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

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

Set-up ◮ V = { ( x 1 , . . . , x n ) ∈ R n | x 1 + . . . + x n = 0 } ◮ α i = e i − e i +1 ∈ V , where 1 ≤ i ≤ n − 1 and { e i } is the standard basis. ◮ α ij = α i + · · · + α j = e i − e j +1 ∈ V , where 1 ≤ i ≤ j ≤ n − 1. ◮ θ = α 1 + · · · + α n − 1 = e 1 − e n ◮ H α, k = { x ∈ V |� x | α � = k } , H + α, k = { x ∈ V |� x | α � ≥ k } , where α = α i , θ , or α ij . Introduction 2/48

Roots and hyperplanes H α 1 , 0 α 2 θ H α 2 , 0 α 1 H θ, 0 The roots α 1 , α 2 , and θ and their reflecting hyperplanes. Introduction 3/48

Extended Shi arrangement For any positive integers n and m , the extended Shi arrangement is { H α ij , 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 H α 1 , − 1 H α 1 , 0 H α 1 , 1 H α 1 , 2 H α 2 , 2 n = 3 and m = 2 H α 2 , 1 H α 2 , 0 H α 2 , − 1 H θ, − 1 H θ, 0 H θ, 1 H θ, 2 Introduction 5/48

Dominant/fundamental chamber The fundamental or dominant chamber is ∩ α ij H + ij , 0 . H α 1 , 0 H α 2 , 0 H θ, 0 Introduction 6/48

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. H α 1 , 0 H α 1 , 1 H α 1 , 2 m = 2 and n = 3 H α 2 , 2 H α 2 , 1 H α 2 , 0 H θ, 1 H θ, 2 Introduction 7/48

Number of regions in the dominant chamber When m = 1, there are the Catalan number � 2 n � 1 C n = n + 1 n regions in the dominant chamber. When m ≥ 1 there are the extended Catalan number � n ( m + 1) � 1 C nm = nm + 1 n regions in the dominant chamber. C n = C n 1 . Introduction 8/48

Number of regions in the dominant chamber H α 1 , 0 H α 1 , 1 H α 1 , 2 � n ( m + 1) � 1 C nm = nm + 1 n H α 2 , 2 H α 2 , 1 C 32 = 12 H α 2 , 0 H θ, 1 H θ, 2 Introduction 9/48

Partitions A partition is a weakly decreasing sequence of positive integers of finite length. (5,3,3,2) has Young diagram Introduction 10/48

Hooks 9 5 3 2 1 5 1 3 2 1 h 21 = 5 Introduction 11/48

n -cores An n -core is an integer partition λ such that n ∤ h ij for all boxes ( i , j ) in λ. Some 3-cores. Boxes contain their hook numbers. 1 2 2 1 4 1 5 2 1 1 2 2 1 1 Not a 3-core. 3 1 1 Introduction 12/48

All 3-cores which are also 7-cores 1 2 2 1 4 2 1 4 1 5 2 5 4 2 1 ∅ 1 1 2 4 1 2 1 1 1 1 8 5 4 2 1 8 5 2 1 11 8 5 4 2 1 5 2 1 5 2 1 5 2 8 5 2 1 2 2 4 1 5 2 1 1 2 4 1 1 2 1 Introduction 13/48

The original question � s + t � In 2002 1 , Jaclyn Anderson showed that there are 1 s + t s partitions which are both s -cores and t -cores when s and t are relatively prime. There are C nm partitions which are n -cores and ( nm + 1)-cores, the same as the number of dominant Shi regions. 1 “Partitions which are simultaneously t 1 - and t 2 -core”, Discrete Mathematics Introduction 14/48

The affine symmetric group The affine symmetric group, denoted � S n , is defined as S n = � s 1 , . . . , s n − 1 , s 0 | s 2 � i = 1 , s i s j = s j s i if i �≡ j ± 1 mod n , s i s j s i = s j s i s j if i ≡ j ± 1 mod n � for n > 2, and � S 2 = � s 1 , s 0 | s 2 i = 1 � . The affine symmetric group contains the symmetric group S n as a subgroup. S n is the subgroup generated by the s i , 0 < i < n . The bijection 15/48

Alcoves Each connected component of V \ � H α ij , k is called an α ij 1 ≤ i ≤ j ≤ n − 1 k ∈ Z alcove. A 0 The fundamental alcove A 0 is yellow. The bijection 16/48

� S n acts on alcoves s i reflects over H α i , 0 for 1 ≤ i ≤ 0 and s 0 reflects over H θ, 1 . s 1 The orbit of A 0 under w − 1 for minimal length w ∈ � S n / S n is the dominant chamber. The bijection 17/48

� S n acts on n -cores The box in row i, column j has residue j − i mod n . 0 1 2 3 0 1 n = 4 3 0 1 s k acts on the n -core λ by removing/adding all boxes with residue k The bijection 18/48

� S n acts on n -cores n = 5 0 1 2 3 4 0 1 2 0 1 2 3 4 0 1 2 3 4 0 1 2 4 0 1 2 3 3 4 0 s 3 3 4 0 2 2 3 1 1 0 0 The bijection 19/48

� S n acts on n -cores n = 5 0 1 2 3 4 0 1 2 0 1 2 3 4 0 1 2 4 0 1 2 4 0 1 2 3 4 0 s 0 3 4 0 2 2 1 1 0 0 The bijection 20/48

