Download Communicating Mathematics Conference

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
Journal of Combinatorial Theory – Series A
1
2
4
3
1
4
3
2
1
3
2
4
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)
x1=x2=x3
x1=x2
x1=x3
3
R
x2=x3
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]
P(4), NN(4), NC(4)
Subposets:
• NN(4)=P(4)\
• NC(4)=P(4)\
How many are there?
| NN (1) | 1
| NN (2) | 2
| NN (3) || P (3) | 5
| NN (4) || P (4) | 1  14
C (2n, n)
| NN (n) || NC (n) |
Catalan number  Cat (n) 
n 1
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 to Coxeter (1934), Witt (1941)
Symmetries
of regular
polytopes
Crystallographic
reflection groups
=Weyl groups
Dn
(n4)
I2(m)
(m3,4,6)
H3
H4
Venn diagram:
Drew Armstrong
A n, B n
(n3)
F4
I2(3)=A 2
I2(4)=B 2
I2(6)=G 2
E6
E7
E8
Root System of type A2
• roots = unit vectors perpendicular to reflecting hyperplanes
• simple roots = basis so each root is positive or negative
A2
a2b3e2e3
a1a2b2e1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)┴
Root poset in type A2
• Express positive bj in ai basis
• Ordering:
a≤b if b-a ═ciai with ci≥0
Root poset for A2
3
1
2
• Connect by an edge if comparable Antichains (ideals) for A
2
• Increases going down
• Pick any set of incomparable roots
(antichain), , and form its
ideal=b  ba for all a
• Leave off bs, just write indices
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
(e1-e2)
(e1-e2)
(e2-e3)

(e1-e3)
R
b1=e1-e2 b3=e2-e3
(e2-e3)
b2=e1-e3=b1+b3
3
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
b 1,(b 2),b 3
b 1,(b 2)
b2
b 2b 3

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}
e1-e2
e2-e3
e3-e4
e1-e3
e2-e4
e1-e2,e2-e3,e3-e4
e1-e4
e1-e2,e2-e3 e1-e2,e2-e4 e1-e3,e2-e4
e1-e2
e1-e3
e1-e2,e3-e4 e1-e3,e3-e4 e2-e3,e3-e4
e1-e4
e2-e3

e2-e4
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
• Name positive roots b1,…,bm
• Add affine hyperplanes defined by
x, bi =1 and label by I
• Important in representation theory
A2
Label each 2-dim region in dominant cone
by all i so that for all x in region, x, bi 1
= all i such that hyperplane is crossed as
move out from origin
1
2
3
123
b3
b2
23
b1
12
2
2
3

1
Regions in hyperplane arrangement
b3
b 1,b 2,b 3
b 1,b 2
b2
b1
b 2b 3

Ideals in the
root poset
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
Noncrystallographic case
•
When m is even roots lie
on reflecting lines so
symmetries break them
into two orbits
• Add affine hyperplanes defined by
x, bi =1 and label by i
• For m even there are two orbits of
hyperplanes and move one of them
1 4
2
I2(4)
b4
a2
a1
b3
b2
b1
3
Indexing dominant regions in I2(4)
Label each 2-dim region by all i such that for all x in region, x, bi ci
= all i such that hyperplane is crossed as move out from origin
12
34
12
34
234
234
234
12
34
123
123
23
123
23
23
2
3

2
2


These subsets of {1,2,3,4} are exactly the ideals in each case
Root posets and ideals
• Express positive bj in ai basis
• Ordering:
a≤b if b-a ═ciai with ci≥0
I2(3)
3
1
• Connect by an edge if
comparable
2
• Increases going down
I2(5)
• Pick any set of incomparable
roots (antichain), , and form
its ideal=b  ba for all a
•
x, bi =c  x, bi /c=1 so
moving hyperplane in orbit 
changing root length in orbit,
and poset changes
I2(4)
1
4
2
3
1
4
3
5
1
2
2
4
3
1
3
2
4
Root poset for I2(5)
1
5
2
4
Ideals index
dominant regions
I2(5)
3
23
45
Ideals for I2(5)
123
45
12
34
12345
2345
1234
234
234
34
5
23
3

4
34
3 23 1
3

2
Correspondence for m even
12
34
12
34
12
34
234
234
234
123
123
23
123
23
23
2
3

1
2
2
2

4
3
1

4
3
2
1
3
2
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 AMS
and www.aimath.org/WWN/braidgroups/