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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

- 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]

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

- 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)┴

3

1

2

Root poset for A2

- Express positive j in i basis
- Ordering: ≤ if -═cii 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

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

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

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1

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

(Stanley, Athanasiadis, Postnikov, Shi)

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

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2

4

I2(4)

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a2

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a1

1

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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

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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2

I2(3)

I2(4)

- Express positive j in i basis
- Ordering: ≤ if -═cii 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

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2

4

3

Root poset for I2(5)

Ideals index

dominant regions

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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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2

- 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/