Let \((\pi, V)\) be a faithful finite-dimensional representation of a finite group \(G\) over \(\mathbb{C}\). Show that, given any irreducible representation \(W\) of \(G\), the \(n\)-th tensor power \(V^{\otimes n}\) (or tensor algebra \(T(V \oplus V^*)\)) contains an isomorphic copy of \(W\) for some integer \(n \ge 0\).
Solution
Let \(\chi\) be the character of the faithful representation \(V\), and let \[ a_1=\chi(1),a_2,\dots,a_r \] be the distinct values taken by \(\chi\) on \(G\). Because \(G\) is finite, we may choose a \(G\)-invariant Hermitian inner product on \(V\), so every \(\pi(g)\) is unitary. Thus \[ |\chi(g)|\le \chi(1), \] and equality \(\chi(g)=\chi(1)\) implies \(\pi(g)=I\). Faithfulness then gives \[ \chi(g)=\chi(1)\iff g=1. \] Hence the level set corresponding to \(a_1=\chi(1)\) is exactly \(\{1\}\).
Let \(\psi\) be the irreducible character of \(W\). Suppose, toward a contradiction, that \(W\) occurs in none of \[ V^{\otimes0},V^{\otimes1},\dots,V^{\otimes(r-1)}. \] Then \[ 0=|G|\langle \chi^n,\psi\rangle =\sum_{g\in G}\chi(g)^n\overline{\psi(g)} =\sum_{j=1}^r a_j^n c_j \] for \(0\le n<r\), where \[ c_j=\sum_{\chi(g)=a_j}\overline{\psi(g)}. \] The Vandermonde matrix \((a_j^n)_{0\le n<r,1\le j\le r}\) is invertible because the \(a_j\) are distinct. Hence all \(c_j=0\). But \[ c_1=\overline{\psi(1)}=\dim W>0, \] a contradiction.
Therefore some \(n\ge0\) satisfies \[ \langle\chi^n,\psi\rangle>0, \] so \(W\) occurs in \(V^{\otimes n}\). In particular it also occurs in the tensor algebra \(T(V\oplus V^*)\).