Group representation theory MOC

Group character

A character 𝜒 of a group 𝐺 over a field 𝕂 is a map 𝜒 :𝐺 →𝕂 that can be defined as the Trace of a finite-degree Group representation 𝔛 :𝐺 →𝖵𝖾𝖼𝗍(𝕂). rep

𝜒(𝑔)=tr⁡𝔛(𝑔)=dim⁡Γ∑𝑗=1𝔛𝑗𝑗(𝑔)

Characters neatly summarize representations. See Character table.

Complex character

Since Trace is invariant under unitary equivalences, unitarily equivalent representations have the same character. If Γ𝜇 is an irrep then 𝜒𝜇 is an irreducible character. The irreducible characters {𝜒𝜇} are class functions and form an orthonormal basis of all such class functions within the group ring 𝑍(ℂ[𝐺]).

Linear character

In the special case of a linear character the vector space is one-dimensional and thus the character is a homomorphism into the multiplicative group of 𝕂, i.e. a 1-dimensional representation.

Properties

• Orthonormality of irreducible characters
• Character irreducibility criterion
• Irreducible character as function of an idempotent
• Finite group character values
• Tensor powers of a faithful representation contain all irreps
• Kernel of a character is the kernel of the representation
• Centre of a character is the “centre” of the representation

See also

• Central character

---

tidy| en | SemBr