statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
submod_mx_repr : mx_repr G (submod_mx Umod). | Proof.
rewrite /submod_mx; split=> [|x y Gx Gy /=].
by rewrite repr_mx1 mulmx1 val_submodK.
rewrite -in_submodJ; last by rewrite repr_mxM ?mulmxA.
by rewrite mxmodule_trans ?val_submodP.
Qed. | Lemma | submod_mx_repr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Umod",
"in_submodJ",
"last",
"mulmx1",
"mulmxA",
"mx_repr",
"mxmodule_trans",
"repr_mx1",
"repr_mxM",
"split",
"submod_mx",
"val_submodK",
"val_submodP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submod_repr | := MxRepresentation submod_mx_repr. | Canonical | submod_repr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"submod_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_factmodJ m (W : 'M_(m, n)) x :
x \in G -> in_factmod (W *m rG x) = in_factmod W *m factmod_mx Umod x. | Proof.
move=> Gx; rewrite -{1}[W]add_sub_fact_mod mulmxDl linearD /=.
apply: (canLR (subrK _)); apply: etrans (_ : 0 = _).
apply/eqP; rewrite in_factmod_eq0 (submx_trans _ (mxmoduleP Umod x Gx)) //.
by rewrite submxMr ?val_submodP.
by rewrite /in_factmod /val_factmod /= !mulmxA mulmx1 ?subrr.
Qed. | Lemma | in_factmodJ | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Umod",
"add_sub_fact_mod",
"apply",
"factmod_mx",
"in_factmod",
"in_factmod_eq0",
"linearD",
"mulmx1",
"mulmxA",
"mulmxDl",
"mxmoduleP",
"rG",
"submxMr",
"submx_trans",
"subrK",
"subrr",
"val_factmod",
"val_submodP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_factmodJ m (W : 'M_(m, \rank (cokermx U))) x :
x \in G ->
val_factmod (W *m factmod_mx Umod x) =
val_factmod (in_factmod (val_factmod W *m rG x)). | Proof. by move=> Gx; rewrite -{1}[W]val_factmodK -in_factmodJ. Qed. | Lemma | val_factmodJ | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Umod",
"cokermx",
"factmod_mx",
"in_factmod",
"in_factmodJ",
"rG",
"rank",
"val_factmod",
"val_factmodK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
factmod_mx_repr : mx_repr G (factmod_mx Umod). | Proof.
split=> [|x y Gx Gy /=].
by rewrite /factmod_mx repr_mx1 mulmx1 val_factmodK.
by rewrite -in_factmodJ // -mulmxA -repr_mxM.
Qed. | Lemma | factmod_mx_repr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Umod",
"factmod_mx",
"in_factmodJ",
"mulmx1",
"mulmxA",
"mx_repr",
"repr_mx1",
"repr_mxM",
"split",
"val_factmodK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
factmod_repr | := MxRepresentation factmod_mx_repr. | Canonical | factmod_repr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"factmod_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_sub_fact_mod x :
\tr (submod_repr x) + \tr (factmod_repr x) = \tr (rG x). | Proof.
rewrite -[submod_repr x]mulmxA mxtrace_mulC -val_submodE addrC.
rewrite -[factmod_repr x]mulmxA mxtrace_mulC -val_factmodE addrC.
by rewrite -mxtraceD add_sub_fact_mod.
Qed. | Lemma | mxtrace_sub_fact_mod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"add_sub_fact_mod",
"addrC",
"factmod_repr",
"mulmxA",
"mxtraceD",
"mxtrace_mulC",
"rG",
"submod_repr",
"val_factmodE",
"val_submodE"
] | For character theory. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
envelop_mx_id x : x \in G -> (rG x \in E_G)%MS. | Proof.
by move=> Gx; rewrite (eq_row_sub (enum_rank_in Gx x)) // rowK enum_rankK_in.
Qed. | Lemma | envelop_mx_id | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"enum_rankK_in",
"enum_rank_in",
"eq_row_sub",
"rG",
"rowK"
] | Properties of enveloping algebra as a subspace of 'rV_(n ^ 2). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
envelop_mx1 : (1%:M \in E_G)%MS. | Proof. by rewrite -(repr_mx1 rG) envelop_mx_id. Qed. | Lemma | envelop_mx1 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"envelop_mx_id",
"rG",
"repr_mx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
envelop_mxP A :
reflect (exists a, A = \sum_(x in G) a x *: rG x) (A \in E_G)%MS. | Proof.
have G_1 := group1 G; have bijG := enum_val_bij_in G_1.
set h := enum_val in bijG; have Gh: h _ \in G by apply: enum_valP.
apply: (iffP submxP) => [[u defA] | [a ->]].
exists (fun x => u 0 (enum_rank_in G_1 x)); apply: (can_inj mxvecK).
rewrite defA mulmx_sum_row linear_sum (reindex h) //=.
by apply: eq_bi... | Lemma | envelop_mxP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"apply",
"enum_rank_in",
"enum_val",
"enum_valK_in",
"enum_valP",
"enum_val_bij_in",
"eq_big",
"group1",
"linearZ",
"linear_sum",
"mulmx_sum_row",
"mxE",
"mxvecK",
"rG",
"reindex",
"rowK",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
envelop_mxM A B : (A \in E_G -> B \in E_G -> A *m B \in E_G)%MS. | Proof.
move=> {A B} /envelop_mxP[a ->] /envelop_mxP[b ->].
rewrite mulmx_suml !linear_sum summx_sub //= => x Gx.
rewrite !linear_sum summx_sub //= => y Gy.
rewrite -scalemxAl 3!linearZ !scalemx_sub//= -repr_mxM //.
by rewrite envelop_mx_id ?groupM.
Qed. | Lemma | envelop_mxM | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"envelop_mxP",
"envelop_mx_id",
"groupM",
"linearZ",
"linear_sum",
"mulmx_suml",
"repr_mxM",
"scalemxAl",
"scalemx_sub",
"summx_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_envelop m1 m2 (U : 'M_(m1, n)) (W : 'M_(m2, n)) A :
(mxmodule U -> mxvec A <= E_G -> W <= U -> W *m A <= U)%MS. | Proof.
move=> modU /envelop_mxP[a ->] sWU; rewrite linear_sum summx_sub //= => x Gx.
by rewrite -scalemxAr scalemx_sub ?mxmodule_trans.
