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
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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/