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Mathlib.LinearAlgebra.RootSystem.Hom

Morphisms of root pairings #

This file defines morphisms of root pairings, following the definition of morphisms of root data given in SGA III Exp. 21 Section 6.

Main definitions: #

TODO #

structure RootPairing.Hom {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) :
Type (max (max (max (max (max u_1 u_3) u_4) u_5) u_6) u_7)

A morphism of root pairings is a pair of mutually transposed maps of weight and coweight spaces that preserves roots and coroots. We make the map of indexing sets explicit.

theorem RootPairing.Hom.ext_iff {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} {inst✝ : CommRing R} {inst✝¹ : AddCommGroup M} {inst✝² : Module R M} {inst✝³ : AddCommGroup N} {inst✝⁴ : Module R N} {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} {inst✝⁵ : AddCommGroup M₂} {inst✝⁶ : Module R M₂} {inst✝⁷ : AddCommGroup N₂} {inst✝⁸ : Module R N₂} {P : RootPairing ι R M N} {Q : RootPairing ι₂ R M₂ N₂} {x y : P.Hom Q} :
theorem RootPairing.Hom.ext {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} {inst✝ : CommRing R} {inst✝¹ : AddCommGroup M} {inst✝² : Module R M} {inst✝³ : AddCommGroup N} {inst✝⁴ : Module R N} {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} {inst✝⁵ : AddCommGroup M₂} {inst✝⁶ : Module R M₂} {inst✝⁷ : AddCommGroup N₂} {inst✝⁸ : Module R N₂} {P : RootPairing ι R M N} {Q : RootPairing ι₂ R M₂ N₂} {x y : P.Hom Q} (weightMap : x.weightMap = y.weightMap) (coweightMap : x.coweightMap = y.coweightMap) (indexEquiv : x.indexEquiv = y.indexEquiv) :
x = y
theorem RootPairing.Hom.weight_coweight_transpose_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (x : N₂) (f : P.Hom Q) :
theorem RootPairing.Hom.root_weightMap_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (i : ι) (f : P.Hom Q) :
f.weightMap (P.root i) = Q.root (f.indexEquiv i)
theorem RootPairing.Hom.coroot_coweightMap_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (i : ι₂) (f : P.Hom Q) :
def RootPairing.Hom.id {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
P.Hom P

The identity morphism of a root pairing.

Equations
@[simp]
theorem RootPairing.Hom.id_indexEquiv_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (a : ι) :
(id P).indexEquiv a = a
@[simp]
theorem RootPairing.Hom.id_coweightMap_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (a : N) :
(id P).coweightMap a = a
@[simp]
theorem RootPairing.Hom.id_weightMap_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (a : M) :
(id P).weightMap a = a
@[simp]
theorem RootPairing.Hom.id_indexEquiv_symm_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (a : ι) :
def RootPairing.Hom.comp {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₁ : Type u_5} {M₁ : Type u_6} {N₁ : Type u_7} {ι₂ : Type u_8} {M₂ : Type u_9} {N₂ : Type u_10} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁] [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂} (g : P₁.Hom P₂) (f : P.Hom P₁) :
P.Hom P₂