Qed. | Lemma | mxmodule_envelop | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"envelop_mxP",
"linear_sum",
"mxmodule",
"mxmodule_trans",
"mxvec",
"scalemxAr",
"scalemx_sub",
"summx_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dom_hom_mx f : 'M_n | :=
kermx (lin1_mx (mxvec \o mulmx (cent_mx_fun E_G f) \o lin_mul_row)). | Definition | dom_hom_mx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"cent_mx_fun",
"kermx",
"lin1_mx",
"lin_mul_row",
"mulmx",
"mxvec"
] | over some domain, namely, dom_hom_mx f. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
hom_mxP m f (W : 'M_(m, n)) :
reflect (forall x, x \in G -> W *m rG x *m f = W *m f *m rG x)
(W <= dom_hom_mx f)%MS. | Proof.
apply: (iffP row_subP) => [cGf x Gx | cGf i].
apply/row_matrixP=> i; apply/eqP; rewrite -subr_eq0 -!mulmxA -!linearB /=.
have:= sub_kermxP (cGf i); rewrite mul_rV_lin1 /=.
move/(canRL mxvecK)/row_matrixP/(_ (enum_rank_in Gx x))/eqP; rewrite !linear0.
by rewrite !row_mul rowK mul_vec_lin /= mul_vec_lin_ro... | Lemma | hom_mxP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"dom_hom_mx",
"enum_rankK_in",
"enum_rank_in",
"enum_valP",
"linear0",
"linearB",
"mul_rV_lin1",
"mul_vec_lin",
"mul_vec_lin_row",
"mulmxA",
"mulmxBr",
"mxvecK",
"rG",
"rowK",
"row_matrixP",
"row_mul",
"row_subP",
"sub_kermxP",
"subr_eq0",
"subrr",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hom_envelop_mxC m f (W : 'M_(m, n)) A :
(W <= dom_hom_mx f -> A \in E_G -> W *m A *m f = W *m f *m A)%MS. | Proof.
move/hom_mxP=> cWfG /envelop_mxP[a ->]; rewrite !linear_sum mulmx_suml.
by apply: eq_bigr => x Gx /=; rewrite -2!scalemxAr -scalemxAl cWfG.
Qed. | Lemma | hom_envelop_mxC | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"apply",
"dom_hom_mx",
"envelop_mxP",
"eq_bigr",
"hom_mxP",
"linear_sum",
"mulmx_suml",
"scalemxAl",
"scalemxAr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dom_hom_invmx f :
f \in unitmx -> (dom_hom_mx (invmx f) :=: dom_hom_mx f *m f)%MS. | Proof.
move=> injf; set U := dom_hom_mx _; apply/eqmxP.
rewrite -{1}[U](mulmxKV injf) submxMr; apply/hom_mxP=> x Gx.
by rewrite -[_ *m rG x](hom_mxP _) ?mulmxKV.
by rewrite -[_ *m rG x](hom_mxP _) ?mulmxK.
Qed. | Lemma | dom_hom_invmx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"dom_hom_mx",
"eqmxP",
"hom_mxP",
"injf",
"invmx",
"mulmxK",
"mulmxKV",
"rG",
"submxMr",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dom_hom_mx_module f : mxmodule (dom_hom_mx f). | Proof.
apply/mxmoduleP=> x Gx; apply/hom_mxP=> y Gy.
rewrite -[_ *m rG y]mulmxA -repr_mxM // 2?(hom_mxP _) ?groupM //.
by rewrite repr_mxM ?mulmxA.
Qed. | Lemma | dom_hom_mx_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"dom_hom_mx",
"groupM",
"hom_mxP",
"mulmxA",
"mxmodule",
"mxmoduleP",
"rG",
"repr_mxM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hom_mxmodule m (U : 'M_(m, n)) f :
(U <= dom_hom_mx f)%MS -> mxmodule U -> mxmodule (U *m f). | Proof.
move/hom_mxP=> cGfU modU; apply/mxmoduleP=> x Gx.
by rewrite -cGfU // submxMr // (mxmoduleP modU).
Qed. | Lemma | hom_mxmodule | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"dom_hom_mx",
"hom_mxP",
"mxmodule",
"mxmoduleP",
"submxMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kermx_hom_module m (U : 'M_(m, n)) f :
(U <= dom_hom_mx f)%MS -> mxmodule U -> mxmodule (U :&: kermx f)%MS. | Proof.
move=> homUf modU; apply/mxmoduleP=> x Gx.
rewrite sub_capmx mxmodule_trans ?capmxSl //=.
apply/sub_kermxP; rewrite (hom_mxP _) ?(submx_trans (capmxSl _ _)) //.
by rewrite (sub_kermxP (capmxSr _ _)) mul0mx.
Qed. | Lemma | kermx_hom_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"capmxSl",
"capmxSr",
"dom_hom_mx",
"hom_mxP",
"kermx",
"mul0mx",
"mxmodule",
"mxmoduleP",
"mxmodule_trans",
"sub_capmx",
"sub_kermxP",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scalar_mx_hom a m (U : 'M_(m, n)) : (U <= dom_hom_mx a%:M)%MS. | Proof. by apply/hom_mxP=> x Gx; rewrite -!mulmxA scalar_mxC. Qed. | Lemma | scalar_mx_hom | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"dom_hom_mx",
"hom_mxP",
"mulmxA",
"scalar_mxC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_mx_hom (U V : 'M_n) :
(U :&: V = 0)%MS -> mxmodule U -> mxmodule V ->
(U + V <= dom_hom_mx (proj_mx U V))%MS. | Proof.
move=> dxUV modU modV; apply/hom_mxP=> x Gx.
rewrite -{1}(add_proj_mx dxUV (submx_refl _)) !mulmxDl addrC.
rewrite {1}[_ *m _]proj_mx_0 ?add0r //.
by rewrite mxmodule_trans ?proj_mx_sub.
by rewrite [_ *m _](proj_mx_id dxUV) // mxmodule_trans ?proj_mx_sub.
Qed. | Lemma | proj_mx_hom | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"add0r",
"add_proj_mx",
"addrC",
"apply",
"dom_hom_mx",
"hom_mxP",
"mulmxDl",
"mxmodule",
"mxmodule_trans",
"proj_mx",
"proj_mx_0",
"proj_mx_id",
"proj_mx_sub",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_mx (H : {set gT}) | :=
let commrH := \matrix_(i < #|H|) mxvec (rG (enum_val i) - 1%:M) in
kermx (lin1_mx (mxvec \o mulmx commrH \o lin_mul_row)). | Definition | rfix_mx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"enum_val",
"gT",
"kermx",
"lin1_mx",
"lin_mul_row",
"mulmx",
"mxvec",
"rG"
] | commute with ring morphisms. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
rfix_mxP m (W : 'M_(m, n)) (H : {set gT}) :
reflect (forall x, x \in H -> W *m rG x = W) (W <= rfix_mx H)%MS. | Proof.
rewrite /rfix_mx; set C := \matrix_i _.
apply: (iffP row_subP) => [cHW x Hx | cHW j].
apply/row_matrixP=> j; apply/eqP; rewrite -subr_eq0 row_mul.
move/sub_kermxP: {cHW}(cHW j); rewrite mul_rV_lin1 /=; move/(canRL mxvecK).
move/row_matrixP/(_ (enum_rank_in Hx x)); rewrite row_mul rowK !linear0.
by rewrit... | Lemma | rfix_mxP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"enum_rankK_in",
"enum_rank_in",
"enum_valP",
"gT",
"linear0",
"mul_rV_lin1",
"mul_vec_lin_row",
"mulmx1",
"mulmxBr",
"mxvecK",
"rG",
"rfix_mx",
"rowK",
"row_matrixP",
"row_mul",
"row_subP",
"sub_kermxP",
"subr_eq0",
"subrr",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_mx_id (H : {set gT}) x : x \in H -> rfix_mx H *m rG x = rfix_mx H. | Proof. exact/rfix_mxP. Qed. | Lemma | rfix_mx_id | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"gT",
"rG",
"rfix_mx",
"rfix_mxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_mxS (H K : {set gT}) : H \subset K -> (rfix_mx K <= rfix_mx H)%MS. | Proof.
by move=> sHK; apply/rfix_mxP=> x Hx; apply: rfix_mxP (subsetP sHK x Hx).
