A bijection between cores and dominant Shiregions

S. Fishel, M. Vazirani

August 6, 2010

arXiv:0904.3118 [math.CO]

To appear in European Journal of CombinatoricsIntroduction

The bijection

Sketch of sketch of proof

Refinements

1/48

Set-up

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

standard basis.

I αij = αi + · · ·+ αj = ei − ej+1 ∈ V , where 1 ≤ i ≤ j ≤ n− 1.

I θ = α1 + · · ·+ αn−1 = e1 − enI Hα,k = {x ∈ V |〈x | α〉 = k}, H+

α,k = {x ∈ V |〈x | α〉 ≥ k},where α = αi , θ, or αij .

Introduction 2/48

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

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

2-Shi arrangement

n = 3 and m = 2

Hα1,0 Hα1,1 Hα1,2Hα1,−1

Hα2,0

Hα2,1

Hα2,2

Hα2,−1

Hθ,0 Hθ,1 Hθ,2Hθ,−1

Introduction 5/48

Dominant/fundamental chamber

The fundamental or dominant chamber is ∩αijH+ij ,0.

Hα1,0

Hα2,0

Hθ,0

Introduction 6/48

Regions

The regions of an arrangement are the connected components ofthe complement of the arrangement. Regions in the dominantchamber 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

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

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

Partitions

A partition is a weakly decreasing sequence of positive integers offinite length.

(5,3,3,2) has Young diagram

Introduction 10/48

Hooks

9 5 3 2 1

5 1

3

2

1

h21 = 5

Introduction 11/48

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

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

The original question

In 20021, Jaclyn Anderson showed that there are 1s+t

(s+ts

)partitions which are both s-cores and t-cores when s and t arerelatively prime.

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

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

Introduction 14/48

The affine symmetric group

The affine symmetric group, denoted Sn, is defined as

Sn = 〈s1, . . . , sn−1, s0 | s2i = 1, si sj = sjsi if i 6≡ j ± 1 mod n,

si sjsi = sjsi sj if i ≡ j ± 1 mod n〉

for n > 2, and S2 = 〈s1, s0 | s2i = 1〉.

The affine symmetric group contains the symmetric group Sn as asubgroup. Sn is the subgroup generated by the si , 0 < i < n.

The bijection 15/48

AlcovesEach connected component of V \

⋃αij 1≤i≤j≤n−1

k∈ZHαij ,k is called an

alcove.

A0

The fundamental alcove A0 is yellow.The bijection 16/48

Sn acts on alcovessi 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 thedominant chamber.

The bijection 17/48

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 1n = 4

sk acts on the n-core λ by removing/adding all boxes with residuek

The bijection 18/48

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

0 1 2 3 4 0 1 2 3

4 0 1 2 3

3 4 0

2 3

1

0

s3

The bijection 19/48

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

0 1 2 3 4 0 1 2

4 0 1 2

3 4 0

2

1

0

s0

The bijection 20/48

Sn acts on cores

∅

n = 3

00

1

2

0 1

2

0

21

0 1 2

2

0

1

0

2

0 1

2

1

0 1 2 0

2 0

0 1 2

2

1

0 1

2 0

1

0

0 1 2 0 1

2 0 1

1

1

0

2

0 1 2 0

2 0

1

0

0 1 2

2 0

1 2

0

2The bijection 21/48

Alcoves ⇐⇒ n-cores

{n − cores} → {alcoves in the dominant chamber}

w∅ 7→ w−1A0

The bijection 22/48

Alcoves ⇐⇒ n-cores

∅

The bijection 23/48

Alcoves ⇐⇒ n-cores

∅

m = 1, n = 33-cores and 4-cores

The bijection 24/48

Alcoves ⇐⇒ n-cores

∅

m = 2, n = 33-cores and 7-cores

The bijection 25/48

m-minimal alcoves

An alcove is m-minimal if it is the alcove in its m-Shi regionseparated from A0 by the least number of hyperplanes in them-Shi arrangement.

We show the m-minimal alcoves have the same characterization asthe n-cores which are also (nm + 1)-cores.

The bijection 26/48

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 6= 0, and

s0(a1, . . . , an) = (an + 1, a2, . . . , an−1, a1 − 1).

Sketch of sketch of proof 27/48

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

m-minimal alcoves

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

Sketch of sketch of proof 29/48

Abacus description of n-coresThe hooklengths from the first column of a partition λ, plus allnegative integers, are a set of β-numbers for λ.

Any set obtained from a set of β-numbers by adding an integerconstant to all its elements is also a set of β-numbers.

Construct an n-abacus for a partition by putting its β-numbers onan n-runner abacus.

