fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
cfconjC_eq1:= cfAut_eq1 conjC. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset fingroup morphism perm",
"From mathcomp Require Import autom... | character/classfun.v | cfconjC_eq1 | |
cfConjg:= Cfun 1 cfConjg_subproof. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjg | |
cfConjgEphi y x : y \in 'N(G) -> (phi ^ y)%CF x = phi (x ^ y^-1)%g.
Proof. by rewrite cfunElock genGid => ->. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgE | |
cfConjgEJphi y x : y \in 'N(G) -> (phi ^ y)%CF (x ^ y) = phi x.
Proof. by move/cfConjgE->; rewrite conjgK. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgEJ | |
cfConjgEoutphi y : y \notin 'N(G) -> (phi ^ y = phi)%CF.
Proof.
by move/negbTE=> notNy; apply/cfunP=> x; rewrite !cfunElock genGid notNy.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgEout | |
cfConjgEinphi y (nGy : y \in 'N(G)) :
(phi ^ y)%CF = cfIsom (norm_conj_isom nGy) phi.
Proof.
apply/cfun_inP=> x Gx.
by rewrite cfConjgE // -{2}[x](conjgKV y) cfIsomE ?memJ_norm ?groupV.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgEin | |
cfConjgMnormphi :
{in 'N(G) &, forall y z, phi ^ (y * z) = (phi ^ y) ^ z}%CF.
Proof.
move=> y z nGy nGz.
by apply/cfunP=> x; rewrite !cfConjgE ?groupM // invMg conjgM.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgMnorm | |
cfConjg_idphi y : y \in G -> (phi ^ y)%CF = phi.
Proof.
move=> Gy; apply/cfunP=> x; have nGy := subsetP (normG G) y Gy.
by rewrite -(cfunJ _ _ Gy) cfConjgEJ.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjg_id | |
cfConjgML phi :
G <| L -> {in L &, forall y z, phi ^ (y * z) = (phi ^ y) ^ z}%CF.
Proof. by case/andP=> _ /subsetP nGL; apply: sub_in2 (cfConjgMnorm phi). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgM | |
cfConjgJ1phi : (phi ^ 1)%CF = phi.
Proof. by apply/cfunP=> x; rewrite cfConjgE ?group1 // invg1 conjg1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgJ1 | |
cfConjgKy : cancel (cfConjg y) (cfConjg y^-1 : 'CF(G) -> 'CF(G)).
Proof.
move=> phi; apply/cfunP=> x; rewrite !cfunElock groupV /=.
by case: ifP => -> //; rewrite conjgKV.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgK | |
cfConjgKVy : cancel (cfConjg y^-1) (cfConjg y : 'CF(G) -> 'CF(G)).
Proof. by move=> phi /=; rewrite -{1}[y]invgK cfConjgK. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgKV | |
cfConjg1phi y : (phi ^ y)%CF 1%g = phi 1%g.
Proof. by rewrite cfunElock conj1g if_same. Qed.
Fact cfConjg_is_linear y : linear (cfConjg y : 'CF(G) -> 'CF(G)).
Proof. by move=> a phi psi; apply/cfunP=> x; rewrite !cfunElock. Qed.
HB.instance Definition _ y := GRing.isSemilinear.Build _ _ _ _ (cfConjg y)
(GRing.semilinear_linear (cfConjg_is_linear y)). | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjg1 | |
cfConjg_cfuniJA y : y \in 'N(G) -> ('1_A ^ y)%CF = '1_(A :^ y) :> 'CF(G).
Proof.
move=> nGy; apply/cfunP=> x; rewrite !cfunElock genGid nGy -sub_conjgV.
by rewrite -class_lcoset -class_rcoset norm_rlcoset ?memJ_norm ?groupV.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjg_cfuniJ | |
cfConjg_cfuniA y : y \in 'N(A) -> ('1_A ^ y)%CF = '1_A :> 'CF(G).
Proof.
by have [/cfConjg_cfuniJ-> /normP-> | /cfConjgEout] := boolP (y \in 'N(G)).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjg_cfuni | |
cfConjg_cfun1y : (1 ^ y)%CF = 1 :> 'CF(G).
Proof.
by rewrite -cfuniG; have [/cfConjg_cfuni|/cfConjgEout] := boolP (y \in 'N(G)).
Qed.
Fact cfConjg_is_monoid_morphism y : monoid_morphism (cfConjg y : _ -> 'CF(G)).
Proof.
split=> [|phi psi]; first exact: cfConjg_cfun1.
by apply/cfunP=> x; rewrite !cfunElock.
Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `cfConjg_is_monoid_morphism` instead")] | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjg_cfun1 | |
cfConjg_is_multiplicativey :=
(fun g => (g.2,g.1)) (cfConjg_is_monoid_morphism y).
HB.instance Definition _ y := GRing.isMonoidMorphism.Build _ _ (cfConjg y)
(cfConjg_is_monoid_morphism y). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjg_is_multiplicative | |
cfConjg_eq1phi y : ((phi ^ y)%CF == 1) = (phi == 1).
Proof. by apply: rmorph_eq1; apply: can_inj (cfConjgK y). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjg_eq1 | |
cfAutConjgphi u y : cfAut u (phi ^ y) = (cfAut u phi ^ y)%CF.
Proof. by apply/cfunP=> x; rewrite !cfunElock. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfAutConjg | |
conj_cfConjgphi y : (phi ^ y)^*%CF = (phi^* ^ y)%CF.
Proof. exact: cfAutConjg. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | conj_cfConjg | |
cfker_conjgphi y : y \in 'N(G) -> cfker (phi ^ y) = cfker phi :^ y.
