Cathy Kriloff Idaho State University Supported in part by NSA grant MDA904-03-1-0093

1. 1 4 2 3 4 1 3 2 1 4 3 2 Indexing regions in dihedral and dodecahedral hyperplane arrangementsMAA Intermountain Sectional Meeting, March 23, 2007 Cathy Kriloff Idaho 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

2. 'Lie group E8' math puzzle solved POSTED: 10:26 a.m. EDT, March 21, 2007 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) (See www.aimath.org/E8)

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)

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) I2(5) I2(7) I2(8)

5. 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)

6. Root systems • roots = unit vectors perpendicular to reflecting lines • simpleroots = basis so each root is positive or negative • When m is even roots lie on reflecting lines so symmetries break them into two orbits I2(3) I2(4) a2 a2 a1 a1

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

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 3 5  34 2 3 3 1 4 1 3  2

9. 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 c= all i such that hyperplane is crossed as move out from origin

10. 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 changing root length, and poset changes I2(5) 5 1 2 4 3

11. 3 1 1 2 3 4 5 2 2 3 4 5 1 2 3 4 2 3 4 5 34 2 3 3 1 4 3  2 Root poset for I2(3) Root poset for I2(5) Ideals index dominant regions 1 5 2 4 3 Ideals for I2(3) Ideals for I2(5) 1 2 3 4 5 2 3 4 5 1 2 3 4 2 3 4 3 4 2 3 3  1 2 3 1 2 2 3 2 

12. 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

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)

14. H3 and H4 • Can generalize I2(5) to:H3 = symmetries of 3-dim dodecahedronH4 = symmetries of regular 4-dimensional solid, hyperdodecahedron or 120-cell (with 120 3-dim dodecahedral faces) • I2(5), H3, and H4 related to quasicrystals

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. 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).

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?