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989cdb8 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | {"id":"sectPuQu","label":"sectPuQu","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["Pu","Qu"],"nlp_text":"Pour un même YoungTableau, l’intersection de \\(P_{\\mu }\\) et \\(Q_{\\mu }\\) est \\({1}\\)","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/sectPuQu","latex_link":"sect0002.html#sectPuQu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/sectPuQu","lean_decl_name":"sectPuQu","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma sectPuQu {μ : YoungDiagram} (Yᵤ : YoungTableau μ):\n (Pu Yᵤ).carrier ∩ (Qu Yᵤ).carrier = {↑1} := by\n rw[Set.eq_singleton_iff_unique_mem, Set.mem_inter_iff]\n constructor\n exact ⟨Subgroup.one_mem (Pu Yᵤ), Subgroup.one_mem (Qu Yᵤ)⟩\n simp\n intro f p q\n rw[Equiv.Perm.ext_iff]\n intro x\n obtain ⟨j, hj, a⟩ := preImYu Yᵤ x\n obtain ⟨k, hk, b⟩ := preImYu Yᵤ (f (Yᵤ.entry j))\n rw[Eq.comm] at hk\n rw[Pu] at p\n rw[Qu] at q\n have h : (j: μ) = (k : μ) := by\n ext\n exact q hk\n exact p hk\n rw[← h] at hk\n rw[Equiv.Perm.coe_one, id_eq, ← hj, hk]\n","line":94,"end_line":113,"proof_status":"formalized"}
{"id":"LinearTransformation","label":"LinearTransformation","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["sectPuQu","YoungSymmetriser","gInPuQu"],"nlp_text":"\\(\\exists l_{\\mu }\\) une fonction linéaire tq\\(\\forall x \\in \\mathbb {C}[S_{n}]\\), \\(a_{\\mu } x b_{\\mu } = l_{\\mu }(x) c_{\\mu }\\)","doc_url":null,"latex_link":"sect0003.html#LinearTransformation","lean_link":null,"lean_decl_name":"LinearTransformation","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"./BlueprintRep/SpechtModules.lean","lean4_code":"lemma LinearTransformation {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :\n ∀x : (PermComplex μ), ∃ y : (PermComplex μ), (au Yᵤ) * x * (bu Yᵤ) = y * (cu Yᵤ) := by\n sorry --TODO\n-/\n\n--SmallerImpZero TODO\n\n/-! Meilleure solution trouvé au problème suivant.\nLean ne me laisse pas multiplier un MonoidAlgebra.single avec un MonoidAlgebra -/\n","line":45,"end_line":53,"proof_status":"statement_formalized"}
{"id":"CuPropIdempotent","label":"CuPropIdempotent","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["LinearTransformation"],"nlp_text":"\\(c_{\\mu }\\) est proportionel à un idempotent. De façon mathématique \\[ \\exists c \\in \\mathbb {C}, c_{\\mu }^{2} = c \\cdot c_{\\mu } \\]","doc_url":null,"latex_link":"sect0003.html#CuPropIdempotent","lean_link":null,"lean_decl_name":"CuPropIdempotent","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"./BlueprintRep/SpechtModules.lean","lean4_code":"lemma CuPropIdempotent {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :\n ∃ c : ℂ, (cu Yᵤ) * (cu Yᵤ) = (ComplexToComplexPerm μ c) * (cu Yᵤ) := by\n sorry --TODO\n-/\n\n","line":58,"end_line":62,"proof_status":"statement_formalized"}
{"id":"IrreductibleRepresentationSn","label":"IrreductibleRepresentationSn","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["CuPropIdempotent","SpechtModules","SmallerImpZero","HomAeMisoeM"],"nlp_text":"\\(\\forall \\mu \\) partition de n, \\(V_{\\mu }\\) est toute les représentations irréductibles de \\(S_n\\)","doc_url":null,"latex_link":"sect0003.html#IrreductibleRepresentationSn","lean_link":null,"lean_decl_name":"IrreductibleRepresentationSn","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"au","label":"au","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["PuCard"],"nlp_text":"\\[ a_{\\mu } := \\frac{1}{|P_{\\mu }|}\\sum _{g \\in P_{\\mu }}g \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/au","latex_link":"sect0003.html#au","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/au","lean_decl_name":"au","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/SpechtModules.lean","lean4_code":"noncomputable def au {μ : YoungDiagram} (Yᵤ : YoungTableau μ) : PermComplex μ :=\n Finset.sum ((Pu Yᵤ).carrier.toFinset) (fun σ => MonoidAlgebra.single σ ((↑1 / ↑(PuCard Yᵤ)) : ℂ))\n","line":31,"end_line":32,"proof_status":"formalized"}
