Fourier transform on real-valued function over symmetric group
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The document discusses the Fourier transform on the space l2(s n). It presents a mathematical definition and notation related to the transformation and its components. The content is highly technical and contains expressions involving functions and parameters pertinent to the Fourier transform.
Fourier transform on real-valued function over symmetric group
- 1. 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚 𝑜𝑛 𝐿2(𝑆 𝑛) July 10 Sunday, 2016 M1 Tamura Takumi
- 2. 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚 𝑜𝑛 𝐿2(𝑆 𝑛) 𝐷𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛. 𝐿2 𝑆 𝑛 ∋ 𝑓 𝜎 ↦ ℱ 𝑓 𝜆 ∈ ℝ 𝑑 𝜆×𝑑 𝜆, 𝜆 ⊢ 𝑛 ℱ 𝑓 𝜆 = 𝑓 𝜆 : = 𝜎∈𝑆 𝑛 𝑓 𝜎 𝜌 𝜆 𝜎 = 𝜎∈𝑆 𝑛 𝑓 𝜎 𝜌𝑠,𝑡 𝜆 𝜎 𝑠,𝑡 = 𝑓|𝜌𝑠,𝑡 𝜆 𝑠, 𝑡 ∈ STab 𝜆 , 𝜆 ⊢ 𝑛
- 3. 𝑓 𝜆 = 𝜎∈𝑆 𝑛 𝑓 𝜎 𝜌11 𝜆 (𝜎) ⋯ 𝜎∈𝑆 𝑛 𝑓 𝜎 𝜌1𝑑 𝜆 𝜆 (𝜎) ⋮ ⋱ ⋮ 𝜎∈𝑆 𝑛 𝑓 𝜎 𝜌 𝑑 𝜆1 𝜆 (𝜎) ⋯ 𝜎∈𝑆 𝑛 𝑓 𝜎 𝜌 𝑑 𝜆 𝑑 𝜆 𝜆 (𝜎) , 𝜆 ⊢ 𝑛