3

1

-2 -6 -5 -4

-1 -3 -2 -1

0 0 1 2

1 3 4 5

1 0 -1

Sketch of sketch of proof 30/48

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

0 0 1 2

1 3 4 5

1 0 -1

-2 -6 -5 -4

-1 -3 -2 -1

0 0 1 2

1 3 4 5

-1 1 0

4

2

1

3

1

n = 3

Sketch of sketch of proof 31/48

Balanced abacus, 5-core

11764321

-2 -10 -9 -8 -7 -6

-1 -5 -4 -3 -2 -1

0 0 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 0 0

-2 -10 -9 -8 -7 -6

-1 -5 -4 -3 -2 -1

0 0 1 2 3 4

1 5 6 7 8 9

2 10 11 12 13 14

3 15 16 17 18 19

2 1 0 0 -2

-2 -10 -9 -8 -7 -6

-1 -5 -4 -3 -2 -1

0 0 1 2 3 4

1 5 6 7 8 9

2 10 11 12 13 14

3 15 16 17 18 19

1 0 0 -2 1

Sketch of sketch of proof 32/48

Balanced abacus, 5-core

11764321

-2 -10 -9 -8 -7 -6

-1 -5 -4 -3 -2 -1

0 0 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 0 0

-2 -10 -9 -8 -7 -6

-1 -5 -4 -3 -2 -1

0 0 1 2 3 4

1 5 6 7 8 9

2 10 11 12 13 14

3 15 16 17 18 19

2 1 0 0 -2

-2 -10 -9 -8 -7 -6

-1 -5 -4 -3 -2 -1

0 0 1 2 3 4

1 5 6 7 8 9

2 10 11 12 13 14

3 15 16 17 18 19

1 0 0 -2 1

Sketch of sketch of proof 32/48

Balanced abacus, 5-core

11764321

-2 -10 -9 -8 -7 -6

-1 -5 -4 -3 -2 -1

0 0 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 0 0

-2 -10 -9 -8 -7 -6

-1 -5 -4 -3 -2 -1

0 0 1 2 3 4

1 5 6 7 8 9

2 10 11 12 13 14

3 15 16 17 18 19

2 1 0 0 -2

-2 -10 -9 -8 -7 -6

-1 -5 -4 -3 -2 -1

0 0 1 2 3 4

1 5 6 7 8 9

2 10 11 12 13 14

3 15 16 17 18 19

1 0 0 -2 1

Sketch of sketch of proof 32/48

Balanced abacus, 5-core

11764321

A vector of level numbers is balanced ifthe sum of levels is 0. N(λ) ∈ V is thebalanced vector of level numbers from theabacus for λ. λ 7→ N(λ) commutes withthe 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

0 0 1 2 3 4

1 5 6 7 8 9

2 10 11 12 13 14

3 15 16 17 18 19

1 0 0 -2 1

Sketch of sketch of proof 33/48

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

-2 -10 -9 -8 -7 -6

-1 -5 -4 -3 -2 -1

0 0 1 2 3 4

1 5 6 7 8 9

2 10 11 12 13 14

3 15 16 17 18 19

1 0 0 -2 1

-1 -10 -9 -8 -7 -6 -5

0 -4 -3 -2 -1 0 1

1 2 3 4 5 6 7

2 8 9 10 11 12 13

1 2 1 1 0 0

(5,2,2,1,1,1,1) is a 5-core but not a 6-core. Runner 4 from the5-abacus has too many more beads than runner 3.

In other words, 〈N(λ), α4〉 < −1.

Sketch of sketch of proof 34/48

n-core and (nm + 1)-core

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

Sketch of sketch of proof 35/48

Bounded regions

There are1

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

Alcoves ⇐⇒ n-cores

∅

m = 1, n = 33-cores and 2-cores

Refinements 37/48

Alcoves ⇐⇒ n-cores

∅

m = 2, n = 33-cores and 5-cores

Refinements 38/48

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 thex-axis, and which have k peaks.

Nn(k) =1

n

(n

k

)(n

k − 1

)n∑

k=1

Nn(k) = Cn

Refinements 39/48

Extended Narayana numbers

Athanasiadas extended and generalized Narayana numbers. For oursituation

Nnm(k) =1

n

(n

k

)(nm

k − 1

)

Refinements 40/48

Separating walls and hyperplanes

A hyperplane H separates two regions if they lie on opposite sidesof it. A hyperplane H is a separating wall for a region R if H is asupporting hyperplane of R and H separates R from A0.

Refinements 41/48

Narayana numbers and m-Shi

Athanasiadis 2 showed that Nnm(k) is the number of dominantregions in the extended Shi arrangement which have n − kseparating 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

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

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

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

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

k = 3 n = 3, m = 2

∅ 0 02

0 1 0 1 221k = 2

0 1 22

0 121

0 12 010

0 1 2 02 0

0 1 2 0 12 0 110

0 1 2 02 01 202k = 1

0 1 2 0 1 22 0 1 21 20 121

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

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

k = 3 n = 3, m = 2

∅ 1 21

2 1 5 2 121k = 2

4 2 11

4 121

5 24 111

5 4 2 12 1

8 5 4 2 15 2 121

8 5 2 15 24 121k = 1

11 8 5 4 2 18 5 2 15 24 121

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

Thank you for your time!

Refinements 48/48