{"id":"YoungSymmetriser","label":"YoungSymmetriser","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["au","bu"],"nlp_text":"Un Young symmetriser est défini par un YoungDiagram \\(\\mu \\)\\(c_{\\mu } := a_{\\mu } b_{\\mu } \\)","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/cu","latex_link":"sect0003.html#YoungSymmetriser","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/cu","lean_decl_name":"cu","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/SpechtModules.lean","lean4_code":"noncomputable def cu {μ : YoungDiagram} (Yᵤ : YoungTableau μ) : PermComplex μ :=\n (au Yᵤ) * (bu Yᵤ)\n","line":39,"end_line":40,"proof_status":"formalized"}
{"id":"SpechtModules","label":"SpechtModules","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["YoungSymmetriser"],"nlp_text":"Soit \\(\\mu \\) un YoungDiagram. \\[ V_{\\mu } := \\mathbb {C}[S_{n}]c_{\\mu } \\]","doc_url":null,"latex_link":"sect0003.html#SpechtModules","lean_link":null,"lean_decl_name":"SpechtModules","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"SmallerImpZero","label":"SmallerImpZero","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["YoungSymmetriser","IneqYoungDiagram"],"nlp_text":"Si \\(\\mu {\\gt} \\lambda \\), alors \\[ a_{\\mu } \\mathbb {C}[S_{n}] b_{\\lambda } = 0 \\]","doc_url":null,"latex_link":"sect0003.html#SmallerImpZero","lean_link":null,"lean_decl_name":"SmallerImpZero","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"QuCard","label":"QuCard","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["Qu"],"nlp_text":"Le nombre d’élément de \\(Q_{\\mu }\\) est fini.","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/QuCard","latex_link":"sect0002.html#QuCard","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/QuCard","lean_decl_name":"QuCard","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"noncomputable def QuCard {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :=\n Fintype.card ↥(Qu Yᵤ)\n","line":90,"end_line":91,"proof_status":"formalized"}
{"id":"bu","label":"bu","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["QuCard"],"nlp_text":"\\[ b_{\\mu } := \\frac{1}{|Q_{\\mu }|}\\sum _{g \\in Q_{\\mu }}(-1)^{g}g \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/bu","latex_link":"sect0003.html#bu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/bu","lean_decl_name":"bu","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/SpechtModules.lean","lean4_code":"noncomputable def bu {μ : YoungDiagram} (Yᵤ : YoungTableau μ) : PermComplex μ :=\n Finset.sum ((Qu Yᵤ).carrier.toFinset) (fun σ => MonoidAlgebra.single σ ((↑(Equiv.Perm.sign σ) / ↑(PuCard Yᵤ)) : ℂ))\n","line":35,"end_line":36,"proof_status":"formalized"}
{"id":"qu","label":"qu","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["staysInY"],"nlp_text":"Soit g une permutation rowToCol,\\(q_{\\mu }\\) est une permutation de [0, n-1] défini comme \\[ q_{\\mu }(m)= Y_{\\mu }((Y_{\\mu }^{-1}(m)).x,(Y_{\\mu }^{-1} \\circ g(m)).y) \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/qu","latex_link":"sect0002.html#qu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/qu","lean_decl_name":"qu","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"def qu {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) (n : Fin μ.card) :=\n Yᵤ.entry ⟨((YuInv Yᵤ n).val.fst, (YuInv Yᵤ (g n)).val.snd), staysInY rtc n⟩\n","line":140,"end_line":141,"proof_status":"formalized"}
