0
$\begingroup$

I understand the example below

"the natural representation of the symmetric group $S_n$ in $n$ dimensions by permutation matrices, which is certainly faithful. Here the order of the group is n! while the $n×n$ matrices form a vector space of dimension n2. As soon as $n$ is at least $4$, dimension counting means that some linear dependence must occur between permutation matrices (since $24 > 16$); this relation means that the module for the group algebra is not faithful."

My question -- furthermore, what are the simple ways to explain the distinction between Faithful representation v.s. unFaithful representation? By simple words to a high school or undergrad level math learner?

  • 0
    Faithful mean that the morphism $G \to GL(V)$ is injective. For example, The representation $S_n \to \mathbb C, a \to \sigma(a)$ where $\sigma$ is the signature is not faithful for $n > 2$. – 2017-01-03

3 Answers 3

1

If you think about a representation as a linear action of a group $G$ on a vector space $V$, then a faithful action means that every non-identity element of your group moves the vector space-- so for every $g \in G$, there is some $v \in V$ so that $g.v\not= v$. So if a representation is not faithful, there is some element of your group that does nothing at all.

As the other answers indicate, if you think about a representation as a map $G \to GL(V)$, then the above criteria for faithfulness says that a representation is faithful if the map $G \to GL(V)$ is injective, so a faithful representation realizes $G$ as a subgroup of $GL(V)$, so as a group of matrices.

4

Faithful means that every distinct element of the group corresponds to a distinct representation matrix.

A trivial example of a nonfaithful representation is the trivial representation where every group element maps to 1. This is a representation, but not very 'faithful' in that it doesn't have all the information in the group in it.

By contrast, the natural representation of a finite group like $S_4$ is faithful. If you write it out, you will see that there are 24 distinct matrices in the representation.

  • 0
    Nothing bad intended, but if you don't yet know the basics of rep theory, the paragraph you quoted seems advanced. I'd look for a more introductory treatment. – 2017-01-04
1

Faithful representation is "A representation that is a monomorphism."