Matrix coefficients and characters

Let $\pi :G\to \mathrm{GL}(V)$ be a finite-dimensional continuous complex representation of a compact Lie group $G$ (Continuous and unitary representations). Fix a positive-definite Hermitian inner product on $V$ for which $\pi$ is unitary, which exists by Finite-dimensional compact-group representations are unitarizable, and fix an orthonormal basis $e_1,\dots ,e_d$ of $V$.

• A matrix coefficient of $\pi$ is the continuous function $g⟼⟨\pi (g)v,w⟩,v,w\in V,$ and the matrix coefficient functions in the chosen basis are ${\pi }_{ij}(g):=⟨\pi (g)e_j,e_i⟩,i,j\in \{1,\dots ,d\}\mathrm{.}$ With this convention the matrix of $\pi (g)$ in the basis $(e_j)$ has $(i,j)$-entry ${\pi }_{ij}(g)$, so that $\pi (g)e_j={\sum }_{i=1}^d{\pi }_{ij}(g)e_i$. The functions ${\pi }_{ij}$ are continuous because the representation is continuous, and they span a finite-dimensional space stable under left and right translation.
• The character of $\pi$ is ${\chi }_{\pi }(g):=\mathrm{tr}\pi (g)={\sum }_{i=1}^d{\pi }_{ii}(g),$ the trace of the linear operator $\pi (g)$; it is independent of the orthonormal basis used to compute it, because the trace of a linear operator is basis-independent and the trace of similar matrices is equal (Similar matrices have the same trace).
• The dimension is $d_{\pi }:={\dim }_{\mathbb{C}}V$, and ${\chi }_{\pi }(e)=d_{\pi }$.

Two equivalent representations have equal characters, and a character is a class function: for all $g,h\in G$, ${\chi }_{\pi }(gh{g}^{-1})=\mathrm{tr}(\pi (g)\pi (h)\pi (g{)}^{-1})=\mathrm{tr}\pi (h)={\chi }_{\pi }(h),$ again by invariance of the trace under similarity. Direct sums and tensor products of representations have characters equal to the sum and the product of the characters, respectively, and the conjugate of a unitary representation has character $\overline{{\chi }_{\pi }}$.

• The symbol ${\pi }_{ij}$ always refers to the convention ${\pi }_{ij}(g)=⟨\pi (g)e_j,e_i⟩$ fixed above; this is the convention used by Schur orthogonality and by the Peter–Weyl theorem on this page.
• For the trivial representation on $\mathbb{C}$ the only matrix coefficient is the constant function $1$ and the character is constantly $1$.