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 |
|---|---|---|---|---|---|---|---|---|---|---|
conjg_IirrK y : cancel (conjg_Iirr^~ y) (conjg_Iirr^~ y^-1%g). | Proof. by move=> i; apply/irr_inj; rewrite !conjg_IirrE cfConjgK. Qed. | Lemma | conjg_IirrK | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgK",
"conjg_Iirr",
"conjg_IirrE",
"irr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjg_IirrKV y : cancel (conjg_Iirr^~ y^-1%g) (conjg_Iirr^~ y). | Proof. by rewrite -{2}[y]invgK; apply: conjg_IirrK. Qed. | Lemma | conjg_IirrKV | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"conjg_Iirr",
"conjg_IirrK",
"invgK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjg_Iirr_inj y : injective (conjg_Iirr^~ y). | Proof. exact: can_inj (conjg_IirrK y). Qed. | Lemma | conjg_Iirr_inj | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"conjg_Iirr",
"conjg_IirrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjg_Iirr_eq0 i y : (conjg_Iirr i y == 0) = (i == 0). | Proof. by rewrite -!irr_eq1 conjg_IirrE cfConjg_eq1. Qed. | Lemma | conjg_Iirr_eq0 | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjg_eq1",
"conjg_Iirr",
"conjg_IirrE",
"irr_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjg_Iirr0 x : conjg_Iirr 0 x = 0. | Proof. by apply/eqP; rewrite conjg_Iirr_eq0. Qed. | Lemma | conjg_Iirr0 | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"conjg_Iirr",
"conjg_Iirr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_irr_conjg i y :
H <| G -> y \in G -> '['chi_i, 'chi_i ^ y]_H = (y \in 'I_G['chi_i])%:R. | Proof.
move=> nsHG Gy; rewrite -conjg_IirrE cfdot_irr -(inj_eq irr_inj) conjg_IirrE.
by rewrite -{1}['chi_i]cfConjgJ1 cfConjg_eqE ?mulg1.
Qed. | Lemma | cfdot_irr_conjg | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgJ1",
"cfConjg_eqE",
"cfdot_irr",
"conjg_IirrE",
"inj_eq",
"irr_inj",
"mulg1",
"nsHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfclass (A : {set gT}) (phi : 'CF(A)) (B : {set gT}) | :=
[seq (phi ^ repr Tx)%CF | Tx in rcosets 'I_B[phi] B]. | Definition | cfclass | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"gT",
"rcosets",
"repr",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"phi ^: G" | := (cfclass phi G) : cfun_scope. | Notation | phi ^: G | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfclass"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_cfclass i : size ('chi[H]_i ^: G)%CF = #|G : 'I_G['chi_i]|. | Proof. by rewrite size_map -cardE. Qed. | Lemma | size_cfclass | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cardE",
"chi",
"size",
"size_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfclassP (A : {group gT}) phi psi :
reflect (exists2 y, y \in A & psi = phi ^ y)%CF (psi \in phi ^: A)%CF. | Proof.
apply: (iffP imageP) => [[_ /rcosetsP[y Ay ->] ->] | [y Ay ->]].
by case: repr_rcosetP => z /setIdP[Az _]; exists (z * y)%g; rewrite ?groupM.
without loss nHy: y Ay / y \in 'N(H).
have [nHy | /cfConjgEout->] := boolP (y \in 'N(H)); first exact.
by move/(_ 1%g); rewrite !group1 !cfConjgJ1; apply.
exists ('I... | Lemma | cfclassP | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgEout",
"cfConjgJ1",
"cfConjgMnorm",
"gT",
"group",
"group1",
"groupM",
"imageP",
"imset_f",
"rcosetE",
"rcosetsP",
"repr_rcosetP",
"setIP",
"setIdP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfclassInorm phi : (phi ^: 'N_G(H) =i phi ^: G)%CF. | Proof.
move=> xi; apply/cfclassP/cfclassP=> [[x /setIP[Gx _] ->] | [x Gx ->]].
by exists x.
have [Nx | /cfConjgEout-> //] := boolP (x \in 'N(H)).
by exists x; first apply/setIP.
by exists 1%g; rewrite ?group1 ?cfConjgJ1.
Qed. | Lemma | cfclassInorm | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgEout",
"cfConjgJ1",
"cfclassP",
"group1",
"setIP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfclass_refl phi : phi \in (phi ^: G)%CF. | Proof. by apply/cfclassP; exists 1%g => //; rewrite cfConjgJ1. Qed. | Lemma | cfclass_refl | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgJ1",
"cfclassP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfclass_transr phi psi :
(psi \in phi ^: G)%CF -> (phi ^: G =i psi ^: G)%CF. | Proof.
rewrite -cfclassInorm; case/cfclassP=> x Gx -> xi; rewrite -!cfclassInorm.
have nHN: {subset 'N_G(H) <= 'N(H)} by apply/subsetP; apply: subsetIr.
apply/cfclassP/cfclassP=> [[y Gy ->] | [y Gy ->]].
by exists (x^-1 * y)%g; rewrite -?cfConjgMnorm ?groupM ?groupV ?nHN // mulKVg.
by exists (x * y)%g; rewrite -?cfCo... | Lemma | cfclass_transr | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgMnorm",
"cfclassInorm",
"cfclassP",
"groupM",
"groupV",
"mulKVg",
"subsetIr",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfclass_sym phi psi : (psi \in phi ^: G)%CF = (phi \in psi ^: G)%CF. | Proof. by apply/idP/idP=> /cfclass_transr <-; apply: cfclass_refl. Qed. | Lemma | cfclass_sym | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfclass_refl",
"cfclass_transr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfclass_uniq phi : H <| G -> uniq (phi ^: G)%CF. | Proof.
move=> nsHG; rewrite map_inj_in_uniq ?enum_uniq // => Ty Tz; rewrite !mem_enum.
move=> {Ty}/rcosetsP[y Gy ->] {Tz}/rcosetsP[z Gz ->] /eqP.
case: repr_rcosetP => u Iphi_u; case: repr_rcosetP => v Iphi_v.
have [[Gu _] [Gv _]] := (setIdP Iphi_u, setIdP Iphi_v).
rewrite cfConjg_eqE ?groupM // => /rcoset_eqP.
by rewr... | Lemma | cfclass_uniq | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjg_eqE",
"enum_uniq",
"groupM",
"map_inj_in_uniq",
"mem_enum",
"nsHG",
"rcosetM",
"rcoset_eqP",
"rcoset_id",
"rcosetsP",
"repr_rcosetP",
"setIdP",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfclass_invariant phi : G \subset 'I[phi] -> (phi ^: G)%CF = phi. | Proof.
move/setIidPl=> IGphi; rewrite /cfclass IGphi // rcosets_id.
by rewrite /(image _ _) enum_set1 /= repr_group cfConjgJ1.
