## General Characters and the Uniqueness of Decomposition Into Irreps

**Point of post: **In this post we use our results about irreducible characters to show that any decomposition of a representation into a direct sum of irreps is unique. We do this by introducing the notion of a character for a general (not necessarily irreducible) representation.

*Motivation*

In our last few posts we saw that every irrep of a finite group produces a class function namely the irreducible character given by that representation. That said, one can plainly see that the methodology of producing this class function had nothing to do with the fact that the representation was irreducible(this was important to prove that any set of irreducible characters, one from each equivalency class , forms a basis for the space of class functions ). Thus, using the same definition we will define the character of a general representation and use this notion to show that for any arbitrary but fixed set of representative irreps for each the decomposition of (where denotes how many times appears in the direct sum) is unique.

*General Characters*

Let be a finite group. Then for any representation we define the *character *of , denoted to be for every . It’s clear that since the trace function is invariant under conjugation and basis choice one has that whenever . In particular it’s easy to see that although we fixed some representative in that any representative will give the same character (we fixed it for clerical convenience–although on occasion we shall most likely invoke this independence). Some things are, just as before, immediately clear. Namely:

**Theorem: ***Let be a finite group and any representation with character . Then for every *

The proof of this theorem is precisely the same as in the case for irreducible characters. Another very obvious theorem (one we have implicitly used before) is:

**Theorem: ***Let be a finite group and and be two representations of with characters and respectively. Then, if is the tensor product of and then . If is the direct sum of and then .*

**Proof: **This follows almost immediately by looking at the matrix representations for and with respect to the tensor basis (lexicographically ordered) and the direct sum basis (any ordering) respectively.

*Uniqueness of Decomposition*

We now show that given any representation and distinguished representatives from each then the (guaranteed) decomposition is unique. Indeed:

**Theorem: ***Let be a finite group and a representation of . Then, given distinguished representatives for each the decomposition of is unique (up to reordering).*

**Proof: **Suppose that

then by taking the trace we may conclude that

and since forms an orthonormal basis for we may conclude that for every and since everything was arbitrary the conclusion follows.

*Remark: *It’s also clear from this that two representations are equivalent if and only if they admit the same character.

**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

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