1. Indexing regions in dihedral and dodecahedral hyperplane arrangements MAA Intermountain Sectional Meeting, March 23, 2007
Cathy Kriloff
Idaho State University
Supported in part by NSA grant MDA
Joint work with Yu Chen, Idaho State University
to appear in Journal of Combinatorial Theory – Series A 3 2 4 1

2. 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)
                                      
I usually study quite abstract (infinite-dimensional) graded Hecke algebras. Today will talk about a more accessible combinatorial detour
First must introduce the setting of noncrystallographic reflection groups.
Result will provide a bijection between two types of objects.
On the one hand, geometric regions in hyperplane arrangements associated to root systems of the reflection groups
On the other hand, combinatorial objects like ideals or antichains in a partial order on root vectors.
Will illustrate and state results on bijections for dihedral types and H3 and H4.
Original motivation thanks to Peter Trapa is to study unitary representations of noncrystallographic graded Hecke algebras as an outgrowth of the Atlas of Lie Groups project he is involved in (and that just made the news on CNN!) – in fact H4 root system sits inside E8
But also led to interesting combinatorial questions.
'Lie group E8' math puzzle solved
POSTED: 10:26 a.m. EDT, March 21, 2007
(See

3. 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)
Begin with the setting we will work in. Start with something simple…
Reflection groups are finite groups generated by reflections.
Crystallographic because can tile space with analogues of eq. triangles, squares, regular hexagons.
Generalizations to n dimensional standard simplex and hypercube with symmetry groups the permutations of 1 to n and an extension of S_n by n copies of Z2.
Lying slightly as not all crystallographic reflection groups are symmetries of regular polytopes.
If instead stay in 2 dimensions can generalize in a different way… by increasing the number of sides

4. 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)
Then symmetry group is the dihedral group of order 2n
Setting for next while will be symmetries of regular n-gon.
Our combinatorial result provides a perfect matching or pairing (a bijective correspondence, one-to-one and onto) between dominant regions in a hyperplane arrangement and ideals in a partial ordering of root vectors.
So begin to introduce examples to illustrate these terms.
In place of reflecting lines, consider vectors perpendicular to them.
Call these roots.
I2(5)
I2(7)
I2(8)

5. Reflection groups
There is a classification (Coxeter , 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)

6. Root systems
roots = unit vectors perpendicular to reflecting lines
simple roots = basis so each root is positive or negative
I2(3)
I2(4) a2 a2 a1 a1
Will call triangle example I2(3) and use it despite being crystallographic since simpler.
Two vectors perpendicular to each line so 6.
For I2(4), there are 8.
What difference do you notice?
Roots lie on reflecting lines or do not. When they do, has important mathematical consequences.
Next introduce some additional lines.
When m is even roots lie on reflecting lines so symmetries break them into two orbits

7. Hyperplane arrangement
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 1 1 2 4 2 3 3 4 3 b2 2 3 b1 1
Hyperplane is an n-1-dim subspace of an n-dim space, so in 2-dim space, are just lines.
Add shifted hyperplanes and number roots from 1 to n
Notice that other roots can be written in terms of basis alpha_1 and alpha_n. –e.g. alpha 2=alpha 1+alpha 3, etc.
Interested in the 2-d regions formed as the lines intersect.
Notice that symmetry helps reduce to a simpler problem – focus on just one cone, called dominant chamber,
bounded by black lines. Interested in understanding the 2-dim regions in the dominant chamber for all n!
Easy for I2(3), not as easy for I2(4), both because larger and because when n is even, two orbits of hyperplanes are independent of one another and can move one of them relative to the other so number of regions changes.
Just as reflecting hyperplane is vectors with inner product 0 with a root vector, affine hyperplane is vectors with inner product c with a root vector for some fixed c. Can view moving the hyperplane either as changing c or as scaling the root/changing its length. Will be important later.
This is our hyperplane arrangement. Now consider how to keep track of regions.

8. Indexing dominant regions
Label 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 2 3
4 5
I2(3)
I2(5)
1 2 3
2 3
4 5
2 3
1 2
3 4
1 2 2 2
2 3 4
To keep track of dominant regions, start near center point, walk out along rays in cone, keep track of which lines you cross.
Details…
being on one side of a line is equivalent to satisfying a linear inequality, so a region is described by a system of linear inequalities.
When finish, go back to slide 6, then cut out to GSP, then return and click on slide 8 to start again. 3 5 
3 4
2 3 3 1 4 1 3  2

9. 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 2
3 4 2 3
1 2 3
2 4
2 3
2 3 4
Same recording principle. Remember that we must keep

10. Root posets and ideals I2(3) I2(4) I2(5)
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  changing root length, and poset changes
I2(3)
I2(4) 1 3 2 3 2 4 1
I2(5) 5 1
Poset is short for partially ordered set. 2 4 3

11. Root poset for I2(3) Root poset for I2(5) Ideals index
dominant regions 1 3 2 1 5 2 4  2 1 3 4 5
1 2 3
4 5
1 2
3 4
2 3
2 3 4 3
Ideals for I2(3)
Ideals for I2(5)
2 3 4 3 
1 2 3 2 

12. Correspondence for m even
1 2
3 4 2 3
1 2 3
2 4
2 3
2 3 4 1 4 1 4 4 1 3 3 2 3 2 2

13. 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)
In remaining time, want to discuss the only two other noncrystallographic cases by generalizing symmetries of the regular pentagon to higher dimensions. Turns out, can only generalize to dimensions 3 and 4!

14. 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 dim dodecahedral faces)
I2(5), H3, and H4 related to quasicrystals
Recall we said there are analogues of the eq. triangle and square in all dimensions.
SHOW MODELS
Here are the 3-d versions
Can also find analogue of I2(5) in dimension 3

15. H3 root system Roots = edge midpoints of dodecahedron or icosahedron
Source: cage.ugent.be/~hs/polyhedra/dodeicos.html

16. H3 hyperplane arrangement
Dominant regions are enclosed by yellow, pink, and light gray planes

17. H3 root poset
Has 41 ideals

18. 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).

19. A 3-d projection of the 120-cell
Source: en.wikipedia.org

20. Another view of the120-cell
Source: home.inreach.com

21. Taken by Jim King at the Park City Mathematics Institute, Summer, 2004
A truly 3-d projection!
Taken by Jim King at the Park City Mathematics Institute, Summer, 2004

22. A 2-d projection of the 120-cell
Source: mathworld.wolfram.com

23. H4 root poset (sideways)
Has 429 ideals

24. 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).
Proof for I2(m), H3: Interplay between various antichains and ideals. Relate solution of linear equations from antichains to solution of linear inequalities from ideals to get criterion for region associated to a smaller ideal to be nonempty.
Proof for H4: Verify all but 28 of 429 regions are nonempty as before. Show 16 are empty by fairly simple calculation for contradiction. Show 12 are nonempty by solving systems of linear equations.
Other objects counted by Catalan number:
Vertices in generalized associahedra introduced in recent work of Fomin-Zelevinsky.
Noncrossing partitions in geometric group theory.
These have been generalized to noncrystallographic cases and still counted by product formula.

25. 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?