Do you know any representation theory? What about representations of \(A_4\)?
Give a nontrivial one. What else is there? How many irreducible representations do we have? What are their degrees? Write the character table of \(A_4\).
Solution
The conjugacy classes of \(A_4\) are \[ \{1\},\qquad \{(12)(34),(13)(24),(14)(23)\}, \] and two conjugacy classes of \(3\)-cycles, each of size \(4\). Hence \(A_4\) has four irreducible complex characters.
Let \[ V_4=\{1,(12)(34),(13)(24),(14)(23)\}\trianglelefteq A_4. \] Since \[ A_4/V_4\cong C_3, \] there are three one-dimensional characters. If \(\omega=e^{2\pi i/3}\), they are the trivial character and the two characters taking values \(\omega,\omega^2\) on the two classes of \(3\)-cycles.
The remaining irreducible has degree \(3\), because \[ 12=1^2+1^2+1^2+d^2 \] forces \(d=3\). It is the standard representation obtained from the permutation representation on four letters by removing the trivial summand. Its character is \[ (3,-1,0,0) \] on the four classes above.
Thus, choosing representatives \(1,(12)(34),(123),(132)\) for the four classes, \[ \begin{array}{c|cccc} &1&(12)(34)&(123)&(132)\\\hline \chi_1&1&1&1&1\\ \chi_2&1&1&\omega&\omega^2\\ \chi_3&1&1&\omega^2&\omega\\ \chi_4&3&-1&0&0 \end{array} \] is the character table of \(A_4\).