# Generalized Catalan numbers and hyperplane arrangements Communicating Mathematics, July, 2007 - PowerPoint PPT Presentation

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1. 4. 2. 3. 4. 1. 3. 2. 1. 4. 3. 2. Generalized Catalan numbers and hyperplane arrangements Communicating Mathematics, July, 2007. Cathy Kriloff Idaho State University Supported in part by NSA grant MDA904-03-1-0093 Joint work with Yu Chen, Idaho State University

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Generalized Catalan numbers and hyperplane arrangements Communicating Mathematics, July, 2007

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## Generalized Catalan numbers and hyperplane arrangementsCommunicating Mathematics, July, 2007

Cathy Kriloff

Idaho State University

Supported in part by NSA grant MDA904-03-1-0093

Joint work with Yu Chen, Idaho State University

Journal of Combinatorial Theory – Series A

### Outline

• Partitions counted by Cat(n)

• Real reflection groups

• Generalized partitions counted by Cat(W)

• Regions in hyperplane arrangements and the dihedral noncrystallographic case

### Poset of partitions of [n]

• Let P(n)=partitions of [n]={1,2,…,n}

• Order by: P1≤P2 if P1 refinesP2

• Same as intersection lattice of Hn={xi=xj | 1≤i<j≤n} in Rn under reverse inclusion

• Example: P(3)

### Nonnesting partitions of [n]

Nonnesting partitions have no nested arcs = NN(n)

Examples in P(4):

Nonnesting partition of [4]

Nesting partition of [4]

Noncrossing partitions have no crossing arcs = NC(n)

Examples in P(4):

Noncrossing partition of [4]

Crossing partition of [4]

Subposets:

• NN(4)=P(4)\

• NC(4)=P(4)\

### How many are there?

Catalan number

See #6.19(pp,uu) in Stanley, Enumerative Combinatorics II, 1999

or www-math.mit.edu/~rstan/

NN(n) Postnikov – 1999

NC(n) Becker - 1948, Kreweras - 1972

These posets are all naturally related to the permutation group Sn

### Some crystallographic reflection groups

• Symmetries of these shapes are crystallographic reflection groups of types A2, B2, G2

• First two generalize to n-dim simplex and hypercube

• Corresponding groups: Sn+1=An and Sn⋉(Z2)n=Bn

• (Some crystallographic groups are not symmetries of regular polytopes)

### Some noncrystallographic reflection groups

• Generalize to 2-dim regular m-gons

• Get dihedral groups,I2(m), for any m

• Noncrystallographic unless m=3,4,6 (tilings)

I2(5)

I2(7)

I2(8)

### Real reflection groups

Classification of finite groups generated by reflections = finite Coxeter groups due toCoxeter (1934), Witt (1941)

Symmetries

of regular

polytopes

Crystallographic

reflection groups

=Weyl groups

Venn diagram:

Drew Armstrong

### Root System of type A2

• roots = unit vectors perpendicular to reflecting hyperplanes

• simpleroots = basis so each root is positive or negative

A2

a1+a2=b2=e1-e3

a2=b3=e2-e3

a1=b1=e1-e2

• ai are simple roots

• bi are positive roots

• work in plane x1+x2+x3=0

• ei-ej connect to NN(3) since hyperplane xi=xj is (ei-ej)┴

3

1

2

### Root poset in type A2

Root poset for A2

• Express positive j in i basis

• Ordering: ≤ if -═cii with ci≥0

• Connect by an edge if comparable

• Increases going down

• Pick any set of incomparable roots (antichain), , and form its ideal=   for all 

• Leave off bs, just write indices

Antichains (ideals) for A2

1 (2) 3

1 (2) (2) 3

2

### NN(n) as antichains

Let e1,e2,…,en be an orthonormal basis of Rn

n=3, type A2

Subposet of intersection lattice of hyperplane arrangement

{xi-xj=0 | 1≤i<j≤n} in type An-1,

{<x,bi>=0 | 1≤j≤n} in general

Antichains (ideals)

in Int(n-1) in type An-1 (Stanley-Postnikov 6.19(bbb)), root poset in general

### Case when n=4

Using antichains/ideals in the root poset excludes {e1-e4,e2-e3}

### Generalized Catalan numbers

• For W=crystallographic reflection group define NN(W) to be antichains in the root poset (Postnikov)Get |NN(W)|=Cat(W)= (h+di)/|W|, where h = Coxeter number, di=invariant degreesNote: for W=Sn (type An-1), Cat(W)=Cat(n)

• What if W=noncrystallographic reflection group?

