/-
Copyright 2024 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
module
public import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs@[expose] public sectionopen Matrixopen scoped MatrixGroupsvariable (n : Type*) [DecidableEq n] [Fintype n] (R : Type*) [CommRing R]
The multiplicative group homomorphism (Rˣ)ⁿ →* GL(n, R) given by mapping a vector x to
Matrix.diagonal x.
def Matrix.GeneralLinearGroup.diagonalHom : (n → Rˣ) →* GeneralLinearGroup n R where
toFun := fun d ↦ ⟨Matrix.diagonal (fun i ↦ d i), Matrix.diagonal (fun i ↦ (d i).inv), n:Type u_1inst✝²:DecidableEq ninst✝¹:Fintype nR:Type u_2inst✝:CommRing Rd:n → Rˣ⊢ ((diagonal fun i ↦ ↑(d i)) * diagonal fun i ↦ (d i).inv) = 1 All goals completed! 🐙, n:Type u_1inst✝²:DecidableEq ninst✝¹:Fintype nR:Type u_2inst✝:CommRing Rd:n → Rˣ⊢ ((diagonal fun i ↦ (d i).inv) * diagonal fun i ↦ ↑(d i)) = 1 All goals completed! 🐙⟩
map_mul' x y := n:Type u_1inst✝²:DecidableEq ninst✝¹:Fintype nR:Type u_2inst✝:CommRing Rx:n → Rˣy:n → Rˣ⊢ { val := diagonal fun i ↦ ↑((x * y) i), inv := diagonal fun i ↦ ((x * y) i).inv, val_inv := ⋯, inv_val := ⋯ } =
{ val := diagonal fun i ↦ ↑(x i), inv := diagonal fun i ↦ (x i).inv, val_inv := ⋯, inv_val := ⋯ } *
{ val := diagonal fun i ↦ ↑(y i), inv := diagonal fun i ↦ (y i).inv, val_inv := ⋯, inv_val := ⋯ } n:Type u_1inst✝²:DecidableEq ninst✝¹:Fintype nR:Type u_2inst✝:CommRing Rx:n → Rˣy:n → Rˣi✝:nj✝:n⊢ ↑{ val := diagonal fun i ↦ ↑((x * y) i), inv := diagonal fun i ↦ ((x * y) i).inv, val_inv := ⋯, inv_val := ⋯ } i✝ j✝ =
↑({ val := diagonal fun i ↦ ↑(x i), inv := diagonal fun i ↦ (x i).inv, val_inv := ⋯, inv_val := ⋯ } *
{ val := diagonal fun i ↦ ↑(y i), inv := diagonal fun i ↦ (y i).inv, val_inv := ⋯, inv_val := ⋯ })
i✝ j✝; All goals completed! 🐙
map_one' := n:Type u_1inst✝²:DecidableEq ninst✝¹:Fintype nR:Type u_2inst✝:CommRing R⊢ { val := diagonal fun i ↦ ↑(1 i), inv := diagonal fun i ↦ (1 i).inv, val_inv := ⋯, inv_val := ⋯ } = 1 n:Type u_1inst✝²:DecidableEq ninst✝¹:Fintype nR:Type u_2inst✝:CommRing Ri✝:nj✝:n⊢ ↑{ val := diagonal fun i ↦ ↑(1 i), inv := diagonal fun i ↦ (1 i).inv, val_inv := ⋯, inv_val := ⋯ } i✝ j✝ = ↑1 i✝ j✝; All goals completed! 🐙The group of invertible diagonal matrices.
def Matrix.GeneralLinearGroup.diagonalSubgroup : Subgroup (GeneralLinearGroup n R) :=
Subgroup.map (Matrix.GeneralLinearGroup.diagonalHom n R) ⊤