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

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 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

• Partitions counted by Cat(n)

• Real reflection groups

• Generalized partitions counted by Cat(W)

• Regions in hyperplane arrangements and the dihedral noncrystallographic case

• 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 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]

P(4), NN(4), NC(4)

Subposets:

• NN(4)=P(4)\

• NC(4)=P(4)\

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)

• 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)

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

• 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)┴

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}

• 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?

• 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

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3

1 2 3

3

b2

2 3

b1

1 2

2

2

3

1

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

• 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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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

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3

2

2

Correspondence for m even

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4

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4

4

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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)

• 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/