{"id":"quInQu","label":"quInQu","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["qu"],"nlp_text":"\\[ q_{\\mu } \\in Q_{\\mu } \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/quPermInQu","latex_link":"sect0002.html#quInQu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/quPermInQu","lean_decl_name":"quPermInQu","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma quPermInQu {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) :\n (quPerm rtc) ∈ (Qu Yᵤ) := by\n rw[quPerm, Qu]\n simp only [Subtype.forall, Prod.forall, Subgroup.mem_mk, Set.mem_setOf_eq, Equiv.ofBijective_apply]\n intros ix iy _ jx jy _ h\n rw[qu, Function.Injective.eq_iff (injYu Yᵤ)] at h\n simp only [YuInvYu, Subtype.mk.injEq, Prod.mk.injEq] at h\n obtain ⟨h, _⟩ := h\n exact h\n","line":175,"end_line":183,"proof_status":"formalized"}
{"id":"quInv","label":"quInv","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["qu"],"nlp_text":"\\(q_{\\mu }^{-1}\\) est la fonction inverse de \\(q_{\\mu }\\)","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/quInv","latex_link":"sect0002.html#quInv","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/quInv","lean_decl_name":"quInv","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"def quInv {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) :=\n Fintype.bijInv (bijqu rtc)\n","line":186,"end_line":187,"proof_status":"formalized"}
{"id":"gInPuQu","label":"gInPuQu","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["quInQu","puInPu","PuQu"],"nlp_text":"Soit g une permutation rowToCol \\[ g \\in P_{\\mu }Q_{\\mu } \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/gInPuQu","latex_link":"sect0002.html#gInPuQu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/gInPuQu","lean_decl_name":"gInPuQu","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma gInPuQu {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) :\n g ∈ (PuQu Yᵤ) := by\n rw[PuQu, Set.mem_setOf_eq]\n exists (puPerm rtc)\n constructor\n exact puPermInPu rtc\n exists (quPerm rtc)\n constructor\n exact quPermInQu rtc\n rw[puPerm,quPerm, ← puqug]\n rfl\n exact rtc\n","line":261,"end_line":272,"proof_status":"formalized"}
{"id":"staysInX","label":"staysInX","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["quInv"],"nlp_text":"Soit g une permutation rowToCol, \\[ \\forall m \\in [0,n-1], ((Y_{\\mu }^{-1} \\circ g \\circ q_{\\mu }^{-1}(m)).x, (Y_{\\mu }^{-1}(m)).y) \\in \\mu \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/staysInX","latex_link":"sect0002.html#staysInX","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/staysInX","lean_decl_name":"staysInX","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma staysInX {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) (n : Fin μ.card) :\n ((YuInv Yᵤ (g (quInv rtc n))).val.fst,(YuInv Yᵤ n).val.snd ) ∈ μ := by\n have h : ∀ (n' : Fin μ.card), ∃! m', (qu rtc) m' = n' := by\n rw[← Function.bijective_iff_existsUnique _]\n exact (bijqu rtc)\n obtain ⟨m, hm, _⟩ := h n\n obtain ⟨i, hi, _⟩ := preImYu Yᵤ m\n obtain ⟨j, hj, _⟩ := preImYu Yᵤ (g (Yᵤ.entry i))\n rw[← hm, ← hi]\n simp only [quInvqu]\n rw[← hj, qu]\n simp only [YuInvYu, ← hj, Prod.mk.eta, SetLike.coe_mem]\n","line":195,"end_line":206,"proof_status":"formalized"}
{"id":"pu","label":"pu","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["staysInX"],"nlp_text":"Soit g une permutation rowToCol,\\(p_{\\mu }\\) est une permutation de [0, n-1] défini comme \\[ p_{\\mu }(m)= Y_{\\mu }((Y_{\\mu }^{-1} \\circ g \\circ q^{-1}(m)).x,(Y_{\\mu }^{-1}(m)).y) \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/pu","latex_link":"sect0002.html#pu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/pu","lean_decl_name":"pu","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"def pu {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) (n : Fin μ.card) :=\n Yᵤ.entry ⟨((YuInv Yᵤ (g (quInv rtc n))).val.fst,(YuInv Yᵤ n).val.snd ), staysInX rtc n⟩\n","line":209,"end_line":210,"proof_status":"formalized"}