Composition of morphisms

Equations
  • One or more equations did not get rendered due to their size.
@[simp]
theorem RootPairing.Hom.comp_weightMap_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₁ : Type u_5} {M₁ : Type u_6} {N₁ : Type u_7} {ι₂ : Type u_8} {M₂ : Type u_9} {N₂ : Type u_10} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁] [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂} (g : P₁.Hom P₂) (f : P.Hom P₁) (a✝ : M) :
(g.comp f).weightMap a✝ = g.weightMap (f.weightMap a✝)
@[simp]
theorem RootPairing.Hom.comp_indexEquiv_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₁ : Type u_5} {M₁ : Type u_6} {N₁ : Type u_7} {ι₂ : Type u_8} {M₂ : Type u_9} {N₂ : Type u_10} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁] [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂} (g : P₁.Hom P₂) (f : P.Hom P₁) (a✝ : ι) :
(g.comp f).indexEquiv a✝ = g.indexEquiv (f.indexEquiv a✝)
@[simp]
theorem RootPairing.Hom.comp_indexEquiv_symm_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₁ : Type u_5} {M₁ : Type u_6} {N₁ : Type u_7} {ι₂ : Type u_8} {M₂ : Type u_9} {N₂ : Type u_10} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁] [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂} (g : P₁.Hom P₂) (f : P.Hom P₁) (a✝ : ι₂) :
@[simp]
theorem RootPairing.Hom.comp_coweightMap_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₁ : Type u_5} {M₁ : Type u_6} {N₁ : Type u_7} {ι₂ : Type u_8} {M₂ : Type u_9} {N₂ : Type u_10} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁] [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂} (g : P₁.Hom P₂) (f : P.Hom P₁) (a✝ : N₂) :
(g.comp f).coweightMap a✝ = f.coweightMap (g.coweightMap a✝)
@[simp]
theorem RootPairing.Hom.id_comp {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (f : P.Hom Q) :
f.comp (id P) = f
@[simp]
theorem RootPairing.Hom.comp_id {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (f : P.Hom Q) :
(id Q).comp f = f
@[simp]
theorem RootPairing.Hom.comp_assoc {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₁ : Type u_5} {M₁ : Type u_6} {N₁ : Type u_7} {ι₂ : Type u_8} {M₂ : Type u_9} {N₂ : Type u_10} {ι₃ : Type u_11} {M₃ : Type u_12} {N₃ : Type u_13} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁] [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] [AddCommGroup M₃] [Module R M₃] [AddCommGroup N₃] [Module R N₃] {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂} {P₃ : RootPairing ι₃ R M₃ N₃} (h : P₂.Hom P₃) (g : P₁.Hom P₂) (f : P.Hom P₁) :
(h.comp g).comp f = h.comp (g.comp f)
instance RootPairing.Hom.instMonoid {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
Monoid (P.Hom P)

The endomorphism monoid of a root pairing.

Equations
@[simp]
theorem RootPairing.Hom.weightMap_one {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
@[simp]
theorem RootPairing.Hom.coweightMap_one {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
@[simp]
theorem RootPairing.Hom.indexEquiv_one {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
@[simp]
theorem RootPairing.Hom.weightMap_mul {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (x y : P.Hom P) :
@[simp]
theorem RootPairing.Hom.coweightMap_mul {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (x y : P.Hom P) :
@[simp]
theorem RootPairing.Hom.indexEquiv_mul {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (x y : P.Hom P) :
(x * y).indexEquiv = x.indexEquiv y.indexEquiv
@[reducible, inline]
abbrev RootPairing.End {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
Type (max (max (max (max (max u_1 u_3) u_4) u_1) u_3) u_4)

The endomorphism monoid of a root pairing.

Equations
def RootPairing.Hom.weightHom {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :

The weight space representation of endomorphisms

Equations
theorem RootPairing.Hom.weightHom_injective {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
def RootPairing.Hom.coweightHom {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :

The coweight space representation of endomorphisms

Equations
theorem RootPairing.Hom.coweightHom_injective {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
def RootPairing.Hom.indexHom {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
P.End →* ι ι

The permutation representation of the endomorphism monoid on the root index set

Equations
structure RootPairing.Equiv {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) extends P.Hom Q :
Type (max (max (max (max (max u_1 u_3) u_4) u_5) u_6) u_7)

An equivalence of root pairings is a morphism where the maps of weight and coweight spaces are bijective.

See also RootPairing.Equiv.toEndUnit.

theorem RootPairing.Equiv.ext {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} {inst✝ : CommRing R} {inst✝¹ : AddCommGroup M} {inst✝² : Module R M} {inst✝³ : AddCommGroup N} {inst✝⁴ : Module R N} {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} {inst✝⁵ : AddCommGroup M₂} {inst✝⁶ : Module R M₂} {inst✝⁷ : AddCommGroup N₂} {inst✝⁸ : Module R N₂} {P : RootPairing ι R M N} {Q : RootPairing ι₂ R M₂ N₂} {x y : P.Equiv Q} (weightMap : (↑x).weightMap = (↑y).weightMap) (coweightMap : (↑x).coweightMap = (↑y).coweightMap) (indexEquiv : (↑x).indexEquiv = (↑y).indexEquiv) :
x = y
theorem RootPairing.Equiv.ext_iff {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} {inst✝ : CommRing R} {inst✝¹ : AddCommGroup M} {inst✝² : Module R M} {inst✝³ : AddCommGroup N} {inst✝⁴ : Module R N} {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} {inst✝⁵ : AddCommGroup M₂} {inst✝⁶ : Module R M₂} {inst✝⁷ : AddCommGroup N₂} {inst✝⁸ : Module R N₂} {P : RootPairing ι R M N} {Q : RootPairing ι₂ R M₂ N₂} {x y : P.Equiv Q} :
x = y (↑x).weightMap = (↑y).weightMap (↑x).coweightMap = (↑y).coweightMap (↑x).indexEquiv = (↑y).indexEquiv
def RootPairing.Equiv.weightEquiv {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (e : P.Equiv Q) :
M ≃ₗ[R] M₂