Qed. | Lemma | cfclass_invariant | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgJ1",
"cfclass",
"enum_set1",
"image",
"rcosets_id",
"repr_group",
"setIidPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfclass1 : H <| G -> (1 ^: G)%CF = [:: 1 : 'CF(H)]. | Proof. by move/normal_norm=> nHG; rewrite cfclass_invariant ?inertia1. Qed. | Lemma | cfclass1 | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfclass_invariant",
"inertia1",
"nHG",
"normal_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfclass_Iirr (A : {set gT}) i | := conjg_Iirr i @: A. | Definition | cfclass_Iirr | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"conjg_Iirr",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfclass_IirrE i j :
(j \in cfclass_Iirr G i) = ('chi_j \in 'chi_i ^: G)%CF. | Proof.
apply/imsetP/cfclassP=> [[y Gy ->] | [y]]; exists y; rewrite ?conjg_IirrE //.
by apply: irr_inj; rewrite conjg_IirrE.
Qed. | Lemma | cfclass_IirrE | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfclassP",
"cfclass_Iirr",
"conjg_IirrE",
"imsetP",
"irr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_cfclass_IirrE i j :
(cfclass_Iirr G j == cfclass_Iirr G i) = (j \in cfclass_Iirr G i). | Proof.
apply/eqP/idP=> [<- | iGj]; first by rewrite cfclass_IirrE cfclass_refl.
by apply/setP=> k; rewrite !cfclass_IirrE in iGj *; apply/esym/cfclass_transr.
Qed. | Lemma | eq_cfclass_IirrE | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfclass_Iirr",
"cfclass_IirrE",
"cfclass_refl",
"cfclass_transr",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_cfclass_Iirr i :
H <| G -> perm_eq [seq 'chi_j | j in cfclass_Iirr G i] ('chi_i ^: G)%CF. | Proof.
move=> nsHG; have UchiG := cfclass_uniq 'chi_i nsHG.
apply: uniq_perm; rewrite ?(map_inj_uniq irr_inj) ?enum_uniq // => phi.
apply/imageP/idP=> [[j iGj ->] | /cfclassP[y]]; first by rewrite -cfclass_IirrE.
by exists (conjg_Iirr i y); rewrite ?imset_f ?conjg_IirrE.
Qed. | Lemma | im_cfclass_Iirr | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfclassP",
"cfclass_Iirr",
"cfclass_IirrE",
"cfclass_uniq",
"conjg_Iirr",
"conjg_IirrE",
"enum_uniq",
"imageP",
"imset_f",
"irr_inj",
"map_inj_uniq",
"nsHG",
"perm_eq",
"seq",
"uniq_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_cfclass_Iirr i : H <| G -> #|cfclass_Iirr G i| = #|G : 'I_G['chi_i]|. | Proof.
move=> nsHG; rewrite -size_cfclass -(perm_size (im_cfclass_Iirr i nsHG)).
by rewrite size_map -cardE.
Qed. | Lemma | card_cfclass_Iirr | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cardE",
"cfclass_Iirr",
"im_cfclass_Iirr",
"nsHG",
"perm_size",
"size_cfclass",
"size_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reindex_cfclass R idx (op : Monoid.com_law idx) (F : 'CF(H) -> R) i :
H <| G ->
\big[op/idx]_(chi <- ('chi_i ^: G)%CF) F chi
= \big[op/idx]_(j | 'chi_j \in ('chi_i ^: G)%CF) F 'chi_j. | Proof.
move/im_cfclass_Iirr/(perm_big _) <-; rewrite big_image /=.
by apply: eq_bigl => j; rewrite cfclass_IirrE.
Qed. | Lemma | reindex_cfclass | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"big_image",
"cfclass_IirrE",
"chi",
"com_law",
"eq_bigl",
"im_cfclass_Iirr",
"perm_big"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfResInd j:
H <| G ->
'Res[H] ('Ind[G] 'chi_j) = #|H|%:R^-1 *: (\sum_(y in G) 'chi_j ^ y)%CF. | Proof.
case/andP=> [sHG /subsetP nHG].
rewrite (reindex_inj invg_inj); apply/cfun_inP=> x Hx.
rewrite cfResE // cfIndE // ?cfunE ?sum_cfunE; congr (_ * _).
by apply: eq_big => [y | y Gy]; rewrite ?cfConjgE ?groupV ?invgK ?nHG.
Qed. | Lemma | cfResInd | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgE",
"cfIndE",
"cfResE",
"cfunE",
"cfun_inP",
"eq_big",
"groupV",
"invgK",
"invg_inj",
"nHG",
"reindex_inj",
"sHG",
"subsetP",
"sum_cfunE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Clifford_Res_sum_cfclass i j :
H <| G -> j \in irr_constt ('Res[H, G] 'chi_i) ->
'Res[H] 'chi_i =
'['Res[H] 'chi_i, 'chi_j] *: (\sum_(chi <- ('chi_j ^: G)%CF) chi). | Proof.
move=> nsHG chiHj; have [sHG /subsetP nHG] := andP nsHG.
rewrite reindex_cfclass //= big_mkcond.
rewrite {1}['Res _]cfun_sum_cfdot linear_sum /=; apply: eq_bigr => k _.
have [[y Gy ->] | ] := altP (cfclassP _ _ _); first by rewrite cfdot_Res_conjg.
apply: contraNeq; rewrite scaler0 scaler_eq0 orbC => /norP[_ chi... | Lemma | Clifford_Res_sum_cfclass | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Cnat_cfdot_char",
"apply",
"big1",
"big_mkcond",
"cfInd_char",
"cfResInd",
"cfclassP",
"cfdotC",
"cfdotZl",
"cfdot_Res_conjg",
"cfdot_Res_l",
"cfdot_irr",
"cfdot_sum_irr",
"cfdot_suml",
"cfun_sum_cfdot",
"chi",
"conjC_eq0",
"conjg_IirrE",
"contraNeq",
"contraNneq",
"eq_bigr"... | This is Isaacs, Theorem (6.2) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfRes_Ind_invariant psi :
H <| G -> G \subset 'I[psi] -> 'Res ('Ind[G, H] psi) = #|G : H|%:R *: psi. | Proof.
case/andP=> sHG _ /subsetP IGpsi; apply/cfun_inP=> x Hx.
rewrite cfResE ?cfIndE ?natf_indexg // cfunE -mulrA mulrCA; congr (_ * _).
by rewrite mulr_natl -sumr_const; apply: eq_bigr => y /IGpsi/inertia_valJ->.
Qed. | Lemma | cfRes_Ind_invariant | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfIndE",
"cfResE",
"cfunE",
"cfun_inP",
"eq_bigr",
"inertia_valJ",
"mulrA",
"mulrCA",
"mulr_natl",
"natf_indexg",
"sHG",
"subsetP",
"sumr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
constt0_Res_cfker i :
H <| G -> 0 \in irr_constt ('Res[H] 'chi[G]_i) -> H \subset cfker 'chi[G]_i. | Proof.
move=> nsHG /(Clifford_Res_sum_cfclass nsHG); have [sHG nHG] := andP nsHG.
rewrite irr0 cfdot_Res_l cfclass1 // big_seq1 cfInd_cfun1 //.
rewrite cfdotZr conjC_nat => def_chiH.
apply/subsetP=> x Hx; rewrite cfkerEirr inE -!(cfResE _ sHG) //.
by rewrite def_chiH !cfunE cfun11 cfun1E Hx.
