Group theory MOC

Class function

A class function 𝑓 ∈𝑋[𝐺] :𝐺 →𝑋 from a group 𝐺 to some set 𝑋 is a function constant for each conjugacy class, group i.e. for all 𝑥,𝑦 ∈𝐺

𝑓(𝑦𝑥𝑦−1)=𝑓(𝑥)

Examples

• Group character

Properties

• If 𝑋 is a ring, we consider functions as elements of the Group ring
  • Centre of the group ring
• Tensor powers of a faithful representation contain all irreps

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