Complex Conjugate Representations
Point of post: In this post we discuss the notion of complex conjugates on pre-Hilbert spaces and how to produce new representations based on them
Anyone acquainted with basic concepts of complex numbers is well aware of the power of the conjugation map. It would then seem fruitful to try to abstractify the important concepts of this map to general pre-Hilbert spaces. It then seems natural to figure out if one can take these generalizations and produce from them new representations. It turns out that this is indeed fruitful, but much more can be said than is at first apparent. In fact, it turns out that these new representations can be classified into one of three types which model, roughly, the difference between real, complex, and quaternionic numbers.
Let be a pre-Hilbert space. Then a map is called a complex conjugate if
This is evidently a generalization of the conjugation map on since the map is a complex conjugate.
Remark: It should be noted that the set of complex conjugates does not form an algebra under composition since the composition of two antilinear maps is not antilinear (in general).
Our first result is that considering complex conjugates gives us the possibility of building many new representations out of old ones. Namely, if is a specified complex conjugate on a pre-Hilbert space , a finite group, and a representation we define the conjugation representation of with respect to , denoted , to be the map
It’s not immediately apparent that this mapping is well-defined (in the sense that is always unitary) or that it is representation, that is what we shall now show:
Theorem: Let be a pre-Hilbert space with complex conjugate , a finite group, and a representation. Then, for every one has that is unitary, and that the resulting map is a homomorphism
Proof: To see that is unitary we merely note that for any one has that
To see that the map is a homomorphism it suffices to note that
from where the conclusion follows.
Corollary: If is a representation then so is is a representation.
Our next theorem shows how the irreducibility of a conjugation representation and the irreducibility of the original representation interact, namely:
Theorem: Let be a pre-Hilbert space with complex conjugate , a finite group, and a representation. Then, is an irrep if and only if is.
Proof: Suppose first that is an irrep and let be -invariant. Then we see that is -invariant. Indeed, for any and one has that so that is invariant. It follows then that or . Thus, appealing to a dimension argument (and using that is invertible) we may conclude that or . And, since was arbitrary it follows that is an irrep as desired.
To prove the converse we merely note that if were an irrep, then by the previous part of the problem we know that is irreducible. Note though that for every
and thus from where the conclusion follows.
It’s clear that different choices of complex conjugates will produce different conjugate representations. More directly, if are distinct complex conjugates and a representation there’s no reason to believe that . That said, our next theorem will show that there equivalence class (under the usual definition of representation equivalence)
Theorem: Let be a pre-Hilbert space with complex conjugates and . Then, for any finite group and representation one has that .
Proof: Merely note that is unitary since for any one has that
and that . Thus for any we have that
And since was arbitrary it follows that if we define that is a unitary endomorphism on and for ever every and so trivially .
It follows then that if is an irrep and and are any complex conjugates on then where is the equivalency class in .
With this information in mind we separate irreps into two classes. Namely, we say that an irrep is self-conjugate if for any complex conjugate on one has that . We call if for every complex conjugate , complex if . Because of our previous theorem we see that both of these universal quantifiers may be changed to existential ones.
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