Generalized catalan numbers and hyperplane arrangements communicating mathematics july 2007
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1. 4. 2. 3. 4. 1. 3. 2. 1. 4. 3. 2. 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

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

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Generalized catalan numbers and hyperplane arrangements communicating mathematics july 2007

1

4

2

3

4

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3

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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


Outline

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

Poset of partitions of [n]

  • 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 of n

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

P(4), NN(4), NC(4)

Subposets:

  • NN(4)=P(4)\

  • NC(4)=P(4)\


How many are there

How many are there?

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


Some crystallographic reflection groups

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

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

Real reflection groups

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


Root system of type a 2

Root System of type A2

  • 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)┴


Root poset in type a 2

3

1

2

Root poset in type A2

Root poset for A2

  • Express positive j in i basis

  • Ordering: ≤ if -═cii 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


Nn n as antichains

NN(n) as antichains

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


Case when n 4

Case when n=4

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


Generalized catalan numbers

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

  • What if W=noncrystallographic reflection group?


Hyperplane arrangement

Hyperplane arrangement

  • 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

A2

2

3

1 2 3

3

b2

2 3

b1

1 2

2

2

3

1


Regions in hyperplane arrangement

Regions in hyperplane arrangement

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


Noncrystallographic case

Noncrystallographic case

  • 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

1

2

4

I2(4)

3

4

a2

2

3

a1

1


Indexing dominant regions in i 2 4

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

2

3

2

2

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

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


Root posets and ideals

1

4

3

1

3

2

2

1

4

3

2

1

4

3

2

Root posets and ideals

I2(3)

I2(4)

  • Express positive j in i basis

  • Ordering: ≤ if -═cii 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

1

2

4

3


Generalized catalan numbers and hyperplane arrangements communicating mathematics july 2007

Root poset for I2(5)

Ideals index

dominant regions

1

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


Correspondence for m even

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

2

3

2

2

Correspondence for m even

1

4

1

4

4

1

3

3

2

3

2

2


Result for i 2 m

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 numbers1

Generalized Catalan numbers

  • 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/


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