Given a finite group \(G\), recall that its regular representation is the representation on the complex group algebra \(\mathbb C[G]\) induced by left multiplication of \(G\) on itself and its adjoint representation is the representation on the complex group algebra \(\mathbb C[G]\) induced by conjugation of \(G\) on itself.
Let \(G=\GL_2(\mathbb F_2)\). Describe the number and dimensions of the irreducible representations of \(G\). Then describe the decomposition of its regular representation as a direct sum of irreducible representations.
Let \(H\) be a group of order 12. Show that its adjoint representation is reducible; that is, there is an \(H\)-invariant subspace of \(\mathbb C[H]\) besides 0 and \(\mathbb C[H]\).
Solution
1. The group \(\operatorname{GL}_2(\mathbb F_2)\) is isomorphic to \(S_3\).
There are exactly three nonzero vectors in \(\mathbb F_2^2\). Every invertible linear map permutes them, giving a homomorphism \[ \operatorname{GL}_2(\mathbb F_2)\longrightarrow S_3. \] If an invertible linear map fixes all three nonzero vectors, then in particular it fixes a basis, hence is the identity. Thus the homomorphism is injective. Also \[ |\operatorname{GL}_2(\mathbb F_2)|=(2^2-1)(2^2-2)=3\cdot2=6=|S_3|, \] so it is an isomorphism.
2. Up to isomorphism, \(G=\operatorname{GL}_2(\mathbb F_2)\) has exactly three irreducible complex representations, of dimensions\[ 1,\quad1,\quad2. \]
By 1 it is enough to work with \(S_3\). The trivial representation and the sign representation are distinct irreducible representations of dimension \(1\). The permutation representation of \(S_3\) on \(\mathbb C^3\) has the invariant decomposition \[ \mathbb C^3 =\mathbb C(1,1,1)\oplus V, \qquad V=\{(z_1,z_2,z_3):z_1+z_2+z_3=0\}. \] The \(2\)-dimensional summand \(V\) is irreducible: a proper nonzero invariant subspace would be one-dimensional, hence would give a common eigenline for a transposition and a \(3\)-cycle; direct inspection of their actions on \(V\) shows no such common eigenline exists.
The number of irreducible complex representations of a finite group equals its number of conjugacy classes. The group \(S_3\) has three conjugacy classes: the identity, the transpositions, and the \(3\)-cycles. Hence these three irreducibles are all of them.
3. The regular representation decomposes as\[ \mathbb C[G] \cong \mathbf 1\oplus\operatorname{sgn}\oplus V\oplus V. \]
For a finite group over \(\mathbb C\), each irreducible representation occurs in the regular representation with multiplicity equal to its dimension. Applying this to the irreducible dimensions \(1,1,2\) from 2 gives multiplicities \(1,1,2\). The total dimension is \[ 1^2+1^2+2^2=6=|G|, \] as required.
4. If \(H\) is any group of order \(12\), its adjoint representation on \(\mathbb C[H]\) is reducible.
Let \[ u=\sum_{h\in H}h\in\mathbb C[H]. \] For any \(g\in H\), conjugation permutes the elements of \(H\), so \[ gug^{-1} =\sum_{h\in H}ghg^{-1} =\sum_{h\in H}h =u. \] Thus the one-dimensional subspace \(\mathbb C u\) is invariant under the adjoint action. It is nonzero and proper because \[ 1=\dim_\mathbb C(\mathbb C u)<12=\dim_\mathbb C\mathbb C[H]. \] Hence the adjoint representation is reducible.