Qed. | Corollary | constt0_Res_cfker | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_Res_sum_cfclass",
"apply",
"big_seq1",
"cfInd_cfun1",
"cfResE",
"cfclass1",
"cfdotZr",
"cfdot_Res_l",
"cfker",
"cfkerEirr",
"cfun11",
"cfun1E",
"cfunE",
"chi",
"conjC_nat",
"inE",
"irr0",
"irr_constt",
"nHG",
"nsHG",
"sHG",
"subsetP"
] | This is Isaacs, Corollary (6.7). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
dvdn_constt_Res1_irr1 i j :
H <| G -> j \in irr_constt ('Res[H, G] 'chi_i) ->
exists n, 'chi_i 1%g = n%:R * 'chi_j 1%g. | Proof.
move=> nsHG chiHj; have [sHG nHG] := andP nsHG; rewrite -(cfResE _ sHG) //.
rewrite {1}(Clifford_Res_sum_cfclass nsHG chiHj) cfunE sum_cfunE.
have /natrP[n ->]: '['Res[H] 'chi_i, 'chi_j] \in Num.nat.
by rewrite Cnat_cfdot_char ?cfRes_char ?irr_char.
exists (n * size ('chi_j ^: G)%CF)%N; rewrite natrM -mulrA; c... | Lemma | dvdn_constt_Res1_irr1 | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_Res_sum_cfclass",
"Cnat_cfdot_char",
"apply",
"big_tnth",
"card_ord",
"cfConjg1",
"cfResE",
"cfRes_char",
"cfclassP",
"cfunE",
"eq_bigr",
"in_tuple",
"irr_char",
"irr_constt",
"mem_tnth",
"mulrA",
"mulr_natl",
"nHG",
"nat",
"natrM",
"natrP",
"nsHG",
"sHG",
"si... | This is Isaacs, Lemma (6.8). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfclass_Ind phi psi :
H <| G -> psi \in (phi ^: G)%CF -> 'Ind[G] phi = 'Ind[G] psi. | Proof.
move=> nsHG /cfclassP[y Gy ->]; have [sHG /subsetP nHG] := andP nsHG.
apply/cfun_inP=> x Hx; rewrite !cfIndE //; congr (_ * _).
rewrite (reindex_acts 'R _ (groupVr Gy)) ?astabsR //=.
by apply: eq_bigr => z Gz; rewrite conjgM cfConjgE ?nHG.
Qed. | Lemma | cfclass_Ind | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"astabsR",
"cfConjgE",
"cfIndE",
"cfclassP",
"cfun_inP",
"conjgM",
"eq_bigr",
"groupVr",
"nHG",
"nsHG",
"reindex_acts",
"sHG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''I[' phi ] " | := (inertia phi) : group_scope. | Notation | ''I[' phi ] | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"inertia"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''I[' phi ] " | := (inertia_group phi) : Group_scope. | Notation | ''I[' phi ] | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"inertia_group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''I_' G [ phi ] " | := (G%g :&: 'I[phi]) : group_scope. | Notation | ''I_' G [ phi ] | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''I_' G [ phi ] " | := (G :&: 'I[phi])%G : Group_scope. | Notation | ''I_' G [ phi ] | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgRes_norm phi y :
y \in 'N(K) -> y \in 'N(H) -> ('Res[K, H] phi ^ y)%CF = 'Res (phi ^ y)%CF. | Proof.
move=> nKy nHy; have [sKH | not_sKH] := boolP (K \subset H); last first.
by rewrite !cfResEout // rmorph_alg cfConjg1.
by apply/cfun_inP=> x Kx; rewrite !(cfConjgE, cfResE) ?memJ_norm ?groupV.
Qed. | Lemma | cfConjgRes_norm | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjg1",
"cfConjgE",
"cfResE",
"cfResEout",
"cfun_inP",
"groupV",
"last",
"memJ_norm",
"rmorph_alg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgRes phi y :
H <| G -> K <| G -> y \in G -> ('Res[K, H] phi ^ y)%CF = 'Res (phi ^ y)%CF. | Proof.
move=> /andP[_ nHG] /andP[_ nKG] Gy.
by rewrite cfConjgRes_norm ?(subsetP nHG) ?(subsetP nKG).
Qed. | Lemma | cfConjgRes | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgRes_norm",
"nHG",
"nKG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_inertia_Res phi :
G \subset 'N(K) -> 'I_G[phi] \subset 'I_G['Res[K, H] phi]. | Proof.
move=> nKG; apply/subsetP=> y /setIP[Gy /setIdP[nHy /eqP Iphi_y]].
by rewrite 2!inE Gy cfConjgRes_norm ?(subsetP nKG) ?Iphi_y /=.
Qed. | Lemma | sub_inertia_Res | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgRes_norm",
"inE",
"nKG",
"setIP",
"setIdP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgInd_norm phi y :
y \in 'N(K) -> y \in 'N(H) -> ('Ind[H, K] phi ^ y)%CF = 'Ind (phi ^ y)%CF. | Proof.
move=> nKy nHy; have [sKH | not_sKH] := boolP (K \subset H).
by rewrite !cfConjgEin (cfIndIsom (norm_conj_isom nHy)).
rewrite !cfIndEout // linearZ -(cfConjg_iso y) rmorph1 /=; congr (_ *: _).
by rewrite cfConjg_cfuni ?norm1 ?inE.
Qed. | Lemma | cfConjgInd_norm | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgEin",
"cfConjg_cfuni",
"cfConjg_iso",
"cfIndEout",
"cfIndIsom",
"inE",
"linearZ",
"norm1",
"norm_conj_isom",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgInd phi y :
H <| G -> K <| G -> y \in G -> ('Ind[H, K] phi ^ y)%CF = 'Ind (phi ^ y)%CF. | Proof.
move=> /andP[_ nHG] /andP[_ nKG] Gy.
by rewrite cfConjgInd_norm ?(subsetP nHG) ?(subsetP nKG).
Qed. | Lemma | cfConjgInd | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgInd_norm",
"nHG",
"nKG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_inertia_Ind phi :
G \subset 'N(H) -> 'I_G[phi] \subset 'I_G['Ind[H, K] phi]. | Proof.
move=> nHG; apply/subsetP=> y /setIP[Gy /setIdP[nKy /eqP Iphi_y]].
by rewrite 2!inE Gy cfConjgInd_norm ?(subsetP nHG) ?Iphi_y /=.
Qed. | Lemma | sub_inertia_Ind | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgInd_norm",
"inE",
"nHG",
"setIP",
"setIdP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
T | := 'I_G['chi_i]. | Let | T | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_id : 'I_T['chi_i] = T. | Proof. by rewrite -setIA setIid. Qed. | Lemma | inertia_id | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"setIA",
"setIid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfclass_inertia : ('chi[H]_i ^: T)%CF = [:: 'chi_i]. | Proof.
rewrite /cfclass inertia_id rcosets_id /(image _ _) enum_set1 /=.
by rewrite repr_group cfConjgJ1.
Qed. | Lemma | cfclass_inertia | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgJ1",
"cfclass",
"chi",
"enum_set1",
"image",
"inertia_id",
"rcosets_id",
"repr_group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgMorph (phi : 'CF(f @* H)) y :
y \in D -> y \in 'N(H) -> (cfMorph phi ^ y)%CF = cfMorph (phi ^ f y). | Proof.
move=> Dy nHy; have [sHD | not_sHD] := boolP (H \subset D); last first.
by rewrite !cfMorphEout // rmorph_alg cfConjg1.
apply/cfun_inP=> x Gx; rewrite !(cfConjgE, cfMorphE) ?memJ_norm ?groupV //.
by rewrite (subsetP (morphim_norm _ _)) ?mem_morphim.
by rewrite morphJ ?morphV ?groupV // (subsetP sHD).