### Hyperplane arrangement

• Name positive roots 1,…,m

• Add affine hyperplanes defined by x, i=1 and label by I

• Important in representation theory

Label each 2-dim region in dominant coneby all i so that for all x in region, x, i 1= all i such that hyperplane is crossed as move out from origin

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A2

2

3

1 2 3

3

b2

2 3

b1

1 2

2

2

3

1

### Regions in hyperplane arrangement

Regions into which the cone x1≥x2≥…≥xnis divided by xi-xj=1, 1≤i<j≤n #6.19(lll)

Regions in the dominant cone in general

Ideals in the

root poset

### Noncrystallographic case

• Add affine hyperplanes defined by x, i=1 and label by i

• For m even there are two orbits of hyperplanes and move one of them

• When m is even roots lie on reflecting lines so symmetries break them into two orbits

1

2

4

I2(4)

3

4

a2

2

3

a1

1

12

34

12

34

12

34

2 3 4

1 2 3

2 3 4

2 3 4

1 2 3

2 3

1 2 3

2 3

2 4

2 3

2

3

2

2

### Indexing dominant regions in I2(4)

Label each 2-dim region by all i such that for all x in region, x, i ci= all i such that hyperplane is crossed as move out from origin

These subsets of {1,2,3,4} are exactly the ideals in each case

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4

3

1

3

2

2

1

4

3

2

1

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### Root posets and ideals

I2(3)

I2(4)

• Express positive j in i basis

• Ordering: ≤ if -═cii with ci≥0

• Connect by an edge if comparable

• Increases going down

• Pick any set of incomparable roots (antichain), , and form its ideal=   for all 

• x, i=c  x, i /c=1 so moving hyperplane in orbit changing root length in orbit, and poset changes

I2(5)

5

1

2

4

3

Root poset for I2(5)

Ideals index

dominant regions

1

5

2

4

1 2 3

4 5

I2(5)

3

2 3

4 5

Ideals for I2(5)

1 2

3 4

1 2 3 4 5

2 3 4 5 1 2 3 4

2 3 4

3 4 2 3

3

2 3 4

5

34

2 3

3

1

4

3

2

12

34

12

34

12

34

2 3 4

1 2 3

2 3 4

2 3 4

1 2 3

2 3

1 2 3

2 3

2 4

2 3

2

3

2

2

1

4

1

4

4

1

3

3

2

3

2

2

### Result for I2(m)

• Theorem (Chen, K): There is a bijection between dominant regions in this hyperplane arrangement and ideals in the poset of positive roots for the root system of type I2(m) for every m.If m is even, the correspondence is maintained as one orbit of hyperplanes is dilated.

• Was known for crystallographic root systems,- Shi (1997), Cellini-Papi (2002)and for certain refined counts.- Athanasiadis (2004), Panyushev (2004), Sommers (2005)

### Generalized Catalan numbers

• Cat(I2(5))=7 but I2(5) has 8 antichains!

• Except in crystallographic cases, # of antichains is notCat(I2(m))

• For any reflection group, W, Brady & Watt, Bessis define NC(W) Get |NC(W)|=Cat(W)= (h+di)/|W|, where h = Coxeter number, di=invariant degrees

• But no bijection known from NC(W) to NN(W)!Open: What is a noncrystallographic nonnesting partition?

• See Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups – will appear in Memoirs AMSand www.aimath.org/WWN/braidgroups/