English
Given representations ρV on V and ρW on W, there is a natural representation on the tensor product V ⊗ W given by g ↦ TensorProduct.map(ρV g, ρW g).
Русский
Задано представления ρ_V на V и ρ_W на W; существует естественное представление на тензорном произведении V ⊗ W: g действует как TensorProduct.map(ρ_V(g), ρ_W(g)).
LaTeX
$$$ (ρ_V \\otimes ρ_W) g = TensorProduct.map(ρ_V g, ρ_W g) $$$
Lean4
/-- Given representations of `G` on a family `V i` indexed by `i`, there is a
natural representation of `G` on their direct sum `⨁ i, V i`.
-/
@[simps]
noncomputable def directSum : Representation k G (⨁ i, V i)
where
toFun g := DirectSum.lmap (fun _ => ρ _ g)
map_one' := by ext; simp
map_mul' g h := by ext; simp
