Download H - Idaho State University

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 MDA904-03-1-0093
Joint work with Yu Chen, Idaho State University
to appear in Journal of Combinatorial Theory – Series A
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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)
'Lie group E8' math puzzle solved
POSTED: 10:26 a.m. EDT, March 21, 2007
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
• 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
• When m is even roots lie on reflecting lines so
symmetries break them into two orbits
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
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3
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4
2
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3
2
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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
I2(3)
123
I2(5)
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23
123
45
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234
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
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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
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34
12
34
234
234
234
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123
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Root posets and ideals
• Express positive j in ai basis
• Ordering:
a≤ if -a ═ciai with ci≥0
I2(3)
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1
• Connect by an edge if
comparable
2
• Increases going down
I2(5)
• Pick any set of incomparable
roots (antichain), , and form
its ideal=  a for all a
•
x, i =c  x, i /c=1 so
moving hyperplane 
changing root length,
and poset changes
I2(4)
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Root poset for I2(3)
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Root poset for I2(5) Ideals index
dominant regions
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Ideals for I2(3)
Ideals for I2(5)
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12345
2345
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1234
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Correspondence for m even
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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 arrangement
Dominant 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?