Complex Conjugates (Quaternionic Irreps Are of Even Degree)
Point of post: In this post we show that every quaternionic irrep has even degree.
We are now able to give the final theorem regarding quaternionic representations. We will require the knowledge of the quaternion algebra (if this is unfamiliar to you, see here). We then can consider as a left -module with the usual operations. We claim that every representation space of a quaternionic representation can be give the structure of such a left -module. But, since is a division ring every finitely generated left -module is free and also have invariant basis number. It follows then that if is a representation space of a quaternionic irrep then where (defined below) and thus in particular the degree of every quaternionic irrep is even.
Theorem: Let be a finite dimensional vector space such that there exists some antilinear map such that . Then, can be given the structure of a left -module.
Proof: It’s clear that for every we have that can be written uniquely as for . Indeed, uniqueness is clear and existence follows since we can note that so that for one has that
So, for with we define for by . We claim that this along with the already existing abelian group structure on defines a left -module structure on , which we shall denote . Indeed, we need to prove that this definition of scalar multiplication satisfies vector distributivity , left scalar distributivity , associativity , and . Indeed, to prove we note that if and
To prove scalar distributivity we need that for any and ones that
To prove we merely note that if and then
Note though that
and thus comparing these shows that . Lastly, axiom is verified trivially. Thus, all the axioms are satisfied and thus the conclusion follows.
Remark: The converse is clearly also true. Indeed, if is such that it admits a quaternionic vector space structure then the map is a antilinear map which squares to .
We now claim that if a vector space above satisfies these properties then it must have even complex dimension.
Theorem: Let be a vector space such that there exists an antilinear map such that . Then, .
Proof: By our previous theorem we know that admits a quaternionic structure with multiplication . We claim then that if is a basis for then is a basis for . Indeed, it’s clear that is linearly independent since if are such that
but this implies that but this is only true if from where linear independence follows. Now, to see that we merely note that for any there exists such that
But, there exists such that and thus
from where . It follows then that from where the conclusion follows.
With this and our previous theorem regarding characterization of quaternionic irreps we may conclude that every quaternonic irrep has even degree.
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