[PPT] - A bijection between cores and dominant Shi regions S. Fishel, M. PowerPoint Presentation

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A bijection between cores and dominant Shi regions

S. Fishel, M. Vazirani

August 6, 2010 arXiv:0904.3118 [math.CO] To appear in European Journal of Combinatorics Introduction The bijection Sketch of sketch of proof Refinements

1/48

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Set-up

◮ V = {(x1, . . . , xn) ∈ Rn|x1 + . . . + xn = 0} ◮ αi = ei − ei+1 ∈ V , where 1 ≤ i ≤ n − 1 and {ei} is the

standard basis.

◮ αij = αi + · · · + αj = ei − ej+1 ∈ V , where 1 ≤ i ≤ j ≤ n − 1. ◮ θ = α1 + · · · + αn−1 = e1 − en ◮ Hα,k = {x ∈ V |x | α = k}, H+ α,k = {x ∈ V |x | α ≥ k},

where α = αi, θ, or αij.

Introduction 2/48

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Roots and hyperplanes

α1 α2 θ Hα1,0 Hα2,0 Hθ,0

The roots α1, α2, and θ and their reflecting hyperplanes.

Introduction 3/48

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Extended Shi arrangement

For any positive integers n and m, the extended Shi arrangement is {Hαij,k|k ∈ Z, − m < k ≤ m and 1 ≤ i ≤ j ≤ n − 1}. We also call it the m-Shi arrangement.

Introduction 4/48

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2-Shi arrangement

n = 3 and m = 2 Hα1,0 Hα1,1 Hα1,2 Hα1,−1 Hα2,0 Hα2,1 Hα2,2 Hα2,−1 Hθ,0 Hθ,1 Hθ,2 Hθ,−1

Introduction 5/48

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Dominant/fundamental chamber

The fundamental or dominant chamber is ∩αijH+

ij,0.

Hα1,0 Hα2,0 Hθ,0

Introduction 6/48

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Regions

The regions of an arrangement are the connected components of the complement of the arrangement. Regions in the dominant chamber are called dominant regions.

Hα1,0 Hα2,0 Hα1,1 Hα2,1 Hθ,1 Hα1,2 Hα2,2 Hθ,2 m = 2 and n = 3

Introduction 7/48

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Number of regions in the dominant chamber

When m = 1, there are the Catalan number Cn = 1 n + 1 2n n

regions in the dominant chamber.

When m ≥ 1 there are the extended Catalan number Cnm = 1 nm + 1 n(m + 1) n

regions in the dominant chamber. Cn = Cn1.

Introduction 8/48

SLIDE 9

Number of regions in the dominant chamber

Hα1,0 Hα2,0 Hα1,1 Hα2,1 Hθ,1 Hα1,2 Hα2,2 Hθ,2

Cnm = 1 nm + 1 n(m + 1) n

C32 = 12

Introduction 9/48

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Partitions

A partition is a weakly decreasing sequence of positive integers of finite length. (5,3,3,2) has Young diagram

Introduction 10/48

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Hooks

9 5 3 2 1 5 1 3 2 1 h21 = 5

Introduction 11/48

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n-cores

An n-core is an integer partition λ such that n ∤ hij for all boxes (i, j) in λ. Some 3-cores. Boxes contain their hook numbers. 1 2 1 2 1 4 1 2 1 5 2 1 2 1 Not a 3-core. 3 1 1

Introduction 12/48

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All 3-cores which are also 7-cores

∅ 1 2 1 2 1 4 2 1 1 4 1 2 1 5 2 4 1 1 1 5 4 2 1 2 1 8 5 4 2 1 5 2 1 2 1 8 5 2 1 5 2 4 1 2 1 11 8 5 4 2 1 8 5 2 1 5 2 4 1 2 1 5 2 1 2 1

Introduction 13/48

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The original question

In 20021, Jaclyn Anderson showed that there are

1 s+t

s+t

s

partitions which are both s-cores and t-cores when s and t are

relatively prime. There are Cnm partitions which are n-cores and (nm + 1)-cores, the same as the number of dominant Shi regions.