{"id":"HomAeM","label":"HomAeM","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["Ae"],"nlp_text":"Soit A un anneau unitaire, M un A-module et e un idempotant de A. \\[ \\forall \\phi \\in \\text{Hom}_{A}(Ae,M), \\exists m \\in e M, \\phi (e)=m \\]","doc_url":null,"latex_link":"sect0001.html#HomAeM","lean_link":null,"lean_decl_name":"HomAeM","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"HomAeMisoeM","label":"HomAeMisoeM","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["HomAeM"],"nlp_text":"Soit A un anneau unitaire, M un A-module et e un idempotant de A. \\[ \\text{Hom}_{A}(A e,M) \\cong e M \\]","doc_url":null,"latex_link":"sect0001.html#HomAeMisoeM","lean_link":null,"lean_decl_name":"HomAeMisoeM","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"staysInY","label":"staysInY","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["YuInv","rowToCol"],"nlp_text":"Soit g une permutation rowToCol, \\[ \\forall m \\in [0,n-1], (Y_{\\mu }^{-1}(m).x,Y_{\\mu }^{-1}(g(m)).y) \\in \\mu \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/staysInY","latex_link":"sect0002.html#staysInY","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/staysInY","lean_decl_name":"staysInY","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma staysInY {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) (n : Fin μ.card) :\n ((YuInv Yᵤ n).val.fst, ((YuInv Yᵤ) (g n)).val.snd) ∈ μ := by\n sorry --TODO\n","line":135,"end_line":137,"proof_status":"statement_formalized"}
{"id":"puqug","label":"puqug","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["pu"],"nlp_text":"Soit g une permutation rowToCol, \\[ g = p_{\\mu } \\circ q_{\\mu } \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/puqug","latex_link":"sect0002.html#puqug","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/puqug","lean_decl_name":"puqug","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma puqug {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) :\n (pu rtc) ∘ (qu rtc) = g := by\n ext n\n obtain ⟨i, hi, _⟩ := preImYu Yᵤ n\n obtain ⟨j, hj, _⟩ := preImYu Yᵤ (g (Yᵤ.entry i))\n rw[← hi, ← hj, Function.comp_apply]\n have hq : (qu rtc (Yᵤ.entry i)) = (qu rtc (Yᵤ.entry i)) := by rfl\n nth_rewrite 2 [qu] at hq\n simp only [YuInvYu, ← hj] at hq\n simp only [pu, quInvqu, hq, YuInvYu]\n have hq' : (quInv rtc) (qu rtc (Yᵤ.entry i)) = (quInv rtc) (qu rtc (Yᵤ.entry i)) := by rfl\n nth_rewrite 1 [quInvqu, hq] at hq'\n simp only[← hq', ← hj, YuInvYu]\n","line":213,"end_line":225,"proof_status":"formalized"}
{"id":"bijpu","label":"bijpu","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["puqug"],"nlp_text":"Soit g une permutation rowToCol, \\(p_{\\mu }\\) est une bijection","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/bijpu","latex_link":"sect0002.html#bijpu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/bijpu","lean_decl_name":"bijpu","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma bijpu {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) :\n Function.Bijective (pu rtc) := by\n have h : g ∘ (quInv rtc) = g ∘ (quInv rtc) := by rfl\n nth_rewrite 1 [← puqug rtc, Function.comp_assoc] at h\n have h' : (qu rtc) ∘ (quInv rtc) = id := by\n ext n\n have j : ((quInv rtc) ((qu rtc) ((quInv rtc) n))) = ((quInv rtc) ((qu rtc) ((quInv rtc) n))) := by rfl\n nth_rewrite 2 [quInvqu] at j\n rw[Function.Injective.eq_iff] at j\n rw[Function.comp_apply, id_eq, j]\n rw[quInv]\n exact (Fintype.bijective_bijInv (bijqu rtc)).injective\n rw[h', Function.comp_id] at h\n rw[h, quInv]\n exact Function.Bijective.comp (Equiv.bijective g) (Fintype.bijective_bijInv (bijqu rtc))\n","line":228,"end_line":242,"proof_status":"formalized"}
