Indexing regions in dihedral and dodecahedral hyperplane arrangements

MAA Intermountain Sectional Meeting, March 23, 2007

Cathy KriloffIdaho State University

Supported in part by NSA grant MDA904-03-1-0093 Joint work with Yu Chen, Idaho State University

to appear in Journal of Combinatorial Theory – Series A 32

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4

4

43

1

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1

1

Outline

• Noncrystallographic reflection groups(motivation: representation theory of graded Hecke algebras)

• Geometry – root systems and hyperplanes

• Combinatorics – root order and ideals• Bijection for I2(m), H3, H4

(motivation: interesting combinatorics, unitary representations of graded Hecke algebras)

                                      

'Lie group E8' math puzzle solvedPOSTED: 10:26 a.m. EDT, March 21, 2007

(See www.aimath.org/E8)

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=An and 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)

Reflection groups• There is a classification (Coxeter - 1934, Witt – 1941) of finite groups generated by reflections = finite Coxeter groups

• Four infinite families, An, Bn, Dn, I2(m), +7 exceptional groups

• Crystallographic reflection groups = Weyl groups from Lie theory - represented by matrices with rational entries

• Noncrystallographic reflection groups need irrational entries - I2(m) = dihedral group of order 2m - H3 = symmetries of the dodecahedron- H4 = symmetries of the hyperdodecahedron

(Good test cases between real and complex reflection groups)

Root systems

• When m is even roots lie on reflecting lines so symmetries break them into two orbits

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

I2(3) I2(4)

Hyperplane arrangement

2

132

• Name positive roots 1,…,m

• 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

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3

4

Indexing dominant regionsLabel each 2-dim region by all i such 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

2

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(3) I2(5)

Indexing dominant regions in I2(4)Label each 2-dim region by all i such that for all x in region, x, i c

= 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

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 changing root length, and poset changes

1

2

1 3

2

3

4

5

32

2

2

4

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1

I2(3)

I2(5)

I2(4)

1 3

2

1 2 3

1 2 2 3

2

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

2

13

4

5

1 2 3 4 5

1 23 4

2 34 5

2 3 4

3 2 33 4

Root poset for I2(3)

Ideals for I2(3)

Root poset for I2(5) Ideals indexdominant regions

Ideals for 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)

H3 and H4

• Can generalize I2(5) to:

H3 = symmetries of 3-dim dodecahedron

H4 = symmetries of regular 4-dimensional solid, hyperdodecahedron or 120-cell (with 120 3-dim dodecahedral faces)

• I2(5), H3, and H4 related to quasicrystals

H3 root system

• Roots = edge midpoints of dodecahedron or icosahedron

Source: cage.ugent.be/~hs/polyhedra/dodeicos.html

H3 hyperplane arrangementDominant regions are enclosed by yellow, pink, and light gray planes

H3 root poset

Has 41 ideals

Result for H3

• 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 H3.

There are 41 dominant regions(29 bounded and 12 unbounded).

A 3-d projection of the 120-cell

Source: en.wikipedia.org

Another view of the120-cell

Source: home.inreach.com

A truly 3-d projection!

Taken by Jim King at the Park City Mathematics Institute, Summer, 2004

A 2-d projection of the 120-cell

Source: mathworld.wolfram.com

H4 root poset (sideways)

Has 429 ideals

Result for H4

• Theorem (Chen, K): There is a bijection between dominant regions in the hyperplane arrangement and all but 16 ideals in the poset of positive roots for the root system of type H4.

(these 16 correspond to empty regions)

• 413 dominant regions (355 bounded, 58 unbounded).

Related combinatorics

• In crystallographic cases, antichains called nonnesting partitions • These and other objects counted by Catalan number:

(h+di)/|W|

where W = Weyl group, h = Coxeter number, di=invariant degrees

• But numbers for I2(m), H3, H4 are not Catalan numbers

• Open question: What is a noncrystallographic nonnesting partition?