Qed. | Lemma | cfConjgMorph | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjg1",
"cfConjgE",
"cfMorph",
"cfMorphE",
"cfMorphEout",
"cfun_inP",
"groupV",
"last",
"memJ_norm",
"mem_morphim",
"morphJ",
"morphV",
"morphim_norm",
"rmorph_alg",
"sHD",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_morph_pre (phi : 'CF(f @* H)) :
H <| G -> G \subset D -> 'I_G[cfMorph phi] = G :&: f @*^-1 'I_(f @* G)[phi]. | Proof.
case/andP=> sHG nHG sGD; have sHD := subset_trans sHG sGD.
apply/setP=> y; rewrite !in_setI; apply: andb_id2l => Gy.
have [Dy nHy] := (subsetP sGD y Gy, subsetP nHG y Gy).
rewrite Dy inE nHy 4!inE mem_morphim // -morphimJ ?(normP nHy) // subxx /=.
rewrite cfConjgMorph //; apply/eqP/eqP=> [Iphi_y | -> //].
by app... | Lemma | inertia_morph_pre | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Dx",
"apply",
"cfConjgMorph",
"cfMorph",
"cfMorphE",
"cfun_inP",
"inE",
"in_setI",
"mem_morphim",
"morphimJ",
"morphimP",
"nHG",
"normP",
"sGD",
"sHD",
"sHG",
"setP",
"subsetP",
"subset_trans",
"subxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_morph_im (phi : 'CF(f @* H)) :
H <| G -> G \subset D -> f @* 'I_G[cfMorph phi] = 'I_(f @* G)[phi]. | Proof.
move=> nsHG sGD; rewrite inertia_morph_pre // morphim_setIpre.
by rewrite (setIidPr _) ?Inertia_sub.
Qed. | Lemma | inertia_morph_im | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Inertia_sub",
"cfMorph",
"inertia_morph_pre",
"morphim_setIpre",
"nsHG",
"sGD",
"setIidPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(isoG : isom G R g) (isoH : isom H S h). | Hypotheses | isoG | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"isom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
(eq_hg : {in H, h =1 g}) (sHG : H \subset G). | Hypotheses | eq_hg | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"sHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
cfConjgIsom phi y :
y \in G -> y \in 'N(H) -> (cfIsom isoH phi ^ g y)%CF = cfIsom isoH (phi ^ y). | Proof.
move=> Gy nHy; have [_ defS] := isomP isoH.
rewrite morphimEdom (eq_in_imset eq_hg) -morphimEsub // in defS.
apply/cfun_inP=> gx; rewrite -{1}defS => /morphimP[x Gx Hx ->] {gx}.
rewrite cfConjgE; first by rewrite -defS inE -morphimJ ?(normP nHy).
by rewrite -morphV -?morphJ -?eq_hg ?cfIsomE ?cfConjgE ?memJ_norm ... | Lemma | cfConjgIsom | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgE",
"cfIsom",
"cfIsomE",
"cfun_inP",
"eq_hg",
"eq_in_imset",
"groupV",
"inE",
"isomP",
"memJ_norm",
"morphJ",
"morphV",
"morphimEdom",
"morphimEsub",
"morphimJ",
"morphimP",
"normP"
] | This does not depend on the (isoG : isom G R g) assumption. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
inertia_isom phi : 'I_R[cfIsom isoH phi] = g @* 'I_G[phi]. | Proof.
have [[_ defS] [injg <-]] := (isomP isoH, isomP isoG).
rewrite morphimEdom (eq_in_imset eq_hg) -morphimEsub // in defS.
rewrite /inertia !setIdE morphimIdom setIA -{1}defS -injm_norm ?injmI //.
apply/setP=> gy /[!inE]; apply: andb_id2l => /morphimP[y Gy nHy ->] {gy}.
rewrite cfConjgIsom // -sub1set -morphim_set1... | Lemma | inertia_isom | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgIsom",
"cfIsom",
"cfIsomE",
"cfun_inP",
"eq_hg",
"eq_in_imset",
"inE",
"inertia",
"injmI",
"injmSK",
"injm_norm",
"isoG",
"isomP",
"morphimEdom",
"morphimEsub",
"morphimIdom",
"morphimP",
"morphim_set1",
"setIA",
"setIdE",
"setP",
"sub1set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgMod_norm H K (phi : 'CF(H / K)) y :
y \in 'N(K) -> y \in 'N(H) -> ((phi %% K) ^ y)%CF = (phi ^ coset K y %% K)%CF. | Proof. exact: cfConjgMorph. Qed. | Lemma | cfConjgMod_norm | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgMorph",
"coset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgMod G H K (phi : 'CF(H / K)) y :
H <| G -> K <| G -> y \in G ->
((phi %% K) ^ y)%CF = (phi ^ coset K y %% K)%CF. | Proof.
move=> /andP[_ nHG] /andP[_ nKG] Gy.
by rewrite cfConjgMod_norm ?(subsetP nHG) ?(subsetP nKG).
Qed. | Lemma | cfConjgMod | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgMod_norm",
"coset",
"nHG",
"nKG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgQuo_norm H K (phi : 'CF(H)) y :
y \in 'N(K) -> y \in 'N(H) -> ((phi / K) ^ coset K y)%CF = (phi ^ y / K)%CF. | Proof.
move=> nKy nHy; have keryK: (K \subset cfker (phi ^ y)) = (K \subset cfker phi).
by rewrite cfker_conjg // -{1}(normP nKy) conjSg.
have [kerK | not_kerK] := boolP (K \subset cfker phi); last first.
by rewrite !cfQuoEout ?rmorph_alg ?cfConjg1 ?keryK.
apply/cfun_inP=> _ /morphimP[x nKx Hx ->].
have nHyb: coset... | Lemma | cfConjgQuo_norm | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjg1",
"cfConjgE",
"cfQuoEnorm",
"cfQuoEout",
"cfker",
"cfker_conjg",
"cfun_inP",
"conjSg",
"coset",
"groupJ",
"groupV",
"inE",
"in_setI",
"last",
"memJ_norm",
"morphJ",
"morphV",
"morphimJ",
"morphimP",
"normP",
"rmorph_alg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgQuo G H K (phi : 'CF(H)) y :
H <| G -> K <| G -> y \in G ->
((phi / K) ^ coset K y)%CF = (phi ^ y / K)%CF. | Proof.
move=> /andP[_ nHG] /andP[_ nKG] Gy.
by rewrite cfConjgQuo_norm ?(subsetP nHG) ?(subsetP nKG).
