Complex Conjugate Representation (A Characterization of Self-Conjugate Maps)
Point of post: This post is a continuation of this one.
We now proceed, as promised, to give a necessary and sufficient conditions for an irrep of a space to be self-conjugate real and self-conjugate quaternonic. Namely:
Theorem: Let be a vector space, a finite group, and an irrep. Then, is self-conjugate if and only if there exists a which is antilienar and antiunitary which commutes with . If there exists such a it is unique up to a constant of modulus one. Moreover, if there exists such a then with iff is real and iff is quaternonic.
Proof: Suppose first that is self-conjugate. Then, for any complex conjugate map on there exists some unitary map such that for every . Note though that is antilinear (indeed, ) and antiunitary (indeed, ) and since it follows from this equation that and thus the conclusion follows.
Suppose then that there existed an antilinear, antiunitary map which commutes with for every . Then, picking some complex conjugate on we see that is unitary (we’ve discussed this before) and by assumption and thus and thus from where it follows that and thus is self-conjugate.
Now, to see that if such a exists then , we merely note that is unitary and for every . It follows by the second form of Schur’s lemma that for some . But, since is unitary we must have that . Note though that since we may apply to both sides to get that and thus , from where it follows that .
Now, to see that such a is unique up to a constant of modulus one, applying the same idea of using Schur’s lemma we can show that if and are any such antiunitary,antilinear commuting maps that is an intertwinor for and thus by the second form of Schur’s lemma for some . Note though that since is unitary we must have that . Note then that by the last part of the proof implies that from where the conclusion follows.
Now, suppose that . Then, by our previous theorem we have that there exists an orthonormal basis for for which . Note though that if and then
Since and was arbitrary it follows that the matrix with respect to this basis is real for each and thus is real. Conversely, suppose that is real. Then, there exists some basis for which the matrix of is real for each . Define by
it’s easily verified then that is an antilinear, antinunitary map and . But, it easily follows from the fact that has a real matrix with respect to the basis that commutes with for each and thus the converse follows.
It follows from this that if is quaternonic then any such is such that , but the above excludes the case and thus . Similarly, if then the above shows that is not real and thus must be quaternonic. The conclusion follows.
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