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

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

2

2

4

4

43

1

3

1

1

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

2

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

3

2

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)

12

3

4

• 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

1

2

1 3

2

3

4

5

32

2

2

4

4

4

3

1

3

1

1

I2(3)

I2(5)

I2(4)

1

2

3

4

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)

2

13

4

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/