{"id":"puInPu","label":"puInPu","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["bijpu"],"nlp_text":"\\[ p_{\\mu } \\in P_{\\mu } \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/puPermInPu","latex_link":"sect0002.html#puInPu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/puPermInPu","lean_decl_name":"puPermInPu","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma puPermInPu {μ : YoungDiagram} {Yᵤ : YoungTableau μ} {g : (Equiv.Perm (Fin μ.card))} (rtc : rowToCol Yᵤ g) :\n (puPerm rtc) ∈ (Pu Yᵤ) := by\n rw[puPerm, Pu]\n simp only [Subtype.forall, Prod.forall, Subgroup.mem_mk, Set.mem_setOf_eq, Equiv.ofBijective_apply]\n intros ix iy _ jx jy _ h\n rw[pu, Function.Injective.eq_iff (injYu Yᵤ)] at h\n simp only [YuInvYu, Subtype.mk.injEq, Prod.mk.injEq] at h\n obtain ⟨_, h⟩ := h\n exact h\n","line":250,"end_line":258,"proof_status":"formalized"}
{"id":"YuInv","label":"YuInv","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["bijYu"],"nlp_text":"\\(Y_{\\mu }^{-1}\\) est une l’inverse de \\(Y_{\\mu }\\)","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/YuInv","latex_link":"sect0002.html#YuInv","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/YuInv","lean_decl_name":"YuInv","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"def YuInv {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :=\n Fintype.bijInv (bijYu Yᵤ)\n","line":126,"end_line":127,"proof_status":"formalized"}
{"id":"IneqYoungDiagram","label":"IneqYoungDiagram","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(\\mu \\) et \\(\\lambda \\) deux YoungDiagram de même cardinalité.On dit que \\(\\mu {\\gt} \\lambda \\) si \\(\\exists i \\in \\mathbb {N}\\) tq \\(\\mu _{i}{\\gt}\\lambda _{i}\\) et \\(\\forall j \\in \\mathbb {N}_{{\\lt}i}\\), \\(\\mu _{j} = \\lambda _{j}\\).","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/IneqYoungDiagram","latex_link":"sect0002.html#IneqYoungDiagram","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/IneqYoungDiagram","lean_decl_name":"IneqYoungDiagram","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/SpechtModules.lean","lean4_code":"def IneqYoungDiagram (μ1 : YoungDiagram) (μ2 : YoungDiagram) :=\n ∃ i : ℕ, (μ1.colLen i) > (μ2.colLen i) ∧ (∀ j : ℕ, (μ1.colLen j) = (μ2.colLen j))\n","line":8,"end_line":9,"proof_status":"formalized"}
{"id":"Ae","label":"Ae","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(A\\) un anneau et \\(e \\in A\\) \\[ Ae := \\{ a' \\in A | \\exists a \\in A, a' = a \\cdot e \\} \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/Ae","latex_link":"sect0001.html#Ae","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/Ae","lean_decl_name":"Ae","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Module.lean","lean4_code":"instance Ae' (A : Type) [Ring A] (e : A) : AddCommMonoid {a' : A | ∃ a : A, a'= a * e} where\n add := fun a b => ⟨a.1 + b.1, by\n obtain ⟨a', ha⟩ := a.2\n obtain ⟨b', hb⟩ := b.2\n use (a' + b')\n rw [ha, hb, add_mul]⟩\n zero := ⟨0, by use 0; rw[zero_mul]⟩\n add_assoc := by\n intros _ _ _\n apply Subtype.ext\n apply add_assoc\n zero_add := by\n intro _\n apply Subtype.ext\n apply zero_add\n add_zero := by\n intro _\n apply Subtype.ext\n apply add_zero\n add_comm := by\n intro _ _\n apply Subtype.ext\n apply add_comm\n --Why do I need to do this?\n nsmul := fun n a => ⟨n • a.1, by\n obtain ⟨a', ha⟩ := a.2\n use n • a'\n rw [ha, smul_mul_assoc]⟩\n nsmul_zero := by\n intro a\n apply Subtype.ext\n simp [nsmul_zero]\n rfl\n nsmul_succ := by\n intro n a\n apply Subtype.ext\n simp\n rw[add_mul, one_mul]\n rfl\n","line":4,"end_line":42,"proof_status":"formalized"}