1“Partitions which are simultaneously t1- and t2-core”, Discrete

Mathematics

Introduction 14/48

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The affine symmetric group

The affine symmetric group, denoted Sn, is defined as

Sn = s1, . . . , sn−1, s0 | s2

i = 1,

sisj = sjsi if i ≡ j ± 1 mod n, sisjsi = sjsisj if i ≡ j ± 1 mod n for n > 2, and S2 = s1, s0 | s2

i = 1.

The affine symmetric group contains the symmetric group Sn as a

subgroup. Sn is the subgroup generated by the si, 0 < i < n.

The bijection 15/48

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Alcoves

Each connected component of V \

αij 1≤i≤j≤n−1

k∈Z

Hαij,k is called an alcove.

A0

The fundamental alcove A0 is yellow.

The bijection 16/48

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Sn acts on alcoves

si reflects over Hαi,0 for 1 ≤ i ≤ 0 and s0 reflects over Hθ,1.

s1

The orbit of A0 under w−1 for minimal length w ∈ Sn/Sn is the dominant chamber.

The bijection 17/48

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Sn acts on n-cores

The box in row i, column j has residue j − i mod n. 0 1 2 3 0 1 3 0 1 n = 4 sk acts on the n-core λ by removing/adding all boxes with residue k

The bijection 18/48

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Sn acts on n-cores

n = 5 0 1 2 3 4 0 1 2 4 0 1 2 3 4 0 2 1 0 1 2 3 4 0 1 2 3 4 0 1 2 3 3 4 0 2 3 1

s3

The bijection 19/48

SLIDE 20

Sn acts on n-cores

n = 5 0 1 2 3 4 0 1 2 4 0 1 2 3 4 0 2 1 0 1 2 3 4 0 1 2 4 0 1 2 3 4 0 2 1

s0

The bijection 20/48

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Sn acts on cores

∅ n = 3

1 2

0 1

2

1

0 1 2 2

1 2

0 1 2 1 0 1 2 0 2 0 0 1 2 2 1 0 1 2 0 1 0 1 2 0 1 2 0 1 1

1 2

0 1 2 0 2 0 1 0 1 2 2 0 1 2 2

The bijection 21/48

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Alcoves ⇐ ⇒ n-cores

{n − cores} → {alcoves in the dominant chamber} w∅ → w−1A0

The bijection 22/48

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Alcoves ⇐ ⇒ n-cores

∅ The bijection 23/48

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Alcoves ⇐ ⇒ n-cores

∅

m = 1, n = 3 3-cores and 4-cores

The bijection 24/48

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Alcoves ⇐ ⇒ n-cores

∅

m = 2, n = 3 3-cores and 7-cores

The bijection 25/48

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m-minimal alcoves

An alcove is m-minimal if it is the alcove in its m-Shi region separated from A0 by the least number of hyperplanes in the m-Shi arrangement. We show the m-minimal alcoves have the same characterization as the n-cores which are also (nm + 1)-cores.

The bijection 26/48

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Sn acts on V

The action of Sn on V is given by si(a1, . . . , ai, ai+1, . . . , an) = (a1, . . . , ai+1, ai, . . . , an) for i = 0, and s0(a1, . . . , an) = (an + 1, a2, . . . , an−1, a1 − 1).

Sketch of sketch of proof 27/48

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Alcoves and their vectors

(0,0,0) (1,0,-1) (0,1,-1) (0,-1,1) (2,-1,-1) (-1,2,-1) (-1,-1,2) (3,-1,-2) (-1,3,-2) (-1,-2,3) (1,-1,0) (-1,1,0) (-1,0,1) (2,0,-2) (0,2,-2) (0,-2,2) (3,-2,-1) (-2,3,-1) (-2,-1,3) (4,-1,-3) (1,1,-2) (1,-2,1) (2,-2,0) (-2,2,0) (-2,0,2) (3,0,-3) (0,3,-3) (0,-3,3) (-2,1,1) (2,1,-3) (1,2,-3) (1,-3,2) (3,-3,0) (-3,3,0) (2,-3,1) (-3,2,1) (-3,1,2) (3,1,-4)

w−1A0 ← → w(0, . . . , 0) A ← → N(A)

