English
For x ∈ Finsupp(G, k), the left-regular norm vanishes on x if and only if the corresponding linear combination of x with all-ones is zero; i.e., (leftRegular k G).norm x = 0 iff x.linearCombination k (fun _ => 1) = 0.
Русский
Для x ∈ Finsupp(G, k) норма левого регулярного действия нулю на x тогда и только тогда, когда соответствующая линейная комбинация x по единицам равна нулю.
LaTeX
$$$ (leftRegular k G).norm\, x = 0 \\iff x.linearCombination k (\\lambda _ => 1) = 0 $$$
Lean4
/-- Given representations of `G` on `V` and `W`, there is a natural representation of `G` on their
tensor product `V ⊗[k] W`.
-/
noncomputable def tprod : Representation k G (V ⊗[k] W)
where
toFun g := TensorProduct.map (ρV g) (ρW g)
map_one' := by simp only [map_one, TensorProduct.map_one]
map_mul' g h := by simp only [map_mul, TensorProduct.map_mul]
