Regular representation

The left regular representation acts on the basis $(e_h{)}_{h\in G}$ by $\rho (g)e_h=e_{gh}$. It is faithful.

The character of the regular representation of a finite group is $|G|$ at the identity and zero elsewhere. Left multiplication permutes its group-indexed basis, and a nonidentity element fixes none of those basis vectors. Thus each complex irreducible representation occurs with representation multiplicity equal to its dimension, by the character inner product.

The permutation matrices of the three displayed double transpositions, together with $I_4$, form a faithful representation of the Klein four-group. They satisfy $P^2=Q^2=I_4$ and $PQ=QP$. Every nonidentity element fixes no basis vector, so this is its four-dimensional regular representation.

The constant vector $(1,1,1,1{)}^T$ spans a nonzero proper invariant subspace, proving reducibility. More completely, the columns ofform a common orthonormal basis of eigenvectors. Direct multiplication gives $S^{T}PS=\mathrm{diag}(1,1,-1,-1)$ and $S^{T}QS=\mathrm{diag}(1,-1,1,-1)$. Thus the representation splits into the four distinct real one-dimensional characters, each once. The basis change is a normalized Hadamard matrix.

If ${\rho }_1,\dots ,{\rho }_k$ are the complex irreducible representations of a finite group $G$ and $n_j=\dim {\rho }_j$, decomposition of the regular representation gives

By Maschke's theorem, the complex regular representation of a finite group is a direct sum of irreducibles. Its kernel is the intersection of their kernels. If the group is simple, each kernel is either trivial or the whole group; faithfulness of the regular representation therefore forces at least one irreducible constituent to be faithful.