Qed. | Lemma | rfix_mxS | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"gT",
"rfix_mx",
"rfix_mxP",
"sHK",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_mx_conjsg (H : {set gT}) x :
x \in G -> H \subset G -> (rfix_mx (H :^ x) :=: rfix_mx H *m rG x)%MS. | Proof.
move=> Gx sHG; pose rf y := rfix_mx (H :^ y).
suffices{x Gx} IH: {in G &, forall y z, rf y *m rG z <= rf (y * z)%g}%MS.
apply/eqmxP; rewrite -/(rf x) -[H]conjsg1 -/(rf 1%g).
rewrite -{4}[x] mul1g -{1}[rf x](repr_mxKV rG Gx) -{1}(mulgV x).
by rewrite submxMr IH ?groupV.
move=> x y Gx Gy; apply/rfix_mxP=> zx... | Lemma | rfix_mx_conjsg | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"actM",
"apply",
"conj_subG",
"conjgC",
"conjsg1",
"eqmxP",
"gT",
"groupM",
"groupV",
"imsetP",
"mul1g",
"mulgV",
"mulmxA",
"rG",
"repr_mxKV",
"repr_mxM",
"rfix_mx",
"rfix_mxP",
"rfix_mx_id",
"sHG",
"submxMr",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
norm_sub_rstabs_rfix_mx (H : {set gT}) :
H \subset G -> 'N_G(H) \subset rstabs (rfix_mx H). | Proof.
move=> sHG; apply/subsetP=> x /setIP[Gx nHx]; rewrite inE Gx.
apply/rfix_mxP=> y Hy; have Gy := subsetP sHG y Hy.
have Hyx: (y ^ x^-1)%g \in H by rewrite memJ_norm ?groupV.
rewrite -mulmxA -repr_mxM // conjgCV repr_mxM ?(subsetP sHG _ Hyx) // mulmxA.
by rewrite (rfix_mx_id Hyx).
Qed. | Lemma | norm_sub_rstabs_rfix_mx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"conjgCV",
"gT",
"groupV",
"inE",
"memJ_norm",
"mulmxA",
"repr_mxM",
"rfix_mx",
"rfix_mxP",
"rfix_mx_id",
"rstabs",
"sHG",
"setIP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normal_rfix_mx_module H : H <| G -> mxmodule (rfix_mx H). | Proof.
case/andP=> sHG nHG.
by rewrite /mxmodule -{1}(setIidPl nHG) norm_sub_rstabs_rfix_mx.
Qed. | Lemma | normal_rfix_mx_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxmodule",
"nHG",
"norm_sub_rstabs_rfix_mx",
"rfix_mx",
"sHG",
"setIidPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_mx_module : mxmodule (rfix_mx G). | Proof. exact: normal_rfix_mx_module. Qed. | Lemma | rfix_mx_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxmodule",
"normal_rfix_mx_module",
"rfix_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_mx_rstabC (H : {set gT}) m (U : 'M[F]_(m, n)) :
H \subset G -> (H \subset rstab rG U) = (U <= rfix_mx H)%MS. | Proof.
move=> sHG; apply/subsetP/rfix_mxP=> cHU x Hx.
by rewrite (rstab_act (cHU x Hx)).
by rewrite !inE (subsetP sHG) //= cHU.
Qed. | Lemma | rfix_mx_rstabC | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"gT",
"inE",
"rG",
"rfix_mx",
"rfix_mxP",
"rstab",
"rstab_act",
"sHG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclic_mx u | := <<E_G *m lin_mul_row u>>%MS. | Definition | cyclic_mx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"lin_mul_row"
] | The cyclic module generated by a single vector. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cyclic_mxP u v :
reflect (exists2 A, A \in E_G & v = u *m A)%MS (v <= cyclic_mx u)%MS. | Proof.
rewrite genmxE; apply: (iffP submxP) => [[a] | [A /submxP[a defA]]] -> {v}.
exists (vec_mx (a *m E_G)); last by rewrite mulmxA mul_rV_lin1.
by rewrite vec_mxK submxMl.
by exists a; rewrite mulmxA mul_rV_lin1 /= -defA mxvecK.
Qed. | Lemma | cyclic_mxP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"apply",
"cyclic_mx",
"genmxE",
"last",
"mul_rV_lin1",
"mulmxA",
"mxvecK",
"submxMl",
"submxP",
"vec_mx",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclic_mx_id u : (u <= cyclic_mx u)%MS. | Proof. by apply/cyclic_mxP; exists 1%:M; rewrite ?mulmx1 ?envelop_mx1. Qed. | Lemma | cyclic_mx_id | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cyclic_mx",
"cyclic_mxP",
"envelop_mx1",
"mulmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclic_mx_eq0 u : (cyclic_mx u == 0) = (u == 0). | Proof.
rewrite -!submx0; apply/idP/idP.
by apply: submx_trans; apply: cyclic_mx_id.
move/submx0null->; rewrite genmxE; apply/row_subP=> i.
by rewrite row_mul mul_rV_lin1 /= mul0mx ?sub0mx.
Qed. | Lemma | cyclic_mx_eq0 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cyclic_mx",
"cyclic_mx_id",
"genmxE",
"mul0mx",
"mul_rV_lin1",
"row_mul",
"row_subP",
"sub0mx",
"submx0",
"submx0null",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclic_mx_module u : mxmodule (cyclic_mx u). | Proof.
apply/mxmoduleP=> x Gx; apply/row_subP=> i; rewrite row_mul.
have [A E_A ->{i}] := @cyclic_mxP u _ (row_sub i _); rewrite -mulmxA.
by apply/cyclic_mxP; exists (A *m rG x); rewrite ?envelop_mxM ?envelop_mx_id.
Qed. | Lemma | cyclic_mx_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cyclic_mx",
"cyclic_mxP",
"envelop_mxM",
"envelop_mx_id",
"mulmxA",
"mxmodule",
"mxmoduleP",
"rG",
"row_mul",
"row_sub",
"row_subP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyclic_mx_sub m u (W : 'M_(m, n)) :
mxmodule W -> (u <= W)%MS -> (cyclic_mx u <= W)%MS. | Proof.
move=> modU Wu; rewrite genmxE; apply/row_subP=> i.
by rewrite row_mul mul_rV_lin1 /= mxmodule_envelop // vec_mxK row_sub.