{"id":"Pu","label":"Pu","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["bijYu"],"nlp_text":"\\(P_{\\mu }\\) est un sous groupe de \\(S_{n}\\), défini de la façon suivante:Un élément de \\(P_{\\mu }\\) permute les entré du YoungDiagram s’ils sont sur la même rangé.","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/Pu","latex_link":"sect0002.html#Pu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/Pu","lean_decl_name":"Pu","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"def Pu {μ : YoungDiagram} (Yᵤ : YoungTableau μ) : Subgroup (Equiv.Perm (Fin μ.card)) where\n carrier := {x : (Equiv.Perm (Fin μ.card)) | ∀ {i j : μ}, (x (Yᵤ.entry i) = Yᵤ.entry j) → i.val.snd = j.val.snd}\n mul_mem' := by\n intros α β a b i j αβ\n rw[Equiv.Perm.coe_mul, Function.comp_apply] at αβ\n obtain ⟨k, hk, _⟩ := preImYu Yᵤ (β (Yᵤ.entry i))\n rw[Eq.comm] at hk\n rw[hk] at αβ\n rw[b hk, a αβ]\n one_mem' := by\n intros i j h\n rw[Equiv.Perm.coe_one, id_eq, Function.Injective.eq_iff] at h\n rw[h]\n exact injYu Yᵤ\n inv_mem' := by\n intros f h1 i j h2\n rw[Eq.comm, Equiv.Perm.eq_inv_iff_eq] at h2\n rw[Eq.comm]\n exact h1 h2\n","line":34,"end_line":52,"proof_status":"formalized"}
{"id":"PuCard","label":"PuCard","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["Pu"],"nlp_text":"Le nombre d’élément de \\(P_{\\mu }\\) est fini.","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/PuCard","latex_link":"sect0002.html#PuCard","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/PuCard","lean_decl_name":"PuCard","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"noncomputable def PuCard {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :=\n Fintype.card ↥(Pu Yᵤ)\n","line":60,"end_line":61,"proof_status":"formalized"}
{"id":"PuQu","label":"PuQu","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["Pu","Qu"],"nlp_text":"\\(P_{\\mu }Q_{\\mu } := \\{ g : [0,n-1] \\to [0,n-1] | \\exists p \\in P_{\\mu } \\land \\exists q \\in Q_{\\mu }, g = p q \\} \\)","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/PuQu","latex_link":"sect0002.html#PuQu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/PuQu","lean_decl_name":"PuQu","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma sectPuQu {μ : YoungDiagram} (Yᵤ : YoungTableau μ):\n (Pu Yᵤ).carrier ∩ (Qu Yᵤ).carrier = {↑1} := by\n rw[Set.eq_singleton_iff_unique_mem, Set.mem_inter_iff]\n constructor\n exact ⟨Subgroup.one_mem (Pu Yᵤ), Subgroup.one_mem (Qu Yᵤ)⟩\n simp\n intro f p q\n rw[Equiv.Perm.ext_iff]\n intro x\n obtain ⟨j, hj, a⟩ := preImYu Yᵤ x\n obtain ⟨k, hk, b⟩ := preImYu Yᵤ (f (Yᵤ.entry j))\n rw[Eq.comm] at hk\n rw[Pu] at p\n rw[Qu] at q\n have h : (j: μ) = (k : μ) := by\n ext\n exact q hk\n exact p hk\n rw[← h] at hk\n rw[Equiv.Perm.coe_one, id_eq, ← hj, hk]\n","line":94,"end_line":113,"proof_status":"formalized"}
{"id":"Qu","label":"Qu","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["bijYu"],"nlp_text":"\\(Q_{\\mu }\\) est un sous groupe de \\(S_{n}\\), défini de la façon suivante:Un élément de \\(Q_{\\mu }\\) permute les entré du YoungDiagram s’ils sont sur la même colonne.","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/Qu","latex_link":"sect0002.html#Qu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/Qu","lean_decl_name":"Qu","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"def Qu {μ : YoungDiagram} (Yᵤ : YoungTableau μ) : Subgroup (Equiv.Perm (Fin μ.card)) where\n carrier := {x : (Equiv.Perm (Fin μ.card)) | ∀ {i j : μ}, (x (Yᵤ.entry i) = Yᵤ.entry j) → i.val.fst = j.val.fst}\n mul_mem' := by\n intros α β a b i j αβ\n rw[Equiv.Perm.coe_mul, Function.comp_apply] at αβ\n obtain ⟨k, hk, _⟩ := preImYu Yᵤ (β (Yᵤ.entry i))\n rw[Eq.comm] at hk\n rw[hk] at αβ\n rw[b hk, a αβ]\n one_mem' := by\n intros i j h\n rw[Equiv.Perm.coe_one, id_eq, Function.Injective.eq_iff] at h\n rw[h]\n exact injYu Yᵤ\n inv_mem' := by\n intros f h1 i j h2\n rw[Eq.comm, Equiv.Perm.eq_inv_iff_eq] at h2\n rw[Eq.comm]\n exact h1 h2\n","line":64,"end_line":82,"proof_status":"formalized"}
