The binary tetrahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism Spin(3) ≅ Sp(1), where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
Elements
Symmetry projections
8-fold
12-fold
24 quaternion elements:
1 order-1: 1
1 order-2: -1
6 order-4: ±i, ±j, ±k
8 order-6: (+1±i±j±k)/2
8 order-3: (-1±i±j±k)/2.
Explicitly, the binary tetrahedral group is given as the group of units in the ring of Hurwitz integers. There are 24 such units given by
with all possible sign combinations.
All 24 units have absolute value 1 and therefore lie in the unit quaternion group Sp(1). The convex hull of these 24 elements in 4-dimensional space form a convex regular 4-polytope called the 24-cell.
This sequence does not split, meaning that 2T is not a semidirect product of {±1} by T. In fact, there is no subgroup of 2T isomorphic to T.
The binary tetrahedral group is the covering group of the tetrahedral group. Thinking of the tetrahedral group as the alternating group on four letters, T ≅ A4, we thus have the binary tetrahedral group as the covering group, 2T ≅ .
where Q is the quaternion group consisting of the 8 Lipschitz units and C3 is the cyclic group of order 3 generated by ω = −1/2(1 + i + j + k). The group Z3 acts on the normal subgroup Q by conjugation. Conjugation by ω is the automorphism of Q that cyclically rotates i, j, and k.
One can show that the binary tetrahedral group is isomorphic to the special linear group SL(2,3) – the group of all 2 × 2 matrices over the finite fieldF3 with unit determinant, with this isomorphism covering the isomorphism of the projective special linear group PSL(2,3) with the alternating group A4.
A Cayley Table with these properties, elements ordered by GAP, is
1 2 r 4 -1 6 7 8 9 10 11 12 13 s 15 16 17 t 19 20 21 22 23 24
2 6 7 8 9 1 13 s 15 16 17 t r 4 -1 20 21 22 23 10 11 12 24 19
r 8 -1 10 11 20 23 9 t 12 1 19 s 21 24 7 16 2 4 15 22 13 17 6
4 16 19 -1 12 13 8 17 23 r 10 1 15 20 21 9 t 7 11 22 6 24 2 s
-1 9 11 12 1 15 17 t 2 19 r 4 21 22 6 23 7 8 10 24 13 s 16 20
6 1 13 s 15 2 r 4 -1 20 21 22 7 8 9 10 11 12 24 16 17 t 19 23
7 s 9 16 17 10 24 15 22 t 2 23 4 11 19 13 20 6 8 -1 12 r 21 1
8 20 23 9 t r s 21 24 7 16 2 -1 10 11 15 22 13 17 12 1 19 6 4
9 15 17 t 2 -1 21 22 6 23 7 8 11 12 1 24 13 s 16 19 r 4 20 10
10 7 4 11 19 s 9 16 17 -1 12 r 24 15 22 t 2 23 1 13 20 6 8 21
11 t 1 19 r 24 16 2 8 4 -1 10 22 13 20 17 23 9 12 6 s 21 7 15
12 23 10 1 4 21 t 7 16 11 19 -1 6 24 13 2 8 17 r s 15 20 9 22
13 4 15 20 21 16 19 -1 12 22 6 24 8 17 23 r 10 1 s 9 t 7 11 2
s 10 24 15 22 7 4 11 19 13 20 6 9 16 17 -1 12 r 21 t 2 23 1 8
15 -1 21 22 6 9 11 12 1 24 13 s 17 t 2 19 r 4 20 23 7 8 10 16
16 13 8 17 23 4 15 20 21 9 t 7 19 -1 12 22 6 24 2 r 10 1 s 11
17 22 2 23 7 19 20 6 s 8 9 16 12 r 10 21 24 15 t 1 4 11 13 -1
t 24 16 2 8 11 22 13 20 17 23 9 1 19 r 6 s 21 7 4 -1 10 15 12
19 17 12 r 10 22 2 23 7 1 4 11 20 6 s 8 9 16 -1 21 24 15 t 13
20 r s 21 24 8 -1 10 11 15 22 13 23 9 t 12 1 19 6 7 16 2 4 17
21 12 6 24 13 23 10 1 4 s 15 20 t 7 16 11 19 -1 22 2 8 17 r 9
22 19 20 6 s 17 12 r 10 21 24 15 2 23 7 1 4 11 13 8 9 16 -1 t
23 21 t 7 16 12 6 24 13 2 8 17 10 1 4 s 15 20 9 11 19 -1 22 r
24 11 22 13 20 t 1 19 r 6 s 21 16 2 8 4 -1 10 15 17 23 9 12 7
There is 1 element of order 1 (element 1), one element of order 2 (), 8 elements of order 3, 6 elements of order 4 (including ), 8 elements of order 6 (which include and ).
All other subgroups of 2T are cyclic groups generated by the various elements, with orders 3, 4, and 6.[4]
Higher dimensions
Just as the tetrahedral group generalizes to the rotational symmetry group of the n-simplex (as a subgroup of SO(n)), there is a corresponding higher binary group which is a 2-fold cover, coming from the cover Spin(n) → SO(n).
The rotational symmetry group of the n-simplex can be considered as the alternating group on n + 1 points, An+1, and the corresponding binary group is a 2-fold covering group. For all higher dimensions except A6 and A7 (corresponding to the 5-dimensional and 6-dimensional simplexes), this binary group is the covering group (maximal cover) and is superperfect, but for dimensional 5 and 6 there is an additional exceptional 3-fold cover, and the binary groups are not superperfect.
Usage in theoretical physics
The binary tetrahedral group was used in the context of Yang–Mills theory in 1956 by Chen Ning Yang and others.[5]
It was first used in flavor physics model building by Paul Frampton and Thomas Kephart in 1994.[6]
In 2012 it was shown [7] that a relation between two neutrino mixing angles,
derived
[8]
by using this binary tetrahedral flavor symmetry, agrees with experiment.
Conway, John H.; Smith, Derek A. (2003). On Quaternions and Octonions. Natick, Massachusetts: AK Peters, Ltd. ISBN1-56881-134-9.
Coxeter, H. S. M. & Moser, W. O. J. (1980). Generators and Relations for Discrete Groups, 4th edition. New York: Springer-Verlag. ISBN0-387-09212-9. 6.5 The binary polyhedral groups, p. 68