The linear equivalence of weight spaces given by an equivalence of root pairings.

Equations
@[simp]
theorem RootPairing.Equiv.weightEquiv_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (e : P.Equiv Q) (m : M) :
(weightEquiv P Q e) m = (↑e).weightMap m
@[simp]
theorem RootPairing.Equiv.weightEquiv_symm_weightMap {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (e : P.Equiv Q) (m : M) :
(weightEquiv P Q e).symm ((↑e).weightMap m) = m
@[simp]
theorem RootPairing.Equiv.weightMap_weightEquiv_symm {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (e : P.Equiv Q) (m : M₂) :
(↑e).weightMap ((weightEquiv P Q e).symm m) = m
def RootPairing.Equiv.coweightEquiv {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (e : P.Equiv Q) :
N₂ ≃ₗ[R] N

The contravariant equivalence of coweight spaces given by an equivalence of root pairings.

Equations
@[simp]
theorem RootPairing.Equiv.coweightEquiv_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (e : P.Equiv Q) (n : N₂) :
(coweightEquiv P Q e) n = (↑e).coweightMap n
@[simp]
theorem RootPairing.Equiv.coweightEquiv_symm_coweightMap {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (e : P.Equiv Q) (n : N₂) :
(coweightEquiv P Q e).symm ((↑e).coweightMap n) = n
@[simp]
theorem RootPairing.Equiv.coweightMap_coweightEquiv_symm {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_5} {M₂ : Type u_6} {N₂ : Type u_7} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (e : P.Equiv Q) (n : N) :
(↑e).coweightMap ((coweightEquiv P Q e).symm n) = n
def RootPairing.Equiv.id {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
P.Equiv P

The identity equivalence of a root pairing.

Equations
@[simp]
theorem RootPairing.Equiv.id_weightMap_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (a : M) :
(↑(id P)).weightMap a = a
@[simp]
theorem RootPairing.Equiv.id_coweightMap_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (a : N) :
(↑(id P)).coweightMap a = a
@[simp]
theorem RootPairing.Equiv.id_indexEquiv_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (a : ι) :
(↑(id P)).indexEquiv a = a
@[simp]
theorem RootPairing.Equiv.id_indexEquiv_symm_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (a : ι) :
(↑(id P)).indexEquiv.symm a = a
def RootPairing.Equiv.comp {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₁ : Type u_8} {M₁ : Type u_9} {N₁ : Type u_10} {ι₂ : Type u_11} {M₂ : Type u_12} {N₂ : Type u_13} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁] [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂} (g : P₁.Equiv P₂) (f : P.Equiv P₁) :
P.Equiv P₂

