## 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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