subrepresentation — Mathlib

English

If W ≤ V is a G-invariant k-submodule under ρ, then restricting ρ to W yields a representation of G on W.

Русский

Если W ≤ V является G-инвариантным подмодулем над ρ, то ограничение ρ на W gives представление G на W.

LaTeX

$$$\\rho|_W: G \\to \\mathrm{End}_k(W), \\quad (\\rho|_W)(g) = (\\rho(g))|_{W}$$$

Lean4

/-- Given a `k`-linear `G`-representation `(V, ρ)`, this is the representation defined by
restricting `ρ` to a `G`-invariant `k`-submodule of `V`. -/
@[simps]
def subrepresentation (W : Submodule k V) (le_comap : ∀ g, W ≤ W.comap (ρ g)) : Representation k G W
    where
  toFun g := (ρ g).restrict <| le_comap g
  map_one' := by ext; simp
  map_mul' _ _ := by ext; simp

Похожие утверждения