

The Waldspurger Transform applied to other Matrices

From an n × n matrix M, deﬁne the n − 1 × n − 1 matrix, WT (M) where ⎧ ⎪∑ M i ≤ j ⎪ a,b ⎪a≤i WT ( ) ∶= ⎨b>j M i,j ⎪ . ⎪∑ M i ≥ j ⎪ a,b ⎪a>i ⎩b≤j

Proposition: WT (M) is well-deﬁned if and only if the ith row sum of M equals the ith column sum of M, for 1 ≤ i ≤ n.

17 The map is linear and surjective, with kernel the diagonal matrices.

WT ∶ SSn ↠ Matn−1

The Waldspurger Transform applied to other Matrices

From an n × n matrix M, deﬁne the n − 1 × n − 1 matrix, WT (M) where ⎧ ⎪∑ M i ≤ j ⎪ a,b ⎪a≤i WT ( ) ∶= ⎨b>j M i,j ⎪ . ⎪∑ M i ≥ j ⎪ a,b ⎪a>i ⎩b≤j

Proposition: WT (M) is well-deﬁned if and only if the ith row sum of M equals the ith column sum of M, for 1 ≤ i ≤ n.

If an n × n matrix M has this property, we will say it is sum-symmetric

M ∈ SSn

17 The Waldspurger Transform applied to other Matrices

From an n × n matrix M, deﬁne the n − 1 × n − 1 matrix, WT (M) where ⎧ ⎪∑ M i ≤ j ⎪ a,b ⎪a≤i WT ( ) ∶= ⎨b>j M i,j ⎪ . ⎪∑ M i ≥ j ⎪ a,b ⎪a>i ⎩b≤j

Proposition: WT (M) is well-deﬁned if and only if the ith row sum of M equals the ith column sum of M, for 1 ≤ i ≤ n.

If an n × n matrix M has this property, we will say it is sum-symmetric

M ∈ SSn The map is linear and surjective, with kernel the diagonal matrices.

WT ∶ SSn ↠ Matn−1

17 Part 2: Permutations ⇒ Alternating Sign Matrices

Waldspurger, Meinrenken and ???

18 Waldspurger, Meinrenken and ???

Part 2: Permutations ⇒ Alternating Sign Matrices

18 These matrices generalize permutation matrices and arise naturally when using Dodgson condensation to compute a determinant. n − 1 × n − 1 matrices with UM columns and rows with their maxes on the diagonal are in bijection with n × n ASMs via the WT map!

ASMs

An alternating sign matrix (or ASM) is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign.

19 n − 1 × n − 1 matrices with UM columns and rows with their maxes on the diagonal are in bijection with n × n ASMs via the WT map!

ASMs

19 ASMs

An alternating sign matrix (or ASM) is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices generalize permutation matrices and arise naturally when using Dodgson condensation to compute a determinant. n − 1 × n − 1 matrices with UM columns and rows with their maxes on the diagonal are in bijection with n × n ASMs via the WT map!

19 1 1 1 0 0

1 2 2 1 0

1 2 3 2 1

0 1 2 2 1

0 0 1 1 1

Fact (A. Lascoux, M. Schützenberger): Half the entropy of a permutation is its rank in the MacNeille completion of the Bruhat order– a distributive lattice with elements the alternating sign matrices, or ASMs.

Height=Entropy

Theorem:

1 n ∑ WT (g) = ∑(g(i) − i)2 1≤i,j≤n−1 2 i=1

20 Fact (A. Lascoux, M. Schützenberger): Half the entropy of a permutation is its rank in the MacNeille completion of the Bruhat order– a distributive lattice with elements the alternating sign matrices, or ASMs.

Height=Entropy

Theorem:

1 n ∑ WT (g) = ∑(g(i) − i)2 1≤i,j≤n−1 2 i=1

1 1 1 0 0

1 2 2 1 0

1 2 3 2 1

0 1 2 2 1

0 0 1 1 1

20 Height=Entropy

Theorem:

1 n ∑ WT (g) = ∑(g(i) − i)2 1≤i,j≤n−1 2 i=1

1 1 1 0 0

1 2 2 1 0

1 2 3 2 1

0 1 2 2 1

0 0 1 1 1

[1 1] 1 1

321 [1 0] [1 1] 1 1 0 1 312 231

[0 0] [1 0] 0 1 0 0 132 213 [0 0] 123 0 0

21 The MacNielle Completion of the Bruhat Order

[1 1] 1 1

321 [1 0] [1 1] 1 1 0 1 312 231 [1 0] 0 1 ? [0 0] [1 0] 0 1 0 0 132 213 [0 0] 123 0 0

22 The MacNielle Completion for S4

23 What does this tells us about the Meinrenken Tile?

Part 3: ASMs ⇒ Geometry

24 A Geometric Realization of the Hasse Diagram of AMS lattice

25 ∙ Is there a good way to study the Meinrenken tile as a polytopal complex? ∙ Is there always a way to rearrange the pieces in the Mienrenken tile to get a convex polytopal complex?

∙ The domain of the WT map is SSn If we restrict ourselves to Gln what is the image? What is its topology? ∙ In experimentation, the WT map seems to preserve both the Birkhoff polytope and the ASM polytope. Is this true in general?

Open Questions

∙ Does this embedding of the Hasse diagram extend to other types?

26 ∙ Is there always a way to rearrange the pieces in the Mienrenken tile to get a convex polytopal complex?

Open Questions

∙ Does this embedding of the Hasse diagram extend to other types? ∙ Is there a good way to study the Meinrenken tile as a polytopal complex?

26 ∙ The domain of the WT map is SSn If we restrict ourselves to Gln what is the image? What is its topology? ∙ In experimentation, the WT map seems to preserve both the Birkhoff polytope and the ASM polytope. Is this true in general?

Open Questions

∙ Does this embedding of the Hasse diagram extend to other types? ∙ Is there a good way to study the Meinrenken tile as a polytopal complex? ∙ Is there always a way to rearrange the pieces in the Mienrenken tile to get a convex polytopal complex?

26 ∙ In experimentation, the WT map seems to preserve both the Birkhoff polytope and the ASM polytope. Is this true in general?

Open Questions

∙ The domain of the WT map is SSn If we restrict ourselves to Gln what is the image? What is its topology?

26 Open Questions

26 Thank You!