In these exercises, \(G\) always denotes a finite group and all vector spaces are assumed to be finite dimensional over the field \(F = \mathbb{C}\).
In these exercises, you may use results stated but not yet proved in class about characters of representations of \(G\).
In this problem, we identify the character \(\chi_\Omega\) of the permutation representation \((\rho,F[\Omega])\) of a group \(G\).
Let \(V\) be a vector space and let \(\Phi:V \to V\) a linear mapping If \(\mathcal{B}\) is a basis for \(V\), recall that the trace of \(\Phi\) is defined by \[\operatorname{tr}(\Phi) = \operatorname{tr}([\Phi]_{\mathcal{B}}).\]
apologies – this is just explanatory; it isn’t actually a question
Recall that the dual of \(V\) is the vector space \(V^\vee = \operatorname{Hom}_F(V,F)\) of linear functionals on \(V\).
If \(b_1,\dots,b_n\) is a basis for \(V\), let \({b_j}^\vee:V \to F\) be defined by \({b_j}^\vee(b_i) = \delta_{i,j}\). Show that \({b_1}^\vee,\dots,{b_n}^\vee\) is a basis for \(V^\vee\); it is known as the dual basis to \(b_1,\dots,b_n\).
Prove that the trace of the linear mapping \(\Phi:V \to V\) is given by the expression \[\operatorname{tr}(\Phi) = \sum_{i=1}^n {b_i}^\vee(\Phi(b_i)).\]
Suppose that the finite group \(G\) acts on the finite set \(\Omega\), and consider the corresponding permutation representation \((\rho,F[\Omega])\) of \(G\). Recall that \(F[\Omega]\) is the vector space of all \(F\)-values functions on \(\Omega\), and that for \(f \in F[\Omega]\) and \(g \in G\), we have \[\rho(g)f(\omega) = f(g^{-1}\omega).\] In particular, we saw in the lecture that \[\rho(g)\delta_\omega) = \delta_{g\omega},\] where \(\delta_\omega\) denotes the Dirac function at \(\omega \in \Omega\).
Show that \[\operatorname{tr}(\rho(g)) = \#\{\omega \in \Omega \mid g\omega = \omega\};\] i.e. the trace of \(\rho(g)\) is the number of fixed points of the action of \(g\) on \(\Omega\).
Let \(V\) be a representation of \(G\), suppose that \(W_1,W_2\) are invariant subspaces, and that \(V\) is the internal direct sum \[V = W_1 \oplus W_2.\]
Show that the character \(\chi_V\) of \(V\) satisfies \[\chi_V = \chi_{W_1} + \chi_{W_2}\] i.e. for \(g \in G\) that \[\chi_V(g) = \chi_{W_1}(g) + \chi_{W_2}(g).\]
Let \(G = A_4\) be the alternating group of order \(\dfrac{4!}{2} = 12\).
We are going to find the character table of this group.
Confirm that the following list gives a representative for each of the conjugacy classes of \(G\):
\[1, (12)(34), (123), (124)\]
(Note that \((123)\) and \((124)\) are conjugate in \(S_4\), but not in \(A_4\)).
What are the sizes of the corresponding conjugacy classes?
Let \(K = \langle (12)(34), (14)(23)\rangle\). Show that \(K\) is a normal subgroup of index \(3\), so that \(G/K \simeq \mathbb{Z}/3\mathbb{Z}\).
Let \(\zeta_3\) be a primitive \(3\)rd root of unity in \(F^\times\) and for \(i=0,1,2\) let \(\rho_i:G \to F^\times\) be the unique homomorphism with the following properties:
- \(\rho_i\left( (123) \right) = \zeta^i\)
- \(K \subseteq \ker \rho_i\).
Explain why \(\rho_0 = \mathbf{1},\rho_1,\rho_2\) determine distinct irreducible (1-dimensional) representations of \(G\).
Let \(\Omega = \{1,2,3,4\}\) on which \(G\) acts by the embedding \(A_4 \subset S_4\).
Compute the character \(\chi_\Omega\) of the representation \(F[\Omega]\). (This means: compute and list the values of \(\chi_\Omega\) at the conjugacy class representatives given in a.)
(Use the result of problem 1 above).
The span of the vector \(\delta_1 + \delta_2 + \delta_3 + \delta_4 \in F[\Omega]\) is an invariant subspace isomorphic to the irreducible representation \(\rho_0\) (the so-called trivial representation).
Thus \(F[\Omega] = \rho_0 \oplus W\) for a \(3\)-dimensional invariant subspace. Explain why problem 2 shows that the character of \(W\) is given by \(\chi_W = \chi_\Omega - \mathbf{1}\).
Now prove that \(\langle \chi_W, \chi_W \rangle = 1\) and conclude that \(W\) is an irreducible representation.
Explain why \[\mathbf{1},\rho_1,\rho_2,W\] is a complete set of non-isomorphic irreducible representations of \(G\).
Display the character table of \(G = A_4\).