ISU by liuqingyan

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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

								
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