## Representation Theory: Definitions and Basics (Pt. II Equivalent Representations)

**Point of post: **This post is a continuation of this one.

*Equivalent Representations*

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

Note that since (this notation means that is positive-definite) by general principles we know that has some unique positive-definite square root . Noting then that is positive-definite and

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.

**References:**

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.

[…] Indeed, by Schur’s lemma we know that must be an isomorphism such that , and thus by earlier theorem it follows that as […]

Pingback by Representation Theory: Schur’s Lemma (First and Second Forms) « Abstract Nonsense | January 27, 2011 |

[…] that (since is -invariant) or for every –and by an earlier theorem we may conclude that is self-conjugate. Thus, there exists some unitary such that for every . […]

Pingback by Representation Theory: A Way of Creating C-representations Satisfying the Real Condition With No (rho,J)-invariant Subspaces (Pt. II) « Abstract Nonsense | April 2, 2011 |

i understand that ρg commutes with the square root of T*T (= P).

but shouldn’t UρgU^-1 be TP^-1ρgPT^-1?

unless T*T = I i don’t see how we can conclude P^-1 = P.

Comment by David Wheeler | April 11, 2011 |

Dear David,

I agree made a typo, but I believe I am missing what you’re saying. We have that and so and and so . Which is what we wanted. Is there something I’m missing?

Best,

Alex

P.S. Thanks for catching the typo! Please let me know if you catch any more!

Comment by drexel28 | April 11, 2011 |

[…] And, since was arbitrary we may appeal to a previous theorem. […]

Pingback by Another Way of Looking at Induced Representations (Pt. II) « Abstract Nonsense | April 24, 2011 |