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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2005

Paper 2

The irreducible <characters> are an <orthonormal basis> of the space of complex class <functions> for the <inner product>
$$
\langle f,h\rangle=\frac1{|G|}\sum_{g\in G}\overline{f(g)}h(g).
$$
Thus the row <orthogonality> relation and <completeness> statement are
$$
\boxed{\frac1{|G|}\sum_{g\in G}\overline{\chi_i(g)}\chi_k(g)=\delta_{ik},
\qquad f=\sum_i\langle\chi_i,f\rangle\chi_i.}
$$
In particular the number of irreducible <characters> equals the number of conjugacy classes. In class-representative notation, the <orthogonality> relation is $\sum_j\overline{\chi_i(g_j)}\chi_k(g_j)/|Z(g_j)|=\delta_{ik}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 19G / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 19G / i / Solution

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Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 19G / i / Solution

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