## A Characterization of Irreducibility

**Point of post: **In this post we give an easy way to check whether or not a given representation of a finite group is irreducible in terms of the representations character.

*Motivation*

Often given a representation of a finite group it is difficult to look at it and decide whether or not it is irreducible. In this post we prove a theorem which enable us to decide whether or not a given representation is irreducible by checking whether or not the inner product of its induced character with itself is unity.

*Numerical Check for Irreducibility*

Let’s get right to it.

**Theorem: ***Let be a finite group and a representation of with character . Then, is irreducible if and only if *

**Proof: **Evidently if is irreducible then for some and thus and the rest follows from the orthonormality relations for the irreducible characters.

Conversely, suppose that . We know that for any chosen set of representations for each there exists and a unitary such that

it follows then that and thus

and thus by assumption that we may conclude that there exists some such that

from where it follows that and thus is irreducible as desired.

**References:**

1. Isaacs, I. Martin. *Character Theory of Finite Groups*. New York: Academic, 1976. Print.

2. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

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