Sketch of sketch of proof 28/48

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m-minimal alcoves

An alcove A is m-minimal if and only if N(A), αi ≥ −m for all i = 1, . . . , n − 1 and N(A), θ ≤ m + 1

Sketch of sketch of proof 29/48

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Abacus description of n-cores

The hooklengths from the first column of a partition λ, plus all negative integers, are a set of β-numbers for λ. Any set obtained from a set of β-numbers by adding an integer constant to all its elements is also a set of β-numbers. Construct an n-abacus for a partition by putting its β-numbers on an n-runner abacus. 3 1

2
6
5
4
1
3
2
1

1 2 1 3 4 5 1

1

Sketch of sketch of proof 30/48

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Abacus description of n-cores

A partition λ is an n-core if and only if its abacus is flush.

2
6
5
4
1
3
2
1

1 2 1 3 4 5 1

1
2
6
5
4
1
3
2
1

1 2 1 3 4 5

1

1

4 2 1 3 1 n = 3

Sketch of sketch of proof 31/48

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Balanced abacus, 5-core

11 7 6 4 3 2 1

2
10
9
8
7
6
1
5
4
3
2
1

1 2 3 4 1 5 6 7 8 9 2 10 11 12 13 14 3 15 16 17 18 19

1

2 1

Sketch of sketch of proof 32/48

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Balanced abacus, 5-core

11 7 6 4 3 2 1

2
10
9
8
7
6
1
5
4
3
2
1

1 2 3 4 1 5 6 7 8 9 2 10 11 12 13 14 3 15 16 17 18 19

1

2 1

2
10
9
8
7
6
1
5
4
3
2
1

1 2 3 4 1 5 6 7 8 9 2 10 11 12 13 14 3 15 16 17 18 19 2 1

2

Sketch of sketch of proof 32/48

SLIDE 34

Balanced abacus, 5-core

11 7 6 4 3 2 1

2
10
9
8
7
6
1
5
4
3
2
1

1 2 3 4 1 5 6 7 8 9 2 10 11 12 13 14 3 15 16 17 18 19

1

2 1

2
10
9
8
7
6
1
5
4
3
2
1

1 2 3 4 1 5 6 7 8 9 2 10 11 12 13 14 3 15 16 17 18 19 2 1

2
2
10
9
8
7
6
1
5
4
3
2
1

1 2 3 4 1 5 6 7 8 9 2 10 11 12 13 14 3 15 16 17 18 19 1

2

1

Sketch of sketch of proof 32/48

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Balanced abacus, 5-core

11 7 6 4 3 2 1 A vector of level numbers is balanced if the sum of levels is 0. N(λ) ∈ V is the balanced vector of level numbers from the abacus for λ. λ → N(λ) commutes with the Sn action on n-cores and V . N(5, 2, 2, 1, 1, 1, 1) = (1, 0, 0, −2, 1)

2
10
9
8
7
6
1
5
4
3
2
1

1 2 3 4 1 5 6 7 8 9 2 10 11 12 13 14 3 15 16 17 18 19 1

2

1

Sketch of sketch of proof 33/48

SLIDE 36

When is an n-core also an (nm + 1)-core?

2
10
9
8
7
6
1
5
4
3
2
1

1 2 3 4 1 5 6 7 8 9 2 10 11 12 13 14 3 15 16 17 18 19 1

2

1

1
10
9
8
7
6
5
4
3
2
1

1 1 2 3 4 5 6 7 2 8 9 10 11 12 13 1 2 1 1

(5,2,2,1,1,1,1) is a 5-core but not a 6-core. Runner 4 from the 5-abacus has too many more beads than runner 3. In other words, N(λ), α4 < −1.