Qed. | Lemma | cyclic_mx_sub | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cyclic_mx",
"genmxE",
"mul_rV_lin1",
"mxmodule",
"mxmodule_envelop",
"row_mul",
"row_sub",
"row_subP",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hom_cyclic_mx u f :
(u <= dom_hom_mx f)%MS -> (cyclic_mx u *m f :=: cyclic_mx (u *m f))%MS. | Proof.
move=> domf_u; apply/eqmxP; rewrite !(eqmxMr _ (genmxE _)).
apply/genmxP; rewrite genmx_id; congr <<_>>%MS; apply/row_matrixP=> i.
by rewrite !row_mul !mul_rV_lin1 /= hom_envelop_mxC // vec_mxK row_sub.
Qed. | Lemma | hom_cyclic_mx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cyclic_mx",
"dom_hom_mx",
"eqmxMr",
"eqmxP",
"genmxE",
"genmxP",
"genmx_id",
"hom_envelop_mxC",
"mul_rV_lin1",
"row_matrixP",
"row_mul",
"row_sub",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
annihilator_mx u | := (E_G :&: kermx (lin_mul_row u))%MS. | Definition | annihilator_mx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"kermx",
"lin_mul_row"
] | The annihilator of a single vector. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
annihilator_mxP u A :
reflect (A \in E_G /\ u *m A = 0)%MS (A \in annihilator_mx u)%MS. | Proof.
rewrite sub_capmx; apply: (iffP andP) => [[-> /sub_kermxP]|[-> uA0]].
by rewrite mul_rV_lin1 /= mxvecK.
by split=> //; apply/sub_kermxP; rewrite mul_rV_lin1 /= mxvecK.
Qed. | Lemma | annihilator_mxP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"annihilator_mx",
"apply",
"mul_rV_lin1",
"mxvecK",
"split",
"sub_capmx",
"sub_kermxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_hom_mx u | :=
(\bigcap_j kermx (vec_mx (row j (annihilator_mx u))))%MS. | Definition | row_hom_mx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"annihilator_mx",
"kermx",
"row",
"vec_mx"
] | The subspace of homomorphic images of a row vector. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
row_hom_mxP u v :
reflect (exists2 f, u <= dom_hom_mx f & u *m f = v)%MS (v <= row_hom_mx u)%MS. | Proof.
apply: (iffP sub_bigcapmxP) => [iso_uv | [f hom_uf <-] i _].
have{iso_uv} uv0 A: (A \in E_G)%MS /\ u *m A = 0 -> v *m A = 0.
move/annihilator_mxP=> /submxP[a defA].
rewrite -[A]mxvecK {A}defA [a *m _]mulmx_sum_row !linear_sum big1 // => i _.
by rewrite !linearZ /= (sub_kermxP _) ?scaler0 ?iso_uv.
... | Lemma | row_hom_mxP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"addmx_sub",
"annihilator_mx",
"annihilator_mxP",
"apply",
"big1",
"cyclic_mx_id",
"cyclic_mx_module",
"dom_hom_mx",
"envelop_mx_id",
"eqmx_opp",
"genmxE",
"hom_envelop_mxC",
"hom_mxP",
"lin_mul_row",
"linearB",
"linearZ",
"linear_sum",
"mul0mx",
"mul_rV_lin1",
"mulmx1... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_iso (U V : 'M_n) : Prop | :=
MxIso f of f \in unitmx & (U <= dom_hom_mx f)%MS & (U *m f :=: V)%MS. | Variant | mx_iso | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"dom_hom_mx",
"unitmx"
] | decided when one of the two modules is known to be simple. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eqmx_iso U V : (U :=: V)%MS -> mx_iso U V. | Proof.
by move=> eqUV; exists 1%:M; rewrite ?unitmx1 ?scalar_mx_hom ?mulmx1.
Qed. | Lemma | eqmx_iso | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mulmx1",
"mx_iso",
"scalar_mx_hom",
"unitmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_iso_refl U : mx_iso U U. | Proof. exact: eqmx_iso. Qed. | Lemma | mx_iso_refl | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"eqmx_iso",
"mx_iso"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_iso_sym U V : mx_iso U V -> mx_iso V U. | Proof.
case=> f injf homUf defV; exists (invmx f); first by rewrite unitmx_inv.
by rewrite dom_hom_invmx // -defV submxMr.
by rewrite -[U](mulmxK injf); apply: eqmxMr (eqmx_sym _).
Qed. | Lemma | mx_iso_sym | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"dom_hom_invmx",
"eqmxMr",
"eqmx_sym",
"injf",
"invmx",
"mulmxK",
"mx_iso",
"submxMr",
"unitmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_iso_trans U V W : mx_iso U V -> mx_iso V W -> mx_iso U W. | Proof.
case=> f injf homUf defV [g injg homVg defW].
exists (f *m g); first by rewrite unitmx_mul injf.
by apply/hom_mxP=> x Gx; rewrite !mulmxA 2?(hom_mxP _) ?defV.
by rewrite mulmxA; apply: eqmx_trans (eqmxMr g defV) defW.
Qed. | Lemma | mx_iso_trans | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqmxMr",
"eqmx_trans",
"hom_mxP",
"injf",
"mulmxA",
"mx_iso",
"unitmx_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_iso U V : mx_iso U V -> \rank U = \rank V. | Proof. by case=> f injf _ <-; rewrite mxrankMfree ?row_free_unit. Qed. | Lemma | mxrank_iso | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"injf",
"mx_iso",
"mxrankMfree",
"rank",
"row_free_unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_iso_module U V : mx_iso U V -> mxmodule U -> mxmodule V. | Proof.
by case=> f _ homUf defV; rewrite -(eqmx_module defV); apply: hom_mxmodule.
Qed. | Lemma | mx_iso_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqmx_module",
"hom_mxmodule",
"mx_iso",
"mxmodule"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple (V : 'M_n) | :=
[/\ mxmodule V, V != 0 &
forall U : 'M_n, mxmodule U -> (U <= V)%MS -> U != 0 -> (V <= U)%MS]. | Definition | mxsimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxmodule"
] | Simple modules (we reserve the term "irreducible" for representations). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mxnonsimple (U : 'M_n) | :=
exists V : 'M_n, [&& mxmodule V, (V <= U)%MS, V != 0 & \rank V < \rank U]. | Definition | mxnonsimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxmodule",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimpleP U :
[/\ mxmodule U, U != 0 & ~ mxnonsimple U] <-> mxsimple U. | Proof.
do [split => [] [modU nzU simU]; split] => // [V modV sVU nzV | [V]].
apply/idPn; rewrite -(ltn_leqif (mxrank_leqif_sup sVU)) => ltVU.
by case: simU; exists V; apply/and4P.
by case/and4P=> modV sVU nzV; apply/negP; rewrite -leqNgt mxrankS ?simU.