Composition of equivalences

Equations
  • g.comp f = { toHom := (↑g).comp f, bijective_weightMap := , bijective_coweightMap := }
@[simp]
theorem RootPairing.Equiv.comp_coweightMap_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₁ : Type u_8} {M₁ : Type u_9} {N₁ : Type u_10} {ι₂ : Type u_11} {M₂ : Type u_12} {N₂ : Type u_13} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁] [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂} (g : P₁.Equiv P₂) (f : P.Equiv P₁) (a✝ : N₂) :
(↑(g.comp f)).coweightMap a✝ = (↑f).coweightMap ((↑g).coweightMap a✝)
@[simp]
theorem RootPairing.Equiv.comp_indexEquiv_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₁ : Type u_8} {M₁ : Type u_9} {N₁ : Type u_10} {ι₂ : Type u_11} {M₂ : Type u_12} {N₂ : Type u_13} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁] [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂} (g : P₁.Equiv P₂) (f : P.Equiv P₁) (a✝ : ι) :
(↑(g.comp f)).indexEquiv a✝ = (↑g).indexEquiv ((↑f).indexEquiv a✝)
@[simp]
theorem RootPairing.Equiv.comp_weightMap_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₁ : Type u_8} {M₁ : Type u_9} {N₁ : Type u_10} {ι₂ : Type u_11} {M₂ : Type u_12} {N₂ : Type u_13} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁] [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂} (g : P₁.Equiv P₂) (f : P.Equiv P₁) (a✝ : M) :
(↑(g.comp f)).weightMap a✝ = (↑g).weightMap ((↑f).weightMap a✝)
@[simp]
theorem RootPairing.Equiv.comp_indexEquiv_symm_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₁ : Type u_8} {M₁ : Type u_9} {N₁ : Type u_10} {ι₂ : Type u_11} {M₂ : Type u_12} {N₂ : Type u_13} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁] [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂} (g : P₁.Equiv P₂) (f : P.Equiv P₁) (a✝ : ι₂) :
(↑(g.comp f)).indexEquiv.symm a✝ = (↑f).indexEquiv.symm ((↑g).indexEquiv.symm a✝)
@[simp]
theorem RootPairing.Equiv.toHom_comp {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₁ : Type u_8} {M₁ : Type u_9} {N₁ : Type u_10} {ι₂ : Type u_11} {M₂ : Type u_12} {N₂ : Type u_13} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁] [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂} (g : P₁.Equiv P₂) (f : P.Equiv P₁) :
(g.comp f) = (↑g).comp f
@[simp]
theorem RootPairing.Equiv.id_comp {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_8} {M₂ : Type u_9} {N₂ : Type u_10} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (f : P.Equiv Q) :
f.comp (id P) = f
@[simp]
theorem RootPairing.Equiv.comp_id {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_8} {M₂ : Type u_9} {N₂ : Type u_10} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (f : P.Equiv Q) :
(id Q).comp f = f
@[simp]
theorem RootPairing.Equiv.comp_assoc {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₁ : Type u_8} {M₁ : Type u_9} {N₁ : Type u_10} {ι₂ : Type u_11} {M₂ : Type u_12} {N₂ : Type u_13} {ι₃ : Type u_14} {M₃ : Type u_15} {N₃ : Type u_16} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁] [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] [AddCommGroup M₃] [Module R M₃] [AddCommGroup N₃] [Module R N₃] {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂} {P₃ : RootPairing ι₃ R M₃ N₃} (h : P₂.Equiv P₃) (g : P₁.Equiv P₂) (f : P.Equiv P₁) :
(h.comp g).comp f = h.comp (g.comp f)
instance RootPairing.Equiv.instMonoid {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :

Equivalences form a monoid.

Equations
@[simp]
theorem RootPairing.Equiv.weightEquiv_one {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
@[simp]
theorem RootPairing.Equiv.coweightEquiv_one {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
@[simp]
theorem RootPairing.Equiv.toHom_one {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
1 = 1
@[simp]
theorem RootPairing.Equiv.mul_eq_comp {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {P : RootPairing ι R M N} (x y : P.Equiv P) :
x * y = x.comp y
@[simp]
theorem RootPairing.Equiv.weightEquiv_comp_toLin {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {P : RootPairing ι R M N} (x y : P.Equiv P) :
@[simp]
theorem RootPairing.Equiv.weightEquiv_mul {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {P : RootPairing ι R M N} (x y : P.Equiv P) :
@[simp]
theorem RootPairing.Equiv.coweightEquiv_comp_toLin {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {P : RootPairing ι R M N} (x y : P.Equiv P) :
@[simp]
theorem RootPairing.Equiv.coweightEquiv_mul {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {P : RootPairing ι R M N} (x y : P.Equiv P) :
def RootPairing.Equiv.symm {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_8} {M₂ : Type u_9} {N₂ : Type u_10} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (f : P.Equiv Q) :
Q.Equiv P

The inverse of a root pairing equivalence.

