Representation theory of finite symmetric groups

Associate representation

Multiplying any irrep by the alternating character 𝜒𝔞 gives another irrep (see Tensor product with a 1-dimensional representation). Two irreps Γ𝜇,Γ˜𝜇 related by

Γ˜𝜇≅𝜒𝔞⊗Γ𝜇

are called associate representations. sym Iff Γ𝜈 ≅𝜒𝔞 ⊗Γ𝜈 then Γ𝜈 is called self-associate.

Properties

• dim⁡Γ˜𝜇 =dim⁡Γ𝜇
• Γ𝜇 is self-associate iff Γ𝜇(𝑝) =𝟎 for odd 𝑝
• 𝜒𝔰 and 𝜒𝔞 are associate to each other

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