Complex Conjugate Representations (An Example)
Point of post: This post is a continuation of this one.
It turns out that this classification is still too broad in the sense that there is a natural way in which self-conjugate maps can be further classified. The self-conjugate class breaks naturally down into whether for a given there exists a basis on ‘s representation space for which ‘s matrix representation has all real entries for each . To illustrate this concept further we consider the following example
Example: Consider the quaternion group with presentation
which has explicit Cayley Table
and define the representation by
This is easily seen to be an irrep. Moreover, a quick computation shows that if is the usual complex conjugate on (the one which conjugates each entry of a tuple, i.e. ) then
and thus is self-conjugate. That said, suppose that is a basis for such that has real entries. Note then that
forms a basis for and thus there must exists constants for which
But, we know that the trace is conjugation (and thus coordinate invariant) so that taking the trace of both sides, using the linearity of the trace function and the fact that the non-identity matrix elements of this matrix basis are traceless we may conclude that and thus . But, note that
from where it easily follows from the assumption that this matrix is real that are real and is pure imaginary. Note though that from where it follows that . But, by assumption
and thus . But, noticing that and one may conclude that for one has that there exists such that and
though a quick check shows that this is impossible. Indeed, noticing that in particular
But this is quickly shown to be inconsistent. It thus follows that no such basis exists.
Thus, it follows that there is a distinction that should be made on the set of self-conjugate irreps. In particular, we call a self-conjugate irrep real if there exists a basis for for which each entry of is real for every . If is self-conjugate but not real it’s called quaternionic. The previous example shows the existence of quaternionic irreps. These shall prove to be interesting in the theory to come.
1.Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.
2. Serre, Jean Pierre. Linear Representations of Finite Groups. New York: Springer-Verlag, 1977. Print