Abstract Nonsense

Crushing one theorem at a time

Dimension of the Space of Class Functions


Point of post: In this post we prove the simple result that the dimension of the space of class functions is equal to the number of conjugacy classes in G.

Motivation

In our last post we hinted that the dimensionality of the space \text{Cl}(G) of class functions of the finite group G shall be used to derive a very interestint result. As a step toward this we prove in this post that the dimension of \text{Cl}(G) thought of as a subspace of the group algebra \mathcal{A}(G) is the number of conjugacy classes of G .

Dimension of \text{Cl}(G)

Let G be a finite group and \text{Cl}(G) the space of all class functions on G. Furthermore, let C_1,\cdots,C_k be the  distinct conjugacy classes of G then:

Theorem: \dim_{\mathbb{C}}\text{Cl}(G)=k.

Proof: For each j\in[k] let (as usual) \mathbf{1}_{C_j}:G\to\mathbb{C} denote the indicator function of C_j. We claim that \left\{\mathbf{1}_{C_j}\right\}_{j\in[k]} forms a basis for \text{Cl}(G) (note that each \mathbf{1}_{C_j} is evidently a class function itself since it’s constant on the conjugacy classes of G). Clearly \left\{\mathbf{1}_{C_j}\right\}_{j\in[k]} is linearly independent since if c_1,\cdots,c_k\in\mathbb{C} were such that

\displaystyle \sum_{j=1}^{k}c_j\mathbf{1}_{C_j}=\bold{0}

then, choosing some representative r_\ell\in C_\ell for \ell\in[k] we may conclude by the pairwise-disjointness of \left\{C_j:j\in[k]\right\} that

\displaystyle 0=\sum_{j=1}^{k}c_j\mathbf{1}_{C_j}(r_\ell)=c_\ell

and thus c_1=\cdots=c_k=0.

Now to see that \displaystyle \text{span}\{\mathbf{1}_{C_j}\}_{j\in[k]}=\text{Cl}(G) we merely note that by previous chatracterizations of class functions for each f\in\text{Cl}(G) one has that f(C_j) is a singleton say s_j. We claim then that

\displaystyle f=\sum_{j=1}^{k}s_j\mathbf{1}_{C_j}

But, this is indeed obvious since given any x\in G we have that x\in C_\ell for precisely one \ell\in [k] and

\displaystyle f(x)=s_\ell=\sum_{j=1}^{k}s_j\mathbf{1}_{C_j}(x)

Since x\in G was arbitrary the conclusion follows. \blacksquare

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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February 24, 2011 - Posted by | Algebra, Representation Theory | , , , ,

2 Comments »

  1. […] we will use them to ascertain that the cardinality of some set is the dimension of (we, in our last post, showed that this was the number of conjugacy classes in ). In this set we shall see that this […]

    Pingback by Representation Theory: Irreducible Characters « Abstract Nonsense | February 25, 2011 | Reply

  2. […] particular this tells us that is entirely determined on the canonical basis for where are the conjugacy classes of […]

    Pingback by Representation Theory: A ‘Lemma’ « Abstract Nonsense | March 10, 2011 | Reply


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