Qed. | Lemma | mxsimpleP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"leqNgt",
"ltn_leqif",
"mxmodule",
"mxnonsimple",
"mxrankS",
"mxrank_leqif_sup",
"mxsimple",
"simU",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple_module U : mxsimple U -> mxmodule U. | Proof. by case. Qed. | Lemma | mxsimple_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxmodule",
"mxsimple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple_exists m (U : 'M_(m, n)) :
mxmodule U -> U != 0 -> classically (exists2 V, mxsimple V & V <= U)%MS. | Proof.
move=> modU nzU [] // simU; move: {2}_.+1 (ltnSn (\rank U)) => r leUr.
elim: r => // r IHr in m U leUr modU nzU simU.
have genU := genmxE U; apply: (simU); exists <<U>>%MS; last by rewrite genU.
apply/mxsimpleP; split; rewrite ?(eqmx_eq0 genU) ?(eqmx_module genU) //.
case=> V; rewrite !genU=> /and4P[modV sVU nzV... | Lemma | mxsimple_exists | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqmx_eq0",
"eqmx_module",
"genmxE",
"last",
"leUr",
"leq_trans",
"ltnSn",
"mxmodule",
"mxsimple",
"mxsimpleP",
"rank",
"simU",
"split",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_iso_simple U V : mx_iso U V -> mxsimple U -> mxsimple V. | Proof.
move=> isoUV [modU nzU simU]; have [f injf homUf defV] := isoUV.
split=> [||W modW sWV nzW]; first by rewrite (mx_iso_module isoUV).
by rewrite -(eqmx_eq0 defV) -(mul0mx n f) (can_eq (mulmxK injf)).
rewrite -defV -[W](mulmxKV injf) submxMr //; set W' := W *m _.
have sW'U: (W' <= U)%MS by rewrite -[U](mulmxK in... | Lemma | mx_iso_simple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"can_eq",
"dom_hom_invmx",
"eqmx_eq0",
"hom_mxmodule",
"injf",
"last",
"mul0mx",
"mulmxK",
"mulmxKV",
"mx_iso",
"mx_iso_module",
"mxsimple",
"simU",
"split",
"submxMr",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple_cyclic u U :
mxsimple U -> u != 0 -> (u <= U)%MS -> (U :=: cyclic_mx u)%MS. | Proof.
case=> [modU _ simU] nz_u Uu; apply/eqmxP; set uG := cyclic_mx u.
have s_uG_U: (uG <= U)%MS by rewrite cyclic_mx_sub.
by rewrite simU ?cyclic_mx_eq0 ?submx_refl // cyclic_mx_module.
Qed. | Lemma | mxsimple_cyclic | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Uu",
"apply",
"cyclic_mx",
"cyclic_mx_eq0",
"cyclic_mx_module",
"cyclic_mx_sub",
"eqmxP",
"mxsimple",
"nz_u",
"simU",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_Schur_onto m (U : 'M_(m, n)) V f :
mxmodule U -> mxsimple V -> (U <= dom_hom_mx f)%MS ->
(U *m f <= V)%MS -> U *m f != 0 -> (U *m f :=: V)%MS. | Proof.
move=> modU [modV _ simV] homUf sUfV nzUf.
apply/eqmxP; rewrite sUfV -(genmxE (U *m f)).
rewrite simV ?(eqmx_eq0 (genmxE _)) ?genmxE //.
by rewrite (eqmx_module (genmxE _)) hom_mxmodule.
Qed. | Lemma | mx_Schur_onto | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"dom_hom_mx",
"eqmxP",
"eqmx_eq0",
"eqmx_module",
"genmxE",
"hom_mxmodule",
"mxmodule",
"mxsimple"
] | The surjective part of Schur's lemma. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mx_Schur_inj U f :
mxsimple U -> (U <= dom_hom_mx f)%MS -> U *m f != 0 -> (U :&: kermx f)%MS = 0. | Proof.
case=> [modU _ simU] homUf nzUf; apply/eqP; apply: contraR nzUf => nz_ker.
rewrite (sameP eqP sub_kermxP) (sameP capmx_idPl eqmxP) simU ?capmxSl //.
exact: kermx_hom_module.
Qed. | Lemma | mx_Schur_inj | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"capmxSl",
"capmx_idPl",
"dom_hom_mx",
"eqmxP",
"kermx",
"kermx_hom_module",
"mxsimple",
"simU",
"sub_kermxP"
] | The injective part of Schur's lemma. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mx_Schur_inj_iso U f :
mxsimple U -> (U <= dom_hom_mx f)%MS -> U *m f != 0 -> mx_iso U (U *m f). | Proof.
move=> simU homUf nzUf; have [modU _ _] := simU.
have eqUfU: \rank (U *m f) = \rank U by apply/mxrank_injP; rewrite mx_Schur_inj.
have{eqUfU} [g invg defUf] := complete_unitmx eqUfU.
suffices homUg: (U <= dom_hom_mx g)%MS by exists g; rewrite ?defUf.
apply/hom_mxP=> x Gx; have [ux defUx] := submxP (mxmoduleP mod... | Lemma | mx_Schur_inj_iso | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"complete_unitmx",
"dom_hom_mx",
"hom_mxP",
"invg",
"mulmxA",
"mx_Schur_inj",
"mx_iso",
"mxmoduleP",
"mxrank_injP",
"mxsimple",
"rank",
"simU",
"submxP"
] | The injectve part of Schur's lemma, stated as isomorphism with the image. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mx_Schur_iso U V f :
mxsimple U -> mxsimple V -> (U <= dom_hom_mx f)%MS ->
(U *m f <= V)%MS -> U *m f != 0 -> mx_iso U V. | Proof.
move=> simU simV homUf sUfV nzUf; have [modU _ _] := simU.
have [g invg homUg defUg] := mx_Schur_inj_iso simU homUf nzUf.
exists g => //; apply: mx_Schur_onto; rewrite ?defUg //.
by rewrite -!submx0 defUg in nzUf *.
