Representations of finite groups; basic examples (2)

Posted: February 8, 2011 in Representations of Finite Groups
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Example 1. Find all degree one representations of the quaternion group Q_8.

Solution. We will show that there are exactly four degree one representations of Q_8. We know that

Q_8 = \{ \pm 1, \pm i, \pm j, \pm k \}. \ \ \ \ \ \ \ \ \ \ \ (1) 

There is a another way to define Q_8 if we use relations:

Q_8 = \langle a,b : \ a^4=b^4=1, \ a^2=b^2, \ aba=b \rangle. \ \ \ \ \ \ \ \ \ \ (2)

Note that a and b in (2) are just i and j in (1). Anyway, suppose that \rho : Q_8 \longrightarrow \mathbb{C}^{\times} is a representation of Q_8. Then \rho(-1) = \pm 1 because (\rho(-1))^2=\rho(1)=1. If \rho(-1)=-1, then since ij=-ji, we will get \rho(i) \rho(j) = -\rho(i) \rho(j), which is absurd. Thus \rho(-1)=1 and so, since (\rho(i))^2=(\rho(j))^2=\rho(-1)=1, we get \rho(i) = \pm 1 and \rho(j)=\pm 1. So there are four possible ways to define \rho. Note that either of these four ways is consistent with the relations in (2) because \rho(-1)=1.

Example 2.  In this example we will give two degree one representations of S_n, the group of permutations of \{1,2, \cdots , n \}. We will see later that these two representations are actually the only degree one representations of S_n. The first representation is a trivial one: define \rho : S_n \longrightarrow \mathbb{C}^{\times} by \rho(\sigma)=1, for all \sigma \in S_n. The other representation is called \text{sgn}, which maps every \sigma \in S_n to \text{sgn}(\sigma), the signature of \sigma. Recall that the signature of a permutation \sigma \in S_n is defined to be 1 if \sigma is even and -1 if \sigma is odd. We know from group theory that \text{sgn}(\sigma_1 \sigma_2)= \text{sgn}(\sigma_1) \text{sgn}(\sigma_2), for all \sigma_1, \sigma_2. So \text{sgn} : S_n \longrightarrow \mathbb{C}^{\times} is a representation.  


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