Qed. | Lemma | cfConjgQuo | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgQuo_norm",
"coset",
"nHG",
"nKG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_mod_pre G H K (phi : 'CF(H / K)) :
H <| G -> K <| G -> 'I_G[phi %% K] = G :&: coset K @*^-1 'I_(G / K)[phi]. | Proof. by move=> nsHG /andP[_]; apply: inertia_morph_pre. Qed. | Lemma | inertia_mod_pre | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"coset",
"inertia_morph_pre",
"nsHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_mod_quo G H K (phi : 'CF(H / K)) :
H <| G -> K <| G -> ('I_G[phi %% K] / K)%g = 'I_(G / K)[phi]. | Proof. by move=> nsHG /andP[_]; apply: inertia_morph_im. Qed. | Lemma | inertia_mod_quo | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inertia_morph_im",
"nsHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_quo G H K (phi : 'CF(H)) :
H <| G -> K <| G -> K \subset cfker phi ->
'I_(G / K)[phi / K] = ('I_G[phi] / K)%g. | Proof.
move=> nsHG nsKG kerK; rewrite -inertia_mod_quo ?cfQuoK //.
by rewrite (normalS _ (normal_sub nsHG)) // (subset_trans _ (cfker_sub phi)).
Qed. | Lemma | inertia_quo | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfQuoK",
"cfker",
"cfker_sub",
"inertia_mod_quo",
"normalS",
"normal_sub",
"nsHG",
"nsKG",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgSdprod phi y :
y \in 'N(K) -> y \in 'N(H) ->
(cfSdprod defG phi ^ y = cfSdprod defG (phi ^ y))%CF. | Proof.
move=> nKy nHy.
have nGy: y \in 'N(G) by rewrite -sub1set -(sdprodW defG) normsM ?sub1set.
rewrite -{2}[phi](cfSdprodK defG) cfConjgRes_norm // cfRes_sdprodK //.
by rewrite cfker_conjg // -{1}(normP nKy) conjSg cfker_sdprod.
Qed. | Lemma | cfConjgSdprod | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfConjgRes_norm",
"cfRes_sdprodK",
"cfSdprod",
"cfSdprodK",
"cfker_conjg",
"cfker_sdprod",
"conjSg",
"defG",
"normP",
"normsM",
"sdprodW",
"sub1set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_sdprod (L : {group gT}) phi :
L \subset 'N(K) -> L \subset 'N(H) -> 'I_L[cfSdprod defG phi] = 'I_L[phi]. | Proof.
move=> nKL nHL; have nGL: L \subset 'N(G) by rewrite -(sdprodW defG) normsM.
apply/setP=> z; rewrite !in_setI ![z \in 'I[_]]inE; apply: andb_id2l => Lz.
rewrite cfConjgSdprod ?(subsetP nKL) ?(subsetP nHL) ?(subsetP nGL) //=.
by rewrite (can_eq (cfSdprodK defG)).
Qed. | Lemma | inertia_sdprod | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"can_eq",
"cfConjgSdprod",
"cfSdprod",
"cfSdprodK",
"defG",
"gT",
"group",
"inE",
"in_setI",
"normsM",
"sdprodW",
"setP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgDprodl phi y :
y \in 'N(K) -> y \in 'N(H) ->
(cfDprodl KxH phi ^ y = cfDprodl KxH (phi ^ y))%CF. | Proof. by move=> nKy nHy; apply: cfConjgSdprod. Qed. | Lemma | cfConjgDprodl | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"KxH",
"apply",
"cfConjgSdprod",
"cfDprodl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgDprodr psi y :
y \in 'N(K) -> y \in 'N(H) ->
(cfDprodr KxH psi ^ y = cfDprodr KxH (psi ^ y))%CF. | Proof. by move=> nKy nHy; apply: cfConjgSdprod. Qed. | Lemma | cfConjgDprodr | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"KxH",
"apply",
"cfConjgSdprod",
"cfDprodr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgDprod phi psi y :
y \in 'N(K) -> y \in 'N(H) ->
(cfDprod KxH phi psi ^ y = cfDprod KxH (phi ^ y) (psi ^ y))%CF. | Proof. by move=> nKy nHy; rewrite rmorphM /= cfConjgDprodl ?cfConjgDprodr. Qed. | Lemma | cfConjgDprod | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"KxH",
"cfConjgDprodl",
"cfConjgDprodr",
"cfDprod",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_dprodl L phi :
L \subset 'N(K) -> L \subset 'N(H) -> 'I_L[cfDprodl KxH phi] = 'I_L[phi]. | Proof. by move=> nKL nHL; apply: inertia_sdprod. Qed. | Lemma | inertia_dprodl | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"KxH",
"apply",
"cfDprodl",
"inertia_sdprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_dprodr L psi :
L \subset 'N(K) -> L \subset 'N(H) -> 'I_L[cfDprodr KxH psi] = 'I_L[psi]. | Proof. by move=> nKL nHL; apply: inertia_sdprod. Qed. | Lemma | inertia_dprodr | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"KxH",
"apply",
"cfDprodr",
"inertia_sdprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_dprod L (phi : 'CF(K)) (psi : 'CF(H)) :
L \subset 'N(K) -> L \subset 'N(H) -> phi 1%g != 0 -> psi 1%g != 0 ->
'I_L[cfDprod KxH phi psi] = 'I_L[phi] :&: 'I_L[psi]. | Proof.
move=> nKL nHL nz_phi nz_psi; apply/eqP; rewrite eqEsubset subsetI.
rewrite -{1}(inertia_scale_nz psi nz_phi) -{1}(inertia_scale_nz phi nz_psi).
rewrite -(cfDprod_Resl KxH) -(cfDprod_Resr KxH) !sub_inertia_Res //=.
by rewrite -inertia_dprodl -?inertia_dprodr // -setIIr setIS ?inertia_mul.
Qed. | Lemma | inertia_dprod | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"KxH",
"apply",
"cfDprod",
"cfDprod_Resl",
"cfDprod_Resr",
"eqEsubset",
"inertia_dprodl",
"inertia_dprodr",
"inertia_mul",
"inertia_scale_nz",
"setIIr",
"setIS",
"sub_inertia_Res",
"subsetI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_dprod_irr L i j :
L \subset 'N(K) -> L \subset 'N(H) ->
'I_L[cfDprod KxH 'chi_i 'chi_j] = 'I_L['chi_i] :&: 'I_L['chi_j]. | Proof. by move=> nKL nHL; rewrite inertia_dprod ?irr1_neq0. Qed. | Lemma | inertia_dprod_irr | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"KxH",
"cfDprod",
"inertia_dprod",
"irr1_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nAy: forall i, P i -> y \in 'N(A i). | Hypothesis | nAy | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
cfConjgBigdprodi i (phi : 'CF(A i)) :
(cfBigdprodi defG phi ^ y = cfBigdprodi defG (phi ^ y))%CF. | Proof.
rewrite cfConjgDprodl; try by case: ifP => [/nAy// | _]; rewrite norm1 inE.
rewrite -sub1set norms_gen ?norms_bigcup // sub1set.
by apply/bigcapP=> j /andP[/nAy].
congr (cfDprodl _ _); case: ifP => [Pi | _].
by rewrite cfConjgRes_norm ?nAy.
by apply/cfun_inP=> _ /set1P->; rewrite !(cfRes1, cfConjg1).
Qed. | Lemma | cfConjgBigdprodi | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"bigcapP",
"cfBigdprodi",
"cfConjg1",
"cfConjgDprodl",
"cfConjgRes_norm",
"cfDprodl",
"cfRes1",
"cfun_inP",
"defG",
"inE",
"nAy",
"norm1",
"norms_bigcup",
"norms_gen",
"set1P",
"sub1set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjgBigdprod phi :
(cfBigdprod defG phi ^ y = cfBigdprod defG (fun i => phi i ^ y))%CF. | Proof.
by rewrite rmorph_prod /=; apply: eq_bigr => i _; apply: cfConjgBigdprodi.