Qed. | Lemma | mx_Schur_iso | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"dom_hom_mx",
"invg",
"mx_Schur_inj_iso",
"mx_Schur_onto",
"mx_iso",
"mxsimple",
"simU",
"submx0"
] | The isomorphism part of Schur's lemma. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
nz_row_mxsimple U : mxsimple U -> nz_row U != 0. | Proof. by case=> _ nzU _; rewrite nz_row_eq0. Qed. | Lemma | nz_row_mxsimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxsimple",
"nz_row",
"nz_row_eq0"
] | modules; this is the only case that matters in practice. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mxsimple_iso (U V : 'M_n) | :=
[&& mxmodule V, (V :&: row_hom_mx (nz_row U))%MS != 0 & \rank V <= \rank U]. | Definition | mxsimple_iso | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxmodule",
"nz_row",
"rank",
"row_hom_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple_isoP U V :
mxsimple U -> reflect (mx_iso U V) (mxsimple_iso U V). | Proof.
move=> simU; pose u := nz_row U.
have [Uu nz_u]: (u <= U)%MS /\ u != 0 by rewrite nz_row_sub nz_row_mxsimple.
apply: (iffP and3P) => [[modV] | isoUV]; last first.
split; last by rewrite (mxrank_iso isoUV).
by case: (mx_iso_simple isoUV simU).
have [f injf homUf defV] := isoUV; apply/rowV0Pn; exists (u *m... | Lemma | mxsimple_isoP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Uu",
"apply",
"can_eq",
"cyclic_mx",
"cyclic_mx_id",
"cyclic_mx_module",
"cyclic_mx_sub",
"defU",
"dom_hom_mx",
"dom_hom_mx_module",
"eqmxMr",
"eqmxP",
"eqmx_iso",
"eqn_leq",
"hom_cyclic_mx",
"injf",
"last",
"mul0mx",
"mulmxK",
"mx_Schur_inj_iso",
"mx_iso",
"mx_iso_simple"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple_iso_simple U V :
mxsimple_iso U V -> mxsimple U -> mxsimple V. | Proof.
by move=> isoUV simU; apply: mx_iso_simple (simU); apply/mxsimple_isoP.
Qed. | Lemma | mxsimple_iso_simple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"mx_iso_simple",
"mxsimple",
"mxsimple_iso",
"mxsimple_isoP",
"simU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsemisimple (V : 'M_n) | :=
MxSemisimple I U (W := (\sum_(i : I) U i)%MS) of
forall i, mxsimple (U i) & (W :=: V)%MS & mxdirect W. | Variant | mxsemisimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxdirect",
"mxsimple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_mxsimple_direct_compl m I W (U : 'M_(m, n)) :
let V := (\sum_(i : I) W i)%MS in
(forall i : I, mxsimple (W i)) -> mxmodule U -> (U <= V)%MS ->
{J : {set I} | let S := U + \sum_(i in J) W i in S :=: V /\ mxdirect S}%MS. | Proof.
move=> V simW modU sUV; pose V_ (J : {set I}) := (\sum_(i in J) W i)%MS.
pose dxU (J : {set I}) := mxdirect (U + V_ J).
have [J maxJ]: {J | maxset dxU J}; last case/maxsetP: maxJ => dxUVJ maxJ.
apply: ex_maxset; exists set0.
by rewrite /dxU mxdirectE /V_ /= !big_set0 addn0 addsmx0 /=.
have modWJ: mxmodule (V... | Lemma | sum_mxsimple_direct_compl | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addn0",
"addnCA",
"addsmx0",
"addsmxA",
"addsmxC",
"addsmxSr",
"addsmx_module",
"addsmx_sub",
"apply",
"big_set0",
"big_setU1",
"capmxSl",
"capmx_idPl",
"capmx_module",
"eqmxP",
"ex_maxset",
"last",
"maxset",
"maxsetP",
"mxdirect",
"mxdirectE",
"mxdirect_addsE",
"mxdirec... | This is a slight generalization of Aschbacher 12.5 for finite sets. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
sum_mxsimple_direct_sub I W (V : 'M_n) :
(forall i : I, mxsimple (W i)) -> (\sum_i W i :=: V)%MS ->
{J : {set I} | let S := \sum_(i in J) W i in S :=: V /\ mxdirect S}%MS. | Proof.
move=> simW defV.
have [|J [defS dxS]] := sum_mxsimple_direct_compl simW (mxmodule0 n).
exact: sub0mx.
exists J; split; last by rewrite mxdirectE /= adds0mx mxrank0 in dxS.
by apply: eqmx_trans defV; rewrite adds0mx_id in defS.
Qed. | Lemma | sum_mxsimple_direct_sub | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"adds0mx",
"adds0mx_id",
"apply",
"eqmx_trans",
"last",
"mxdirect",
"mxdirectE",
"mxmodule0",
"mxrank0",
"mxsimple",
"split",
"sub0mx",
"sum_mxsimple_direct_compl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsemisimple0 : mxsemisimple 0. | Proof.
exists 'I_0 (fun _ => 0); [by case | by rewrite big_ord0 | ].
by rewrite mxdirectE /= !big_ord0 mxrank0.
Qed. | Lemma | mxsemisimple0 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"big_ord0",
"mxdirectE",
"mxrank0",
"mxsemisimple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intro_mxsemisimple (I : Type) r (P : pred I) W V :
(\sum_(i <- r | P i) W i :=: V)%MS ->
(forall i, P i -> W i != 0 -> mxsimple (W i)) ->
mxsemisimple V. | Proof.
move=> defV simW; pose W_0 := [pred i | W i == 0].
have [-> | nzV] := eqVneq V 0; first exact: mxsemisimple0.
case def_r: r => [| i0 r'] => [|{r' def_r}].
by rewrite -mxrank_eq0 -defV def_r big_nil mxrank0 in nzV.
move: defV; rewrite (bigID W_0) /= addsmxC -big_filter !(big_nth i0) !big_mkord.
rewrite addsmxC ... | Lemma | intro_mxsemisimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Sub",
"adds0mx_id",
"addsmxC",
"all_filterP",
"all_nthP",
"apply",
"big1",
"bigID",
"big_filter",
"big_mkord",
"big_nil",
"big_nth",
"big_set0",
"def_r",
"eqVneq",
"eq_bigl",
"eq_filter",
"filter_predI",
"i0",
"insubd",
"insubdK",
"mxdirectE",
"mxrank0",
"mxrank_eq0",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple_semisimple U : mxsimple U -> mxsemisimple U. | Proof.
move=> simU; apply: (intro_mxsemisimple (_ : \sum_(i < 1) U :=: U))%MS => //.
by rewrite big_ord1.
Qed. | Lemma | mxsimple_semisimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"big_ord1",
"intro_mxsemisimple",
"mxsemisimple",
"mxsimple",
"simU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx_semisimple U V :
mxsemisimple U -> mxsemisimple V -> mxsemisimple (U + V)%MS. | Proof.
case=> [I W /= simW defU _] [J T /= simT defV _].
have defUV: (\sum_ij sum_rect (fun _ => 'M_n) W T ij :=: U + V)%MS.
by rewrite big_sumType /=; apply: adds_eqmx.
by apply: intro_mxsemisimple defUV _; case=> /=.
Qed. | Lemma | addsmx_semisimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"adds_eqmx",
"apply",
"big_sumType",
"defU",
"intro_mxsemisimple",
"mxsemisimple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumsmx_semisimple (I : finType) (P : pred I) V :
(forall i, P i -> mxsemisimple (V i)) -> mxsemisimple (\sum_(i | P i) V i)%MS. | Proof.
move=> ssimV; elim/big_ind: _ => //; first exact: mxsemisimple0.
exact: addsmx_semisimple.
