English
The operation to combine representations on V and W yields a representation on the linear maps V → W, where g acts by conjugation: ρV g acts on the input V by ρV g^{-1} on the left and ρW g on the right.
Русский
Объединение представлений на V и W даёт представление на space линейных отображений V → W, где g действует через сопряжение: f ↦ ρ_W g ∘ f ∘ ρ_V g^{-1}.
LaTeX
$$$ (\\rho_V \\rightarrow\\nobreakspace\\nobreakspace \\rho_W) g (f) = ρ_W g \\circ f \\circ ρ_V g^{-1} $$$
Lean4
/-- Given representations of `G` on `V` and `W`, there is a natural representation of `G` on their
product `V × W`.
-/
@[simps!]
noncomputable def prod : Representation k G (V × W)
where
toFun g := (ρV g).prodMap (ρW g)
map_one' := by simp
map_mul' g h := by simp [prodMap_mul]
