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