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Indexing regions in dihedral and dodecahedral hyperplane arrangements ISU Colloquium, April 4, 2007 Cathy Kriloff Idaho State University Supported in part by NSA grant MDA904-03-1-0093 Joint work with Yu Chen 1 4 to appear in Journal of Combinatorial Theory – Series A 2 3 1 4 3 2 1 4 3 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) Symmetries of familiar objects • 3 reflections • 4 reflections • 6 reflections • 3 rotations • 4 rotations • 6 rotations Groups: S3 I2(3) S2 ⋉(Z2)2 I2(4) I2(6) Generated by reflections • These symmetry groups are generated by reflections B reflect in l A reflect in m A l C A C m B B C combination of two reflections = rotating 120 counterclockwise about center point 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 • 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 1 2 1 4 2 3 4 3 3 2 2 3 1 1 Dominant regions • Goal: understand 2-dimensional regions in the dominant cone • Can describe as an intersection of half- spaces formed by bounding hyperplanes i.e., solutions to a system of linear inequalities created using the root vectors • Suffices to only record “>” inequalities 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 123 I2(3) I2(5) 45 123 23 23 45 12 34 2 12 2 234 3 5 34 3 23 1 4 1 3 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 12 12 34 12 34 34 234 234 123 234 123 23 123 23 23 2 3 2 2 Which subsets of {1,2,3,4} appear? Root posets and ideals • Express positive j in ai basis I2(3) I2(4) • Ordering: 3 a≤ if -a ═ciai with ci≥0 1 1 4 2 2 3 • Connect by an edge if comparable 1 4 • Increases going down I2(5) 3 1 5 • Pick any set of incomparable roots (antichain), , and form 2 its ideal= a for all a 2 4 1 4 3 • x, i =c x, i /c=1 so moving hyperplane in orbit 3 changing root length in orbit, and poset changes 2 Root poset for I2(3) Root poset for I2(5) Ideals index 3 1 5 dominant regions 1 2 2 4 123 3 45 23 Ideals for I2(3) Ideals for I2(5) 45 12 12345 34 123 2345 1234 12 23 234 234 5 34 2 34 23 3 23 1 4 3 3 2 Correspondence for m even 12 12 34 12 34 34 234 234 123 234 123 23 123 23 23 2 3 2 2 1 4 1 4 1 4 3 3 2 3 2 2 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 with 12 pentagons, 30 edges, 20 vertices H4 = symmetries of regular 4-dimensional solid, hyperdodecahedron or 120-cell 120 dodecahedra, 720 pentagons, 1200 edges, 600 vertices • I2(5)=H2, H3, H4 used to build 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 ideals not containing any ai 12 unbounded ideals containing some ai Proof sketch: Proof for I2(m), H3: Use interplay between various antichains and ideals. Relate solution of linear equations from antichains to solution of linear inequalities from ideals. Get criterion for region associated to a smaller ideal to be nonempty. Use this to see all ideals give rise to nonempty regions. 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: Drawn by Chilton, in Coxeter’s Introduction to Geometry. Also see photograph of wire model in Regular Polytopes This image taken from online version of Story of the 120-cell References on the 120-cell • The Story of the 120-cell by J. Stillwell, AMS Notices, January, 2001. “One of the most beautiful objects in mathematics” Lives in R4, S3, and quaternions H “Encodes the symmetry of the icosahedron and the structure of the Poincaré homology sphere” • Princess of Polytopia: Alicia Boole Stott and 120-cell by T. Phillips, Feature Column at www.ams.org, October, 2006. Built cardboard models of 3-d sections of 120-cell 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 antichains give empty regions: {a16} {a14,a21} {a16,a25} {a14,a21,a23} {a13,a16} {a14,a25} {a18,a21} {a14,a23,a25} {a13,a20} {a16,a17} {a18,a25} {a14,a25,a28} {a14,a19} {a16,a21} {a21,a22} {a18,a25,a28} • 413 dominant regions (355 bounded, 58 unbounded). Proof sketch Proof for H4: Verify 401 regions are nonempty as for I2(m), H3. Show 16 are empty by simple calculation that yields a contradiction. Show 12 are nonempty by solving systems of linear equations. Why the bijection fails: Theorem (Chen, K): For any root system, the function from regions to ideals is a bijection if and only if for every antichain, A, the system of linear equations, x,=1 for all A, has a solution in the dominant chamber. Related combinatorics • In crystallographic cases, antichains called nonnesting partitions • Generalized Catalan number: Cat(W)=(h+di)/|W| where W = Weyl group, h = Coxeter number, di=invariant degrees • Cat(W) counts antichains & nilpotent ideals for crystallographic W, and other objects, even in noncrystallographic cases: - vertices in simplicial associahedra (Fomin-Zelevinsky) - elements in interval in geometric group theory (Bessis, Brady) • But numbers for I2(m), H3, H4 are not Catalan numbers! • Open question: What is a noncrystallographic nonnesting partition? Use in representation theory • Associated graded Hecke algebra has a unitary representation at x in dominant chamber operator (A(x)) is positive semidefinite for all irreducible representations, , of W. • Affine hyperplanes are where (A(v)) is singular. Signature of (A(v)) constant on regions finite calculation. • In crystallographic case, goal is to understand unitary dual of real Lie groups and p-adic groups – deep research by Vogan, Barbasch, Moy, Ciubotaru,… hard computation due to Adams, DuCloux, Stembridge,… Support of the spherical unitary dual for I2(2m) • Conjecture (Chen, K): 2-dim regions supporting unitary representations of H(I2(2m)) correspond to ideals of even size containing roots of a single parity. Connection to breaking news s5 s6 s7 s8 • E8, the largest exceptional crystallographic reflection group, contains H4 as a subgroup s2 s4 s3 s1 5 'Lie group E8' math puzzle solved s2s5 s4s6 s3s7 s1s8 POSTED: 10:26 a.m. EDT, March 21, 2007 The Scientific Promise of Perfect Symmetry March 20, 2007, Tuesday By KENNETH CHANG (NYT); Science Desk Somewhat more descriptive: www.aimath.org/E8 Even better: Atlas of Lie Groups at www.umd.edu/atlas