Quaternionic representation of the Binary Tetrahedral Group

(The details of this representation are due to David Stone.)

First we choose a convenient embedding of the tetrahedron in R^3. With i,j,k the standard unit vectors (these may also be interpreted correctly as unit quaternions), we consider the regular tetrahedron with vertices A = -i-j-k, B = -i+j+k, C = i-j+k and D = i+j-k.

The symmetries of order 3 of the tetrahedron (in terms of permutations of the vertices) are generated by
(BCD) = 120-degree rotation about A,
(ADC) = 120-degree rotation about B,
(ABD) = 120-degree rotation about C, and
(ACB) = 120-degree rotation about D;

the symmetries of order two are
(AB)(CD) = 180-degree rotation about i,
(AC)(BD) = 180-degree rotation about j, and
(AD)(BC) = 180-degree rotation about k.

The corresponding SU(2) elements of order six are

+/- a, +/- a^2, +/- b, +/- b^2, +/- c +/- c^2, +/- d, +/- d^2
where a = (1/2)(1+A), and similarly for b, c, d. This yields

a = (1/2) ( 1- i- j- k),
b = (1/2) ( 1- i+ j+ k),
c = (1/2) ( 1+ i- j+ k),
d = (1/2) ( 1+ i+ j- k).

(And thus
a^2 = (1/2) (- 1- i- j- k),
b^2 = (1/2) (- 1- i+ j+ k),
c^2 = (1/2) (- 1+ i- j+ k),
d^2 = (1/2) (- 1+ i+ j- k).)

The corresponding elements of order four are

+/- i, +/- j, +/- k;

1 and - 1 complete the list of 24 elements.

The Multiplication Table for the binary tetrahedral group may be conveniently calculated and written out in this notation.
Corrected September 26 2004.


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