Equations
  • One or more equations did not get rendered due to their size.
@[simp]
theorem RootPairing.Equiv.inv_weightMap {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_8} {M₂ : Type u_9} {N₂ : Type u_10} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (f : P.Equiv Q) :
(↑(symm P Q f)).weightMap = (weightEquiv P Q f).symm
@[simp]
theorem RootPairing.Equiv.inv_coweightMap {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_8} {M₂ : Type u_9} {N₂ : Type u_10} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (f : P.Equiv Q) :
(↑(symm P Q f)).coweightMap = (coweightEquiv P Q f).symm
@[simp]
theorem RootPairing.Equiv.inv_indexEquiv {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι₂ : Type u_8} {M₂ : Type u_9} {N₂ : Type u_10} [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂] (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (f : P.Equiv Q) :
(↑(symm P Q f)).indexEquiv = (↑f).indexEquiv.symm
instance RootPairing.Equiv.instGroup {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
Group (P.Equiv P)

Equivalences form a group.

Equations
  • One or more equations did not get rendered due to their size.
@[reducible, inline]
abbrev RootPairing.Aut {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
Type (max (max (max (max (max u_1 u_3) u_4) u_1) u_3) u_4)

The automorphism group of a root pairing.

Equations
def RootPairing.Equiv.toEndUnit {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :

The isomorphism between the automorphism group of a root pairing and the group of invertible endomorphisms.

Equations
  • One or more equations did not get rendered due to their size.
theorem RootPairing.Equiv.toEndUnit_val {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (g : P.Aut) :
((toEndUnit P) g) = g
theorem RootPairing.Equiv.toEndUnit_inv {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (g : P.Aut) :
((toEndUnit P) g).inv = (symm P P g)
def RootPairing.Equiv.weightHom {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :

The weight space representation of automorphisms

Equations
@[simp]
theorem RootPairing.Equiv.weightHom_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (e : P.Equiv P) :
(weightHom P) e = weightEquiv P P e
theorem RootPairing.Equiv.weightHom_toLinearMap {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {P : RootPairing ι R M N} (g : P.Aut) :
((weightHom P) g) = (Hom.weightHom P) g
theorem RootPairing.Equiv.weightHom_injective {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
@[simp]
theorem RootPairing.Equiv.weightEquiv_inv {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {P : RootPairing ι R M N} (g : P.Aut) :
def RootPairing.Equiv.coweightHom {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :

The coweight space representation of automorphisms

Equations
@[simp]
theorem RootPairing.Equiv.coweightHom_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (g : P.Aut) :
theorem RootPairing.Equiv.coweightHom_toLinearMap {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {P : RootPairing ι R M N} (g : P.Aut) :
theorem RootPairing.Equiv.coweightHom_injective {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
theorem RootPairing.Equiv.coweightHom_op {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {P : RootPairing ι R M N} (g : P.Aut) :
@[simp]
theorem RootPairing.Equiv.coweightEquiv_inv {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {P : RootPairing ι R M N} (g : P.Aut) :
def RootPairing.Equiv.indexHom {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
P.Aut →* ι ι

The permutation representation of the automorphism group on the root index set

Equations
@[simp]
theorem RootPairing.Equiv.indexHom_apply {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (g : P.Aut) :
(indexHom P) g = (↑g).indexEquiv
@[simp]
theorem RootPairing.Equiv.indexEquiv_inv {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {P : RootPairing ι R M N} (g : P.Aut) :
def RootPairing.Equiv.reflection {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (i : ι) :
P.Aut

The automorphism of a root pairing given by a reflection.

Equations
  • One or more equations did not get rendered due to their size.
@[simp]
theorem RootPairing.Equiv.reflection_weightEquiv {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (i : ι) :
@[simp]
theorem RootPairing.Equiv.reflection_coweightEquiv {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (i : ι) :
@[simp]
theorem RootPairing.Equiv.reflection_indexEquiv {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (i : ι) :
@[simp]
theorem RootPairing.Equiv.reflection_inv {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (i : ι) :
instance RootPairing.Equiv.instDistribMulActionAut {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
Equations
@[simp]
theorem RootPairing.Equiv.reflection_smul {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (i : ι) (x : M) :
reflection P i x = (P.reflection i) x
@[simp]
theorem RootPairing.Equiv.root_indexEquiv_eq_smul {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (i : ι) (g : P.Aut) :
P.root ((↑g).indexEquiv i) = g P.root i
instance RootPairing.Equiv.instDistribMulActionMulOppositeAut {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
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  • One or more equations did not get rendered due to their size.
instance RootPairing.Equiv.instSMulCommClassAut {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
instance RootPairing.Equiv.instSMulCommClassMulOppositeAut {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
instance RootPairing.Equiv.instMulActionAut {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) :
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