Qed. | Lemma | cfConjgBigdprod | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfBigdprod",
"cfConjgBigdprodi",
"defG",
"eq_bigr",
"rmorph_prod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nAL : forall i, P i -> L \subset 'N(A i). | Hypothesis | nAL | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
inertia_bigdprodi i (phi : 'CF(A i)) :
P i -> 'I_L[cfBigdprodi defG phi] = 'I_L[phi]. | Proof.
move=> Pi; rewrite inertia_dprodl ?Pi ?cfRes_id ?nAL //.
by apply/norms_gen/norms_bigcup/bigcapsP=> j /andP[/nAL].
Qed. | Lemma | inertia_bigdprodi | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"bigcapsP",
"cfBigdprodi",
"cfRes_id",
"defG",
"inertia_dprodl",
"nAL",
"norms_bigcup",
"norms_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_bigdprod phi (Phi := cfBigdprod defG phi) :
Phi 1%g != 0 -> 'I_L[Phi] = L :&: \bigcap_(i | P i) 'I_L[phi i]. | Proof.
move=> nz_Phi; apply/eqP; rewrite eqEsubset; apply/andP; split.
rewrite subsetI Inertia_sub; apply/bigcapsP=> i Pi.
have [] := cfBigdprodK nz_Phi Pi; move: (_ / _) => a nz_a <-.
by rewrite inertia_scale_nz ?sub_inertia_Res //= ?nAL.
rewrite subsetI subsetIl; apply: subset_trans (inertia_prod _ _ _).
apply:... | Lemma | inertia_bigdprod | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Inertia_sub",
"apply",
"bigcap_min",
"bigcapsP",
"bigdprodWY",
"cfBigdprod",
"cfBigdprodK",
"defG",
"eqEsubset",
"inertia_bigdprodi",
"inertia_prod",
"inertia_scale_nz",
"nAL",
"norms_bigcup",
"norms_gen",
"setISS",
"split",
"sub_inertia_Res",
"subsetI",
"subsetIl",
"subsetI... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_bigdprod_irr Iphi (phi := fun i => 'chi_(Iphi i)) :
'I_L[cfBigdprod defG phi] = L :&: \bigcap_(i | P i) 'I_L[phi i]. | Proof.
rewrite inertia_bigdprod // -[cfBigdprod _ _]cfIirrE ?irr1_neq0 //.
by apply: cfBigdprod_irr => i _; apply: mem_irr.
Qed. | Lemma | inertia_bigdprod_irr | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfBigdprod",
"cfBigdprod_irr",
"cfIirrE",
"defG",
"inertia_bigdprod",
"irr1_neq0",
"mem_irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nsHG : H <| G. | Hypothesis | nsHG | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
theta | := 'chi_t. | Notation | theta | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
T | := 'I_G[theta]%G. | Notation | T | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"` 'T'" | := 'I_(gval G)[theta] (format "` 'T'") : group_scope. | Notation | ` 'T' | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
calA | := irr_constt ('Ind[T] theta). | Let | calA | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_constt",
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
calB | := irr_constt ('Ind[G] theta). | Let | calB | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_constt",
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
AtoB | := (Ind_Iirr G). | Notation | AtoB | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Ind_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
constt_Inertia_bijection :
[/\ (*a*) {in calA, forall s, 'Ind[G] 'chi_s \in irr G},
(*b*) {in calA &, injective (Ind_Iirr G)},
Ind_Iirr G @: calA =i calB,
(*c*) {in calA, forall s (psi := 'chi_s) (chi := 'Ind[G] psi),
[predI irr_constt ('Res chi) & calA] =i pred1 s}
& (*d*) {in cal... | Proof.
have [sHG sTG]: H \subset G /\ T \subset G by rewrite subsetIl normal_sub.
have nsHT : H <| T := normal_Inertia theta sHG; have sHT := normal_sub nsHT.
have AtoB_P s (psi := 'chi_s) (chi := 'Ind[G] psi): s \in calA ->
[/\ chi \in irr G, AtoB s \in calB & '['Res psi, theta] = '['Res chi, theta]].
- rewrite cons... | Theorem | constt_Inertia_bijection | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"AtoB",
"Clifford_Res_sum_cfclass",
"Cnat_cfdot_char",
"Ind_Iirr",
"addKr",
"addrC",
"addrK",
"apply",
"big1",
"big_seq1",
"calA",
"calB",
"card_cfclass_Iirr",
"cfConjg1",
"cfIirrE",
"cfInd1",
"cfIndInd",
"cfInd_char",
"cfRes1",
"cfResRes",
"cfRes_char",
"cfclassP",
"cfcl... | This is Isaacs, Theorem (6.11). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
chi | := 'chi_c. | Let | chi | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_Iirr b | := cfIirr ('chi_b * chi). | Definition | mul_Iirr | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"cfIirr",
"chi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_mod_Iirr (b : Iirr (G / N)) | := mul_Iirr (mod_Iirr b). | Definition | mul_mod_Iirr | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Iirr",
"mod_Iirr",
"mul_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(nsNG : N <| G) (cNt : 'Res[N] chi = theta). | Hypotheses | nsNG | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"chi",
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
sNG : N \subset G. | Proof. exact: normal_sub. Qed. | Let | sNG | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"normal_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nNG : G \subset 'N(N). | Proof. exact: normal_norm. Qed. | Let | nNG | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"normal_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extendible_irr_invariant : G \subset 'I[theta]. | Proof.
apply/subsetP=> y Gy; have nNy := subsetP nNG y Gy.
rewrite inE nNy; apply/eqP/cfun_inP=> x Nx; rewrite cfConjgE // -cNt.
by rewrite !cfResE ?memJ_norm ?cfunJ ?groupV.
Qed. | Lemma | extendible_irr_invariant | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cfConjgE",
"cfResE",
"cfunJ",
"cfun_inP",
"groupV",
"inE",
"memJ_norm",
"nNG",
"subsetP",
"theta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
IGtheta | := extendible_irr_invariant. | Let | IGtheta | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"extendible_irr_invariant"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
constt_Ind_mul_ext f (phi := 'chi_f) (psi := phi * theta) :
G \subset 'I[phi] -> psi \in irr N ->
let calS := irr_constt ('Ind phi) in
[/\ {in calS, forall b, 'chi_b * chi \in irr G},
{in calS &, injective mul_Iirr},
irr_constt ('Ind psi) =i [seq mul_Iirr b | b in calS]
& 'Ind psi = \sum_(b in cal... | Proof.
move=> IGphi irr_psi calS.
have IGpsi: G \subset 'I[psi].
by rewrite (subset_trans _ (inertia_mul _ _)) // subsetI IGphi.
pose e b := '['Ind[G] phi, 'chi_b]; pose d b g := '['chi_b * chi, 'chi_g * chi].
have Ne b: e b \in Num.nat by rewrite Cnat_cfdot_char ?cfInd_char ?irr_char.
