## Permutation Representation

**Point of Post: **In this post we discuss the notion of the permutation representation and its relation to induced characters.

*Motivation*

In this post we discuss a very fruitful notion known as the permutation representation of a group . In particular, we will show that given a finite group which acts on a finite set there is a natural way to define a representation of on the free vector space . Namely, we will merely have . This is a very fruitful form of representation which occurs very often, particularly in the representations of .

*Permutation Representation*

Let be a finite group which acts on the set with action denoted, as usual, . Consider then the free vector space . Consider then the map defined by

To see that in fact we merely note that permutes the basis . We then define an inner product on by declaring that is orthonormal and extending by sequilinearity. Clearly then is unitary within this inner product since it carries the orthonormal basis onto itself. Then this representation is known as the *permutation representation of the action on *. The first obvious question is what does the character of look like? What we note then is that for any the diagonal entry of associated to (in the ordered basis is either if and if . In other words

What we note though is this:

**Theorem: ***Let be a finite group which acts on the finite set , suppose that breaks up into oribts as where is transveral for the set of orbits . Then, if is permutation representation of on for every , is -invariant and *

*where is the map *

**Proof: **The fact that each is -invariant is obvious since by definition if is a basis vector for and is arbitrary then . To note the second part it suffices to note that

**Corollary: ***Let act on and let be the associated permutation character. Then, if is a transversal for and as described above then .*

**References:**

Fulton, William, and Joe Harris. *Representation Theory: a First Course*. New York: Springer-Verlag, 1991. Print.

No comments yet.

## Leave a Reply