Sketch of sketch of proof 34/48

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n-core and (nm + 1)-core

λ is an n-core and (nm + 1)-core if and only if N(λ), αi ≥ −m for all i = 1, . . . , n − 1 and N(λ), θ ≤ m + 1

Sketch of sketch of proof 35/48

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Bounded regions

There are 1 mn − 1 (m + 1)n − 2 n

partitions which are both n-cores and (mn − 1)-cores and there are

1 mn − 1 (m + 1)n − 2 n

bounded regions in the m-Shi arrangements.

Refinements 36/48

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Alcoves ⇐ ⇒ n-cores

∅

m = 1, n = 3 3-cores and 2-cores

Refinements 37/48

SLIDE 40

Alcoves ⇐ ⇒ n-cores

∅

m = 2, n = 3 3-cores and 5-cores

Refinements 38/48

SLIDE 41

Narayana numbers

The Narayana number Nn(k) is the number of lattice paths from (0,0) to (2n,0) using NE and SE steps, which stay above the x-axis, and which have k peaks. Nn(k) = 1 n n k

n

k − 1

n
k=1

Nn(k) = Cn

Refinements 39/48

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Extended Narayana numbers

Athanasiadas extended and generalized Narayana numbers. For our situation Nnm(k) = 1 n n k nm k − 1

Refinements

40/48

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Separating walls and hyperplanes

A hyperplane H separates two regions if they lie on opposite sides

f it. A hyperplane H is a separating wall for a region R if H is a

supporting hyperplane of R and H separates R from A0.

Refinements 41/48

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Narayana numbers and m-Shi

Athanasiadis 2 showed that Nnm(k) is the number of dominant regions in the extended Shi arrangement which have n − k separating walls of the form Hαij,m.

2“On a refinement of the generalized Catalan numbers for Weyl groups,”

Transactions of the AMS, 2004

Refinements 42/48

SLIDE 45

Narayana numbers and extended Shi, k = 3

Hα1,0 Hα2,0 Hα1,1 Hα2,1 Hα12,1 Hα1,2 Hα2,2 Hα12,2

N32(3) = 5 Regions with no separating walls of the form Hαij,m.

Refinements 43/48

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Narayana numbers and extended Shi, k = 2

Hα1,0 Hα2,0 Hα1,1 Hα2,1 Hα12,1 Hα1,2 Hα2,2 Hα12,2

N32(2) = 6 Regions with one separating wall of the form Hαij,m.

Refinements 44/48

SLIDE 47

Narayana numbers and extended Shi, k = 1

Hα1,0 Hα2,0 Hα1,1 Hα2,1 Hα12,1 Hα1,2 Hα2,2 Hα12,2

N32(1) = 1 Regions with two separating walls of the form Hαij,m.

Refinements 45/48

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n-cores “with n − k separating walls” (residue version)

k = 3 n = 3, m = 2 ∅ 2 0 1 0 1 2 2 1 k = 2 0 1 2 2 0 1 2 1 0 1 2 0 1 0 1 2 0 2 0 0 1 2 0 1 2 0 1 1 0 1 2 0 2 0 1 2 2 k = 1 0 1 2 0 1 2 2 0 1 2 1 2 0 1 2 1 Nnm(k) is the number of partitions λ which are n-cores and (nm + 1)-cores and for which there are n − k residues (mod n) such that λ has m removable boxes of that residue.

Refinements 46/48

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n-cores “with n − k separating walls” (hook version)

k = 3 n = 3, m = 2 ∅ 1 2 1 2 1 5 2 1 2 1 k = 2 4 2 1 1 4 1 2 1 5 2 4 1 1 1 5 4 2 1 2 1 8 5 4 2 1 5 2 1 2 1 8 5 2 1 5 2 4 1 2 1 k = 1 11 8 5 4 2 1 8 5 2 1 5 2 4 1 2 1 Nnm(k) is the number of partitions λ which are n-cores and (nm + 1)-cores and which have n − k n(m − 1) + 1 hooks.

Refinements 47/48

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Thank you for your time!

Refinements 48/48