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

Cathy KriloffIdaho 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 A32

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43

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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 refines P2

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

• Example: P(3)

R3

x2=x3x1=x3x1=x2

x1=x2=x3

Nonnesting partitions of [n]

Nesting partition of [4]Nonnesting partition of [4]

Nonnesting partitions have no nested arcs = NN(n)

Examples in P(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)\

How many are there?

See #6.19(pp,uu) in Stanley, Enumerative Combinatorics II, 1999or www-math.mit.edu/~rstan/ NN(n) Postnikov – 1999NC(n) Becker - 1948, Kreweras - 1972

These posets are all naturally related to the permutation group Sn

141|)4(||)4(|

5|)3(||)3(|

2|)2(|

1|)1(|

PNN

PNN

NN

NN

|)(||)(|1

),2()( nNCnNN

n

nnCnCat

Catalan number

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⋉(ZZ2)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 groupsClassification of finite groups generated by reflections = finite Coxeter groups due to Coxeter (1934), Witt (1941)

Symmetries of regularpolytopes

Crystallographicreflection groups=Weyl groups

Venn diagram:Drew Armstrong

F 4

I2(3)=A 2

I2(4)=B 2

I2(6)=G 2

A n, B n

(n3)

D n

(n4)

E 6

E 7

E 8H4

H3

I2(m) (m3,4,6)

Root System of type A2

• roots = unit vectors perpendicular to reflecting hyperplanes• simple roots = basis so each root is positive or negative

A2

ee

eeee

• i are simple roots• i are positive roots• work in plane x1+x2+x3=0• ei-ej connect to NN(3) since hyperplane xi=xj is (ei-ej)┴

Root poset in type 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 s, just write indices

1 3

2

1 (2) 3

1 (2) (2) 3

2

Root poset for A2

Antichains (ideals) for A2

NN(n) as antichainsLet e1,e2,…,en be an orthonormal basis of Rn

Subposet of intersection lattice of hyperplane arrangement{xi-xj=0 | 1≤i<j≤n} in type An-1,{<x,i>=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

1,(2),3

1,(2)

23

R3

2=e1-e3=1+3

3=e2-e31=e1-e2(e1-e2) (e2-e3)

(e1-e2)

(e1-e3)

(e2-e3)

n=3, type A2

Case when n=4

e1-e2

e1-e3

e1-e4

e2-e3

e2-e4

e3-e4

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

e3-e4e2-e4e2-e3e1-e4e1-e3e1-e2

e2-e3,e3-e4e1-e3,e3-e4e1-e2,e3-e4e1-e3,e2-e4e1-e2,e2-e4e1-e2,e2-e3

e1-e2,e2-e3,e3-e4

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 degrees

Note: for W=Sn (type An-1), Cat(W)=Cat(n)

• What if W=noncrystallographic reflection group?

Hyperplane arrangement

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132

• 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

1

1 2 3

1 2

2 3

22

3

A2

Regions in hyperplane arrangement

Regions into which the cone x1≥x2≥…≥xn

is divided by xi-xj=1, 1≤i<j≤n #6.19(lll)

(Stanley, Athanasiadis, Postnikov, Shi)

Regions in the dominant cone in general

3

1

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1,2

1,2,3

Ideals in the root poset

Noncrystallographic case• When m is even roots lie

on reflecting lines so symmetries break them into two orbits

I2(4)

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• 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

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

1 23 41 2

3 41 23 4

23

1 2 3

22

2 42 3

2 3 4

2 3

2 3

2 3 4 2 3 4 1 2 3

1 2 3

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

Root posets and ideals• 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

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1 3

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5

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1

I2(3)

I2(5)

I2(4)

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5

1 2 3 4 5

2 3 4 5 1 2 3 4

2 3 4

3 4 2 3

3

Root poset for I2(5) Ideals indexdominant regions

Ideals for I2(5)

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13

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5

1 2 3 4 5

1 23 4

2 34 5

2 3 4

3 2 33 4

I2(5)

Correspondence for m even

1 23 41 2

3 41 23 4

23

1 2 3

22

2 42 3

2 3 4

2 3

2 3

2 3 4 2 3 4 1 2 3

1 2 3

1 11

333

22

2

4 4 4

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 not Cat(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/