have egt0 b: b \in calS -> e b ... | Theorem | constt_Ind_mul_ext | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Cnat_cfdot_char",
"apply",
"big1",
"bigD1",
"cfIirrE",
"cfIndM",
"cfInd_char",
"cfRes_Ind_invariant",
"cfdotZl",
"cfdotZr",
"cfdot_Res_l",
"cfdot_irr",
"cfdot_suml",
"cfdot_sumr",
"cfnorm_gt0",
"cfnorm_irr",
"cfunE",
"cfun_sum_constt",
"char1_eq0",
"character",
"chi",
"con... | This is Isaacs, Theorem (6.16) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
constt_Ind_ext :
[/\ forall b : Iirr (G / N), 'chi_(mod_Iirr b) * chi \in irr G,
injective mul_mod_Iirr,
irr_constt ('Ind theta) =i codom mul_mod_Iirr
& 'Ind theta = \sum_b 'chi_b 1%g *: 'chi_(mul_mod_Iirr b)]. | Proof.
have IHchi0: G \subset 'I['chi[N]_0] by rewrite inertia_irr0.
have [] := constt_Ind_mul_ext IHchi0; rewrite irr0 ?mul1r ?mem_irr //.
set psiG := 'Ind 1 => irrMchi injMchi constt_theta {2}->.
have dot_psiG b: '[psiG, 'chi_(mod_Iirr b)] = 'chi[G / N]_b 1%g.
rewrite mod_IirrE // -cfdot_Res_r cfRes_sub_ker ?cfker_... | Corollary | constt_Ind_ext | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Cnat_irr1",
"Iirr",
"apply",
"cfMod1",
"cfRes_sub_ker",
"cfdotZr",
"cfdot_Res_r",
"cfker",
"cfker_mod",
"cfnorm1",
"chi",
"codom",
"conj_natr",
"constt0_Res_cfker",
"constt_Ind_Res",
"constt_Ind_mul_ext",
"eq_big",
"imageP",
"inertia_irr0",
"irr",
"irr0",
"irr1_neq0",
"i... | This is Isaacs, Corollary (6.17) (due to Gallagher). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
invariant_chief_irr_cases G K L s (theta := 'chi[K]_s) :
chief_factor G L K -> abelian (K / L) -> G \subset 'I[theta] ->
let t := #|K : L| in
[\/ 'Res[L] theta \in irr L,
exists2 e, exists p, 'Res[L] theta = e%:R *: 'chi_p & (e ^ 2)%N = t
| exists2 p, injective p & 'Res[L] theta = \sum_(i < t) 'chi_(p... | Proof.
case/andP=> /maxgroupP[/andP[ltLK nLG] maxL] nsKG abKbar IGtheta t.
have [sKG nKG] := andP nsKG; have sLG := subset_trans (proper_sub ltLK) sKG.
have nsLG: L <| G by apply/andP.
have nsLK := normalS (proper_sub ltLK) sKG nsLG; have [sLK nLK] := andP nsLK.
have [p0 sLp0] := constt_cfRes_irr L s; rewrite -/theta i... | Theorem | invariant_chief_irr_cases | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_Res_sum_cfclass",
"Cnat_cfdot_char",
"IGtheta",
"Lagrange",
"LagrangeMr",
"abelian",
"apply",
"big1_seq",
"bigID",
"big_image",
"big_mkord",
"big_nth",
"big_seq1",
"big_uniq",
"bigcapP",
"can2_eq",
"card_Iirr_abelian",
"card_ord",
"card_quotient",
"cfConjg1",
"cfCon... | This is Isaacs, Theorem (6.19). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfRes_prime_irr_cases G N s p (chi := 'chi[G]_s) :
N <| G -> #|G : N| = p -> prime p ->
[\/ 'Res[N] chi \in irr N
| exists2 c, injective c & 'Res[N] chi = \sum_(i < p) 'chi_(c i)]. | Proof.
move=> /andP[sNG nNG] iGN pr_p.
have chiefGN: chief_factor G N G.
apply/andP; split=> //; apply/maxgroupP.
split=> [|M /andP[/andP[sMG ltMG] _] sNM].
by rewrite /proper sNG -indexg_gt1 iGN prime_gt1.
apply/esym/eqP; rewrite eqEsubset sNM -indexg_eq1 /= eq_sym.
rewrite -(eqn_pmul2l (indexg_gt0 G M)) m... | Corollary | cfRes_prime_irr_cases | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Lagrange_index",
"abelian",
"apply",
"card_quotient",
"chi",
"chief_factor",
"cyclic_abelian",
"eqEsubset",
"eq_sym",
"eqn_pmul2l",
"eqxx",
"indexgS",
"indexg_eq1",
"indexg_gt0",
"indexg_gt1",
"invariant_chief_irr_cases",
"irr",
"logn",
"lognX",
"logn_prime",
"maxgroupP",
... | This is Isaacs, Corollary (6.19). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
prime_invariant_irr_extendible G N s p :
N <| G -> #|G : N| = p -> prime p -> G \subset 'I['chi_s] ->
{t | 'Res[N, G] 'chi_t = 'chi_s}. | Proof.
move=> nsNG iGN pr_p IGchi.
have [t sGt] := constt_cfInd_irr s (normal_sub nsNG); exists t.
have [e DtN]: exists e, 'Res 'chi_t = e%:R *: 'chi_s.
rewrite constt_Ind_Res in sGt.
rewrite (Clifford_Res_sum_cfclass nsNG sGt) cfclass_invariant // big_seq1.
set e := '[_, _]; exists (Num.truncn e).
by rewrite t... | Corollary | prime_invariant_irr_extendible | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_Res_sum_cfclass",
"Cnat_cfdot_char",
"addr0",
"apply",
"big1",
"bigD1",
"big_seq1",
"cfRes_char",
"cfRes_eq0",
"cfRes_prime_irr_cases",
"cfclass_invariant",
"cfdotZl",
"cfdot_irr",
"cfdot_suml",
"cfdotr",
"cfnormZ",
"cfnorm_irr",
"constt_Ind_Res",
"constt_cfInd_irr",
... | This is Isaacs, Corollary (6.20). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
extend_to_cfdet G N s c0 u :
let theta := 'chi_s in let lambda := cfDet theta in let mu := 'chi_u in
N <| G -> coprime #|G : N| (Num.truncn (theta 1%g)) ->
'Res[N, G] 'chi_c0 = theta -> 'Res[N, G] mu = lambda ->
exists2 c, 'Res 'chi_c = theta /\ cfDet 'chi_c = mu
& forall c1, 'Res 'chi_c1 = thet... | Proof.
move=> theta lambda mu nsNG; set e := #|G : N|; set f := Num.truncn _.
set eta := 'chi_c0 => co_e_f etaNth muNlam; have [sNG nNG] := andP nsNG.
have fE: f%:R = theta 1%g by rewrite truncnK ?Cnat_irr1.
pose nu := cfDet eta; have lin_nu: nu \is a linear_char := cfDet_lin_char _.