� S n acts on cores 0 1 2 0 0 1 2 0 1 1 2 0 2 0 1 0 n = 3 0 1 2 1 2 2 0 1 1 1 0 ∅ 0 0 1 2 0 1 2 0 0 2 2 2 0 0 1 1 2 2 0 1 0 1 2 0 1 0 1 0 1 2 2 2 0 2 0 1 1 2 0 0 2 The bijection 21/48

Alcoves ⇐ ⇒ n -cores { n − cores } → { alcoves in the dominant chamber } w ∅ �→ w − 1 A 0 The bijection 22/48

Alcoves ⇐ ⇒ n -cores ∅ The bijection 23/48

Alcoves ⇐ ⇒ n -cores m = 1, n = 3 3-cores and 4-cores ∅ The bijection 24/48

Alcoves ⇐ ⇒ n -cores m = 2, n = 3 3-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 region separated from A 0 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

� S n acts on V The action of � S n on V is given by for i � = 0, and s i ( a 1 , . . . , a i , a i +1 , . . . , a n ) = ( a 1 , . . . , a i +1 , a i , . . . , a n ) s 0 ( a 1 , . . . , a n ) = ( a n + 1 , a 2 , . . . , a n − 1 , a 1 − 1) . Sketch of sketch of proof 27/48

Alcoves and their vectors (-1,-2,3) (4,-1,-3) (-1,3,-2) (-2,-1,3) (3,-1,-2) (-2,3,-1) (0,-3,3) (-1,-1,2) (3,-2,-1) (0,3,-3) w − 1 A 0 ← → w (0 , . . . , 0) A ← → N ( A ) (-1,2,-1) (0,-2,2) (3,0,-3) (-3,3,0) (2,-1,-1) (0,2,-2) (-2,0,2) (3,-3,0) (0,-1,1) (2,0,-2) (-2,2,0) (1,-3,2) (3,1,-4) (0,1,-1) (-1,0,1) (2,-2,0) (1,2,-3) (-3,1,2) (1,0,-1) (-1,1,0) (1,-2,1) (2,1,-3) (-3,2,1) (0,0,0) (1,-1,0) (1,1,-2) (-2,1,1) (2,-3,1) Sketch of sketch of proof 28/48

m -minimal alcoves An alcove A is m -minimal if and only if � N ( A ) , α i � ≥ − m for all i = 1 , . . . , n − 1 and � N ( A ) , θ � ≤ m + 1 Sketch of sketch of proof 29/48

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. -2 -6 -5 -4 3 -1 -3 -2 -1 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 -2 -6 -5 -4 3 -1 -3 -2 -1 4 -1 -3 -2 -1 1 2 0 0 1 2 0 0 1 2 1 1 3 4 5 1 3 4 5 1 0 -1 -1 1 0 n = 3 Sketch of sketch of proof 31/48

Balanced abacus, 5-core -2 -10 -9 -8 -7 -6 11 -1 -5 -4 -3 -2 -1 7 0 0 1 2 3 4 6 1 5 6 7 8 9 4 2 10 11 12 13 14 3 3 15 16 17 18 19 2 -1 2 1 0 0 1 Sketch of sketch of proof 32/48

Balanced abacus, 5-core -2 -10 -9 -8 -7 -6 11 -1 -5 -4 -3 -2 -1 7 0 0 1 2 3 4 6 1 5 6 7 8 9 4 2 10 11 12 13 14 3 3 15 16 17 18 19 2 -1 2 1 0 0 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 2 1 0 0 -2 Sketch of sketch of proof 32/48

Balanced abacus, 5-core -2 -10 -9 -8 -7 -6 11 -1 -5 -4 -3 -2 -1 7 0 0 1 2 3 4 6 1 5 6 7 8 9 4 2 10 11 12 13 14 3 3 15 16 17 18 19 2 -1 2 1 0 0 1 -2 -10 -9 -8 -7 -6 -2 -10 -9 -8 -7 -6 -1 -5 -4 -3 -2 -1 -1 -5 -4 -3 -2 -1 0 0 1 2 3 4 0 0 1 2 3 4 1 5 6 7 8 9 1 5 6 7 8 9 2 10 11 12 13 14 2 10 11 12 13 14 3 15 16 17 18 19 3 15 16 17 18 19 2 1 0 0 -2 1 0 0 -2 1 Sketch of sketch of proof 32/48

Balanced abacus, 5-core 11 7 A vector of level numbers is balanced if 6 the sum of levels is 0. N ( λ ) ∈ V is the 4 balanced vector of level numbers from the 3 abacus for λ . λ �→ N ( λ ) commutes with 2 the � S n action on n -cores and V . 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 N (5 , 2 , 2 , 1 , 1 , 1 , 1) = (1 , 0 , 0 , − 2 , 1) 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 -1 -10 -9 -8 -7 -6 -5 0 0 1 2 3 4 0 -4 -3 -2 -1 0 1 1 5 6 7 8 9 1 2 3 4 5 6 7 2 10 11 12 13 14 2 8 9 10 11 12 13 3 15 16 17 18 19 1 2 1 1 0 0 1 0 0 -2 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 ) , α 4 � < − 1. Sketch of sketch of proof 34/48