Qed. | Lemma | sumsmx_semisimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addsmx_semisimple",
"big_ind",
"mxsemisimple",
"mxsemisimple0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_semisimple U V : (U :=: V)%MS -> mxsemisimple U -> mxsemisimple V. | Proof.
by move=> eqUV [I W S simW defU dxS]; exists I W => //; apply: eqmx_trans eqUV.
Qed. | Lemma | eqmx_semisimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"defU",
"eqmx_trans",
"mxsemisimple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hom_mxsemisimple (V f : 'M_n) :
mxsemisimple V -> (V <= dom_hom_mx f)%MS -> mxsemisimple (V *m f). | Proof.
case=> I W /= simW defV _; rewrite -defV => /sumsmx_subP homWf.
have{defV} defVf: (\sum_i W i *m f :=: V *m f)%MS.
by apply: eqmx_trans (eqmx_sym _) (eqmxMr f defV); apply: sumsmxMr.
apply: (intro_mxsemisimple defVf) => i _ nzWf.
by apply: mx_iso_simple (simW i); apply: mx_Schur_inj_iso; rewrite ?homWf.
Qed. | Lemma | hom_mxsemisimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"dom_hom_mx",
"eqmxMr",
"eqmx_sym",
"eqmx_trans",
"intro_mxsemisimple",
"mx_Schur_inj_iso",
"mx_iso_simple",
"mxsemisimple",
"sumsmxMr",
"sumsmx_subP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsemisimple_module U : mxsemisimple U -> mxmodule U. | Proof.
case=> I W /= simW defU _.
by rewrite -(eqmx_module defU) sumsmx_module // => i _; case: (simW i).
Qed. | Lemma | mxsemisimple_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"defU",
"eqmx_module",
"mxmodule",
"mxsemisimple",
"sumsmx_module"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsplits (V U : 'M_n) | :=
MxSplits (W : 'M_n) of mxmodule W & (U + W :=: V)%MS & mxdirect (U + W). | Variant | mxsplits | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxdirect",
"mxmodule"
] | Completely reducible modules, and Maeschke's Theorem. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mx_completely_reducible V | :=
forall U, mxmodule U -> (U <= V)%MS -> mxsplits V U. | Definition | mx_completely_reducible | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxmodule",
"mxsplits"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_reducibleS U V :
mxmodule U -> (U <= V)%MS ->
mx_completely_reducible V -> mx_completely_reducible U. | Proof.
move=> modU sUV redV U1 modU1 sU1U.
have [W modW defV dxU1W] := redV U1 modU1 (submx_trans sU1U sUV).
exists (W :&: U)%MS; first exact: capmx_module.
by apply/eqmxP; rewrite !matrix_modl // capmxSr sub_capmx defV sUV /=.
by apply/mxdirect_addsP; rewrite capmxA (mxdirect_addsP dxU1W) cap0mx.
Qed. | Lemma | mx_reducibleS | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cap0mx",
"capmxA",
"capmxSr",
"capmx_module",
"eqmxP",
"matrix_modl",
"mx_completely_reducible",
"mxdirect_addsP",
"mxmodule",
"sub_capmx",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_Maschke_pchar : [pchar F]^'.-group G -> mx_completely_reducible 1%:M. | Proof.
rewrite /pgroup pcharf'_nat; set nG := _%:R => nzG U => /mxmoduleP Umod _.
pose phi := nG^-1 *: (\sum_(x in G) rG x^-1 *m pinvmx U *m U *m rG x).
have phiG x: x \in G -> phi *m rG x = rG x *m phi.
move=> Gx; rewrite -scalemxAl -scalemxAr; congr (_ *: _).
rewrite {2}(reindex_acts 'R _ Gx) ?astabsR //= mulmx_s... | Lemma | mx_Maschke_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Umod",
"apply",
"astabsR",
"eq_bigr",
"eqmxP",
"group",
"groupM",
"groupV",
"invMg",
"kermx",
"last",
"mul0mx",
"mulKVg",
"mulVf",
"mulmxA",
"mulmxKpV",
"mulmx_ker",
"mulmx_sub",
"mulmx_suml",
"mulmx_sumr",
"mx_completely_reducible",
"mxdirect_addsP",
"mxmoduleP",
"mxr... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsemisimple_reducible V : mxsemisimple V -> mx_completely_reducible V. | Proof.
case=> [I W /= simW defV _] U modU sUV; rewrite -defV in sUV.
have [J [defV' dxV]] := sum_mxsimple_direct_compl simW modU sUV.
exists (\sum_(i in J) W i)%MS.
- by apply: sumsmx_module => i _; case: (simW i).
- exact: eqmx_trans defV' defV.
by rewrite mxdirect_addsE (sameP eqP mxdirect_addsP) /= in dxV; case/and3... | Lemma | mxsemisimple_reducible | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqmx_trans",
"mx_completely_reducible",
"mxdirect_addsE",
"mxdirect_addsP",
"mxsemisimple",
"sum_mxsimple_direct_compl",
"sumsmx_module"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_reducible_semisimple V :
mxmodule V -> mx_completely_reducible V -> classically (mxsemisimple V). | Proof.
move=> modV redV [] // nssimV; have [r leVr] := ubnP (\rank V).
elim: r => // r IHr in V leVr modV redV nssimV.
have [V0 | nzV] := eqVneq V 0.
by rewrite nssimV ?V0 //; apply: mxsemisimple0.
apply: (mxsimple_exists modV nzV) => [[U simU sUV]]; have [modU nzU _] := simU.
have [W modW defUW dxUW] := redV U modU ... | Lemma | mx_reducible_semisimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"add1n",
"addsmxSr",
"addsmx_semisimple",
"apply",
"eqVneq",
"eqmx_semisimple",
"leq_add2r",
"leq_trans",
"lt0n",
"ltnS",
"mx_completely_reducible",
"mx_reducibleS",
"mxdirectP",
"mxmodule",
"mxrank_eq0",
"mxsemisimple",
"mxsemisimple0",
"mxsimple_exists",
"mxsimple_semisimple",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsemisimpleS U V :
mxmodule U -> (U <= V)%MS -> mxsemisimple V -> mxsemisimple U. | Proof.
move=> modU sUV ssimV.
have [W modW defUW dxUW]:= mxsemisimple_reducible ssimV modU sUV.
move/mxdirect_addsP: dxUW => dxUW.
have defU : (V *m proj_mx U W :=: U)%MS.
by apply/eqmxP; rewrite proj_mx_sub -{1}[U](proj_mx_id dxUW) ?submxMr.
apply: eqmx_semisimple defU _; apply: hom_mxsemisimple ssimV _.
