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icosahedral group

Contents

Contents

Idea

The icosahedral group is the group of symmetries of an icosahedron.

As a symmetry group of one of the Platonic solids, the icosahedral group participates in the ADE pattern:

ADE classification and McKay correspondence

Dynkin diagram/
Dynkin quiver
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
A n1A_{n \geq 1}cyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
SU(n+1)SU(n+1)
D4Klein four-group
D 4 2× 2D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2
quaternion group
2D 42 D_4 \simeq Q8
SO(8)
D n4D_{n \geq 4}dihedron,
hosohedron
dihedral group
D 2(n2)D_{2(n-2)}
binary dihedral group
2D 2(n2)2 D_{2(n-2)}
special orthogonal group
SO(2n)SO(2n)
E 6E_6tetrahedrontetrahedral group
TT
binary tetrahedral group
2T2T
E6
E 7E_7cube,
octahedron
octahedral group
OO
binary octahedral group
2O2O
E7
E 8E_8dodecahedron,
icosahedron
icosahedral group
II
binary icosahedral group
2I2I
E8

More in detail, there are variants of the icosahedral group corresponding to the stages of the Whitehead tower of O(3):

String 2I String SU(2) 2I Spin(3)SU(2) IA 5 SO(3) I hA 5×/2 O(3) \array{ String_{2I} &\hookrightarrow& String_{SU(2)} \\ \downarrow && \downarrow \\ 2 I &\hookrightarrow & Spin(3) \simeq SU(2) \\ \downarrow && \downarrow \\ I \simeq A_5 &\hookrightarrow& SO(3) \\ \downarrow && \downarrow \\ I_h \simeq A_5\times \mathbb{Z}/2 &\hookrightarrow & O(3) }

Definition

Regard the icosahedron, determined uniquely up to isometry on 3\mathbb{R}^3 as a regular convex polyhedron with 2020 faces, as a metric subspace SS of 3\mathbb{R}^3. Then the icosahedral group may be defined as the group of isometries of SS.

(…)

The elements of the binary icosahedral group form the vertices of the 120-cell.

More to be added.

Properties

General properties

The subgroup of orientation-preserving symmetries of the icosahedron is the alternating group A 5A_5 whose order is 60. The full icosahedral group is isomorphic to the Cartesian product A 5× 2A_5 \times \mathbb{Z}_2 (with the group of order 2).

Hence the order of the full icosahedral group is 60×2=120 60 \times 2 = 120 , as is that of the binary icosahedral group 2I2 I.

Proposition

There is an exceptional isomorphism

IPSL 2(𝔽 5) I \;\simeq\; PSL_2(\mathbb{F}_5)

of the icosahedral group with the projective special linear group over the prime field 𝔽 5\mathbb{F}_5.

and, covering this,

2ISL 2(𝔽 5) 2I \;\simeq\; SL_2(\mathbb{F}_5)

of the binary icosahedral group with the special linear group over 𝔽 5\mathbb{F}_5.

Proposition

The binary icosahedral group 2I2I is a perfect group: its abelianization is the trivial group.

In fact, up to isomorphism, the binary icosahedral group is the unique finite group of order 120 which is a perfect group.

Proposition

(quaternion group inside binary icosahedral group)

The binary icosahedral group contains the quaternion group of order 8, hence the binary dihedral group of order 8, as a subgroup (not normal):

2D 4=Q 82I. 2 D_4 =Q_8 \subset 2 I \,.

In fact the only finite subgroups of SU(2) which contain 2D 4=Q 82 D_4 =Q_8 as a proper subgroup are the exceptional ones, hence the binary tetrahedral group, the binary octahedral group and the binary icosahedral group.

See this Prop at quaternion group.

Proposition

(normal subgroups of binary icosahedral group

The only proper normal subgroup of the binary icosahedral group is its center Z(2I)/2Z(2I) \simeq \mathbb{Z}/2.

Character table

linear representation theory of binary icosahedral group 2I2 I

\,

group order: |2I|=120{\vert 2I\vert} = 120

conjugacy classes:12345A5B610A10B
their cardinality:1120301212201212

let ϕ12(1+5)\phi \coloneqq \tfrac{1}{2}( 1 + \sqrt{5} ) (the golden ratio)

character table over the complex numbers \mathbb{C}

irrep12345A5B610A10B
ρ 1\rho_1111111111
ρ 2\rho_22-2-10ϕ1\phi - 1ϕ-\phi1ϕ\phi1ϕ1 - \phi
ρ 3\rho_32-2-10ϕ-\phiϕ1\phi - 111ϕ1-\phiϕ\phi
ρ 4\rho_4330-11ϕ1 - \phiϕ\phi0ϕ\phi1ϕ1-\phi
ρ 5\rho_5330-1ϕ\phi1ϕ1-\phi01ϕ1-\phiϕ\phi
ρ 6\rho_64410-1-11-1-1
ρ 7\rho_74-410-1-1-111
ρ 8\rho_855-1100-100
ρ 9\rho_96-600110-1-1

References

  • Groupnames, SL(2,5)

  • Bockland, Character tables and McKay quivers (pdf)

Quotient spaces

The coset space SU(2)/2ISU(2)/2I is the Poincaré homology sphere.

Group cohomology

For a little bit about the group cohomology (or at least the homology) of the binary icosahedral group SL 2(𝔽 5)SL_2(\mathbb{F}_5), see Groupprops

References

Last revised on December 7, 2018 at 08:05:25. See the history of this page for a list of all contributions to it.