{"id":"YoungTableau","label":"YoungTableau","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Un YoungTableau est une fonction des cellules d’un YoungDiagram de taille n et retourne un naturel de 0 à n-1","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/YoungTableau","latex_link":"sect0002.html#YoungTableau","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/YoungTableau","lean_decl_name":"YoungTableau","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"./BlueprintRep/SpechtModules.lean","lean4_code":"noncomputable instance PuElementFinite {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :\n Fintype (Pu Yᵤ).carrier := Fintype.ofFinite ↑(Pu Yᵤ).carrier\n\n/-! Nécessaire pour indiqué que l'ensemble des éléments de Qᵤ est fini -/\nnoncomputable instance QuElementFinite {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :\n Fintype (Qu Yᵤ).carrier := Fintype.ofFinite ↑(Qu Yᵤ).carrier\n\n/-! Nécessaire pour utilisé Finset.sum -/\nnoncomputable instance Addition (μ : YoungDiagram) : AddCommMonoid (PermComplex μ) :=\n MonoidAlgebra.addCommMonoid ℂ (Equiv.Perm (Fin μ.card))\n\n/-! Nécessaire pour multiplier deux éléments -/\nnoncomputable instance Multiplication (μ : YoungDiagram) : Mul (PermComplex μ) :=\n ⟨fun a b => MonoidAlgebra.mul' a b⟩\n\n/-! Définition de aᵤ -/\n","line":15,"end_line":30,"proof_status":"formalized"}
{"id":"injYu","label":"injYu","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["YoungTableau"],"nlp_text":"Un YoungTableau est injectif sur les entrés qui sont dans le YoungDiagram","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/injYu","latex_link":"sect0002.html#injYu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/injYu","lean_decl_name":"injYu","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma injYu {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :\n Function.Injective Yᵤ.entry := by\n rw [Function.Injective]\n intros i j h\n exact Yᵤ.inj h\n","line":12,"end_line":16,"proof_status":"formalized"}
{"id":"rowToCol","label":"rowToCol","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["YoungTableau"],"nlp_text":"Soit \\(g\\) est une permutation de [0, n-1]. Elle est rowToCol si \\[ \\forall i,j,k,l \\in \\mu , ((i \\neq j) \\land (g \\circ Y_{\\mu } (i) = Y_{\\mu } (k)) \\land (g \\circ Y_{\\mu } (j) = Y_{\\mu } (l))) → ((i.x \\neq j.x) \\lor (k.y \\neq l.y)) \\]","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/rowToCol","latex_link":"sect0002.html#rowToCol","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/rowToCol","lean_decl_name":"rowToCol","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"def rowToCol {μ : YoungDiagram} (Yᵤ : YoungTableau μ) (g : (Equiv.Perm (Fin μ.card))) :=\n ∀ {i j k l : μ}, ((i ≠ j) ∧ (g (Yᵤ.entry i) = (Yᵤ.entry k)) ∧ (g (Yᵤ.entry j) = (Yᵤ.entry l))) → ((i.val.fst ≠ j.val.fst) ∨ (k.val.snd ≠ l.val.snd))\n","line":122,"end_line":123,"proof_status":"formalized"}
{"id":"bijYu","label":"bijYu","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://AlexandreCharland.github.io/BlueprintRep/blueprint/dep_graph_document.html","dependencies":["injYu"],"nlp_text":"Un YoungTableau est une bijection entre les case de son YoungDiagram et les naturels de 0 à n-1","doc_url":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/bijYu","latex_link":"sect0002.html#bijYu","lean_link":"https://AlexandreCharland.github.io/BlueprintRep/docs/find/#doc/bijYu","lean_decl_name":"bijYu","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"BlueprintRep","source_path":"BlueprintRep/Young.lean","lean4_code":"lemma bijYu {μ : YoungDiagram} (Yᵤ : YoungTableau μ) :\n Function.Bijective Yᵤ.entry := by\n rw[Fintype.bijective_iff_injective_and_card]\n simp\n exact injYu Yᵤ\n","line":19,"end_line":23,"proof_status":"formalized"}
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