by rewrite ... | Lemma | mxsemisimpleS | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"defU",
"eqmxP",
"eqmx_semisimple",
"hom_mxsemisimple",
"mxdirect_addsP",
"mxmodule",
"mxsemisimple",
"mxsemisimple_reducible",
"proj_mx",
"proj_mx_hom",
"proj_mx_id",
"proj_mx_sub",
"submxMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hom_mxsemisimple_iso I P U W f :
let V := (\sum_(i : I | P i) W i)%MS in
mxsimple U -> (forall i, P i -> W i != 0 -> mxsimple (W i)) ->
(V <= dom_hom_mx f)%MS -> (U <= V *m f)%MS ->
{i | P i & mx_iso (W i) U}. | Proof.
move=> V simU simW homVf sUVf; have [modU nzU _] := simU.
have ssimVf: mxsemisimple (V *m f).
exact: hom_mxsemisimple (intro_mxsemisimple (eqmx_refl V) simW) homVf.
have [U' modU' defVf] := mxsemisimple_reducible ssimVf modU sUVf.
move/mxdirect_addsP=> dxUU'; pose p := f *m proj_mx U U'.
case: (pickP (fun i =>... | Lemma | hom_mxsemisimple_iso | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"contraNneq",
"dom_hom_mx",
"eqmx_refl",
"hom_mxP",
"hom_mxsemisimple",
"intro_mxsemisimple",
"mul0mx",
"mulmxA",
"mx_Schur_iso",
"mx_iso",
"mxdirect_addsP",
"mxsemisimple",
"mxsemisimple_reducible",
"mxsimple",
"pickP",
"proj_mx",
"proj_mx_hom",
"proj_mx_id",
"proj_mx... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
component_mx_key : unit. | Proof. by []. Qed. | Fact | component_mx_key | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
component_mx_expr (U : 'M[F]_n) | :=
(\sum_i cyclic_mx (row i (row_hom_mx (nz_row U))))%MS. | Definition | component_mx_expr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cyclic_mx",
"nz_row",
"row",
"row_hom_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
component_mx | := locked_with component_mx_key component_mx_expr. | Definition | component_mx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"component_mx_expr",
"component_mx_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
component_mx_unfoldable | := [unlockable fun component_mx]. | Canonical | component_mx_unfoldable | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"component_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simU : mxsimple U. | Hypothesis | simU | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxsimple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
u | := nz_row U. | Let | u | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"nz_row"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iso_u | := row_hom_mx u. | Let | iso_u | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"row_hom_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nz_u : u != 0 | := nz_row_mxsimple simU. | Let | nz_u | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"nz_row_mxsimple",
"simU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Uu : (u <= U)%MS | := nz_row_sub U. | Let | Uu | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"nz_row_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
defU : (U :=: cyclic_mx u)%MS | := mxsimple_cyclic simU nz_u Uu. | Let | defU | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Uu",
"cyclic_mx",
"mxsimple_cyclic",
"nz_u",
"simU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
compU | := (component_mx U). | Notation | compU | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"component_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
component_mx_module : mxmodule compU. | Proof.
by rewrite unlock sumsmx_module // => i; rewrite cyclic_mx_module.
Qed. | Lemma | component_mx_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"compU",
"cyclic_mx_module",
"mxmodule",
"sumsmx_module"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
genmx_component : <<compU>>%MS = compU. | Proof.
by rewrite [in compU]unlock genmx_sums; apply: eq_bigr => i; rewrite genmx_id.
Qed. | Lemma | genmx_component | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"compU",
"eq_bigr",
"genmx_id",
"genmx_sums"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
component_mx_def : {I : finType & {W : I -> 'M_n |
forall i, mx_iso U (W i) & compU = \sum_i W i}}%MS. | Proof.
pose r i := row i iso_u; pose r_nz i := r i != 0; pose I := {i | r_nz i}.
exists I; exists (fun i => cyclic_mx (r (sval i))) => [i|].
apply/mxsimple_isoP=> //; apply/and3P.
split; first by rewrite cyclic_mx_module.
apply/rowV0Pn; exists (r (sval i)); last exact: (svalP i).
by rewrite sub_capmx cyclic... | Lemma | component_mx_def | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Sub",
"apply",
"compU",
"cyclic_mx",
"cyclic_mx_eq0",
"cyclic_mx_id",
"cyclic_mx_module",
"defU",
"eq_bigr",
"genmxP",
"genmx_component",
"genmx_id",
"genmx_sums",
"hom_cyclic_mx",
"i0",
"iso_u",
"last",
"mx_iso",
"mxrankM_maxl",
"mxsimple_isoP",
"row",
"rowV0Pn",
"row_h... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
component_mx_semisimple : mxsemisimple compU. | Proof.
have [I [W isoUW ->]] := component_mx_def.
apply: intro_mxsemisimple (eqmx_refl _) _ => i _ _.
exact: mx_iso_simple (isoUW i) simU.
Qed. | Lemma | component_mx_semisimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"compU",
"component_mx_def",
"eqmx_refl",
"intro_mxsemisimple",
"mx_iso_simple",
"mxsemisimple",
"simU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_iso_component V : mx_iso U V -> (V <= compU)%MS. | Proof.
move=> isoUV; have [f injf homUf defV] := isoUV.
have simV := mx_iso_simple isoUV simU.
have hom_u_f := submx_trans Uu homUf.
have ->: (V :=: cyclic_mx (u *m f))%MS.
apply: eqmx_trans (hom_cyclic_mx hom_u_f).
exact: eqmx_trans (eqmx_sym defV) (eqmxMr _ defU).
have iso_uf: (u *m f <= iso_u)%MS by apply/row_ho... | Lemma | mx_iso_component | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Uu",
"apply",
"compU",
"cyclic_mx",
"cyclic_mxP",
"defU",
"eqmxMr",
"eqmx_sym",
"eqmx_trans",
"genmxE",
"hom_cyclic_mx",
"injf",
"iso_u",
"mul_rV_lin1",
"mx_iso",
"mx_iso_simple",
"row_hom_mxP",
"row_mul",
"row_sub",
"row_subP",
"simU",
"submxMr",
"submx_trans",
"sumsm... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
component_mx_id : (U <= compU)%MS. | Proof. exact: mx_iso_component (mx_iso_refl U). Qed. | Lemma | component_mx_id | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"compU",
"mx_iso_component",
"mx_iso_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hom_component_mx_iso f V :
mxsimple V -> (compU <= dom_hom_mx f)%MS -> (V <= compU *m f)%MS ->
mx_iso U V. | Proof.
have [I [W isoUW ->]] := component_mx_def => simV homWf sVWf.
have [i _ _|i _ ] := hom_mxsemisimple_iso simV _ homWf sVWf.
exact: mx_iso_simple (simU).
exact: mx_iso_trans.
Qed. | Lemma | hom_component_mx_iso | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"compU",
"component_mx_def",
"dom_hom_mx",
"hom_mxsemisimple_iso",
"mx_iso",
"mx_iso_simple",
"mx_iso_trans",
"mxsimple",
"simU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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