have nuNlam: 'Res nu = lambda by re... | Lemma | extend_to_cfdet | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Cnat_irr1",
"addn0",
"alpha",
"apply",
"c0",
"c1",
"c2",
"card_quotient",
"cfDet",
"cfDetRes",
"cfDet_lin_char",
"cfDet_mul_lin",
"cfIirrE",
"cfMod1",
"cfModK",
"cfMod_eq1",
"cfMod_lin_char",
"cfQuoK",
"cfQuo_lin_char",
"cfRes1",
"cfRes_lin_lin",
"cfker",
"cfker_Res",
... | This is Isaacs, Lemma (6.24). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
solvable_irr_extendible_from_det G N s (theta := 'chi[N]_s) :
N <| G -> solvable (G / N) ->
G \subset 'I[theta] -> coprime #|G : N| (Num.truncn (theta 1%g)) ->
[exists c, 'Res 'chi[G]_c == theta]
= [exists u, 'Res 'chi[G]_u == cfDet theta]. | Proof.
set e := #|G : N|; set f := Num.truncn _ => nsNG solG IGtheta co_e_f.
apply/exists_eqP/exists_eqP=> [[c cNth] | [u uNdth]].
have /lin_char_irr/irrP[u Du] := cfDet_lin_char 'chi_c.
by exists u; rewrite -Du -cfDetRes ?irr_char ?cNth.
move: {2}e.+1 (ltnSn e) => m.
elim: m => // m IHm in G u e nsNG solG IGtheta ... | Theorem | solvable_irr_extendible_from_det | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"IGtheta",
"Lagrange",
"apply",
"c0",
"cardG_gt0",
"card_injm",
"card_quotient",
"cfConjgRes",
"cfConjg_id",
"cfDet",
"cfDetConjg",
"cfDetRes",
"cfDet_lin_char",
"cfRes1",
"cfResRes",
"cfRes_id",
"cfRes_lin_char",
"chi",
"conjg_IirrE",
"coprime",
"coprime_dvdl",
"eqsVneq",
... | This is Isaacs, Theorem (6.25). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
extend_linear_char_from_Sylow G N (lambda : 'CF(N)) :
N <| G -> lambda \is a linear_char -> G \subset 'I[lambda] ->
(forall p, p \in \pi('o(lambda)%CF) ->
exists2 Hp : {group gT},
[/\ N \subset Hp, Hp \subset G & p.-Sylow(G / N) (Hp / N)%g]
& exists u, 'Res 'chi[Hp]_u = lambda) ->
exist... | Proof.
set m := 'o(lambda)%CF => nsNG lam_lin IGlam p_ext_lam.
have [sNG nNG] := andP nsNG; have linN := @cfRes_lin_lin _ _ N.
wlog [p p_lam]: lambda @m lam_lin IGlam p_ext_lam /
exists p : nat, \pi(m) =i (p : nat_pred).
- move=> IHp; have [linG [cf [inj_cf _ lin_cf onto_cf]]] := lin_char_group N.
case=> cf1 cfM cf... | Theorem | extend_linear_char_from_Sylow | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"CdivE",
"Clifford_Res_sum_cfclass",
"Cnat_cfdot_char",
"Cnat_irr1",
"Sylow",
"addn0",
"apply",
"big_mkcond",
"big_mkord",
"big_morph",
"big_seq1",
"cfDet",
"cfDetMn",
"cfDetRes",
"cfDet_id",
"cfDet_lin_char",
"cfDet_order_lin",
"cfInd1",
"cfInd_char",
"cfRes1",
"cfResRes",
... | This is Isaacs, Theorem (6.26). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
extend_coprime_linear_char G N (lambda : 'CF(N)) :
N <| G -> lambda \is a linear_char -> G \subset 'I[lambda] ->
coprime #|G : N| 'o(lambda)%CF ->
exists u, [/\ 'Res 'chi[G]_u = lambda, 'o('chi_u)%CF = 'o(lambda)%CF
& forall v,
'Res 'chi_v = lambda -> coprime #|G : N| 'o('chi_v... | Proof.
set e := #|G : N| => nsNG lam_lin IGlam co_e_lam; have [sNG nNG] := andP nsNG.
have [p lam_p | v vNlam] := extend_linear_char_from_Sylow nsNG lam_lin IGlam.
exists N; last first.
by have /irrP[u ->] := lin_char_irr lam_lin; exists u; rewrite cfRes_id.
split=> //; rewrite trivg_quotient /pHall sub1G pgrou... | Corollary | extend_coprime_linear_char | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"card_quotient",
"cfDet_order_lin",
"cfQuoK",
"cfQuo_lin_char",
"cfRes_id",
"cfRes_lin_lin",
"cfker",
"cfker_Res",
"cfker_cfun1",
"cforder_Res",
"cforder_lin_char_dvdG",
"cforder_mod",
"cfunE",
"chi",
"chinese",
"chinese_modr",
"conjC1",
"coprime",
"coprimeMr",
"copr... | This is Isaacs, Corollary (6.27). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
extend_solvable_coprime_irr G N t (theta := 'chi[N]_t) :
N <| G -> solvable (G / N) -> G \subset 'I[theta] ->
coprime #|G : N| ('o(theta)%CF * Num.truncn (theta 1%g)) ->
exists c, [/\ 'Res 'chi[G]_c = theta, 'o('chi_c)%CF = 'o(theta)%CF
& forall d,
'Res 'chi_d = theta -> coprim... | Proof.
set e := #|G : N|; set f := Num.truncn _ => nsNG solG IGtheta.
rewrite coprimeMr => /andP[co_e_th co_e_f].
have [sNG nNG] := andP nsNG; pose lambda := cfDet theta.
have lin_lam: lambda \is a linear_char := cfDet_lin_char theta.
have IGlam: G \subset 'I[lambda].
apply/subsetP=> y /(subsetP IGtheta)/setIdP[nNy /... | Corollary | extend_solvable_coprime_irr | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"IGtheta",
"Uu",
"apply",
"c0",
"cfDet",
"cfDetConjg",
"cfDetRes",
"cfDet_lin_char",
"cfDet_order_lin",
"chi",
"coprime",
"coprimeMr",
"exists_eqP",
"extend_coprime_linear_char",
"extend_to_cfdet",
"inE",
"irrP",
"irr_char",
"lin_char_irr",
"linear_char",
"nNG",
"nsNG",
"... | This is Isaacs, Corollary (6.28). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
frobGK : [Frobenius G with kernel K]. | Hypothesis | frobGK | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [] | state these theorems using the Frobenius property. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inertia_Frobenius_ker i : i != 0 -> 'I_G['chi[K]_i] = K. | Proof.
have [_ _ nsKG regK] := Frobenius_kerP frobGK; have [sKG nKG] := andP nsKG.
move=> nzi; apply/eqP; rewrite eqEsubset sub_Inertia // andbT.
apply/subsetP=> x /setIP[Gx /setIdP[nKx /eqP x_stab_i]].
have actIirrK: is_action G (@conjg_Iirr _ K).
split=> [y j k eq_jk | j y z Gy Gz].
by apply/irr_inj/(can_inj (c... | Theorem | inertia_Frobenius_ker | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Frobenius_kerP",
"Iirr",
"act",
"afix1P",
"apply",
"astabs_ract",
"cardD1",
"card_afix_irr_classes",
"cards1P",
"cfConjgEJ",
"cfConjgK",
"cfConjgM",
"cfunJ",
"chi",
"classG_eq1",
"class_lcoset",
"class_rcoset",
"class_refl",
"classes",
"classes1",
"conjgK",
"conjgM",
"co... | This is Isaacs, Theorem 6.34(a1). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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