Representation Theory: Definitions and Basics (Pt. II Equivalent Representations)
Point of post: This post is a continuation of this one.
Let be a finite group and and be two representations of . We say that is equivalent to if there exists some unitary map such that for every . We denote this relation symbolically by We now prove that since each is unitary this is equivalent to stating that there just exists some isomorphism such that . Indeed:
Theorem: Let and be two unitary representations. Suppose then that there existed some isomorphism such that for each . Then, there exists some unitary map such that for each .
Proof: Note that since we clearly have that and then taking the inverse of both sides (recalling that are unitary and thus etc.) we find that using the fact though that we may put these two together to conclude that
or by rearranging
we may conclude by the uniqueness of the square root that
Note then that we may use the polar decomposition of to write where is unitary. Note then that
from where the conclusion follows.
Both these examples show why we so much desire the unitarity of our representations. It makes everything easier. We’ll see that this is carried further in the next post.
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.