Proof.
move=> nGy; rewrite cfConjgEin // cfker_isom.
by rewrite morphim_conj (setIidPr (cfker_sub _)).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfker_conjg | |
cfDetConjgphi y : cfDet (phi ^ y) = (cfDet phi ^ y)%CF.
Proof.
have [nGy | not_nGy] := boolP (y \in 'N(G)); last by rewrite !cfConjgEout.
by rewrite !cfConjgEin cfDetIsom.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfDetConjg | |
inertia(B : {set gT}) (phi : 'CF(B)) :=
[set y in 'N(B) | (phi ^ y)%CF == phi].
Local Notation "''I[' phi ]" := (inertia phi) : group_scope.
Local Notation "''I_' G [ phi ]" := (G%g :&: 'I[phi]) : group_scope.
Fact group_set_inertia (H : {group gT}) phi : group_set 'I[phi : 'CF(H)].
Proof.
apply/group_setP; split; first by rewrite inE group1 /= cfConjgJ1.
move=> y z /setIdP[nHy /eqP n_phi_y] /setIdP[nHz n_phi_z].
by rewrite inE groupM //= cfConjgMnorm ?n_phi_y.
Qed. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia | |
inertia_groupH phi := Group (@group_set_inertia H phi).
Local Notation "''I[' phi ]" := (inertia_group phi) : Group_scope.
Local Notation "''I_' G [ phi ]" := (G :&: 'I[phi])%G : Group_scope.
Variables G H : {group gT}.
Implicit Type phi : 'CF(H). | Canonical | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_group | |
inertiaJphi y : y \in 'I[phi] -> (phi ^ y)%CF = phi.
Proof. by case/setIdP=> _ /eqP->. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertiaJ | |
inertia_valJphi x y : y \in 'I[phi] -> phi (x ^ y)%g = phi x.
Proof. by case/setIdP=> nHy /eqP {1}<-; rewrite cfConjgEJ. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_valJ | |
Inertia_subphi : 'I_G[phi] \subset G.
Proof. exact: subsetIl. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | Inertia_sub | |
norm_inertiaphi : 'I[phi] \subset 'N(H).
Proof. by rewrite ['I[_]]setIdE subsetIl. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | norm_inertia | |
sub_inertiaphi : H \subset 'I[phi].
Proof.
by apply/subsetP=> y Hy; rewrite inE cfConjg_id ?(subsetP (normG H)) /=.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | sub_inertia | |
normal_inertiaphi : H <| 'I[phi].
Proof. by rewrite /normal sub_inertia norm_inertia. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | normal_inertia | |
sub_Inertiaphi : H \subset G -> H \subset 'I_G[phi].
Proof. by rewrite subsetI sub_inertia andbT. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | sub_Inertia | |
norm_Inertiaphi : 'I_G[phi] \subset 'N(H).
Proof. by rewrite setIC subIset ?norm_inertia. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | norm_Inertia | |
normal_Inertiaphi : H \subset G -> H <| 'I_G[phi].
Proof. by rewrite /normal norm_Inertia andbT; apply: sub_Inertia. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | normal_Inertia | |
cfConjg_eqEphi :
H <| G ->
{in G &, forall y z, (phi ^ y == phi ^ z)%CF = (z \in 'I_G[phi] :* y)}.
Proof.
case/andP=> _ nHG y z Gy; rewrite -{1 2}[z](mulgKV y) groupMr // mem_rcoset.
move: {z}(z * _)%g => z Gz; rewrite 2!inE Gz cfConjgMnorm ?(subsetP nHG) //=.
by rewrite eq_sym (can_eq (cfConjgK y)).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjg_eqE | |
cent_sub_inertiaphi : 'C(H) \subset 'I[phi].
Proof.
apply/subsetP=> y cHy; have nHy := subsetP (cent_sub H) y cHy.
rewrite inE nHy; apply/eqP/cfun_inP=> x Hx; rewrite cfConjgE //.
by rewrite /conjg invgK mulgA (centP cHy) ?mulgK.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cent_sub_inertia | |
cent_sub_Inertiaphi : 'C_G(H) \subset 'I_G[phi].
Proof. exact: setIS (cent_sub_inertia phi). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cent_sub_Inertia | |
center_sub_Inertiaphi : H \subset G -> 'Z(G) \subset 'I_G[phi].
Proof.
by move/centS=> sHG; rewrite setIS // (subset_trans sHG) // cent_sub_inertia.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | center_sub_Inertia | |
conjg_inertiaphi y : y \in 'N(H) -> 'I[phi] :^ y = 'I[phi ^ y].
Proof.
move=> nHy; apply/setP=> z; rewrite !['I[_]]setIdE conjIg conjGid // !in_setI.
apply/andb_id2l=> nHz; rewrite mem_conjg !inE.
by rewrite !cfConjgMnorm ?in_group ?(can2_eq (cfConjgKV y) (cfConjgK y)) ?invgK.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | conjg_inertia | |
inertia0: 'I[0 : 'CF(H)] = 'N(H).
Proof. by apply/setP=> x; rewrite !inE linear0 eqxx andbT. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia0 | |
inertia_addphi psi : 'I[phi] :&: 'I[psi] \subset 'I[phi + psi].
Proof.
rewrite !['I[_]]setIdE -setIIr setIS //.
by apply/subsetP=> x /[!(inE, linearD)]/= /andP[/eqP-> /eqP->].
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_add | |
inertia_sumI r (P : pred I) (Phi : I -> 'CF(H)) :
'N(H) :&: \bigcap_(i <- r | P i) 'I[Phi i]
\subset 'I[\sum_(i <- r | P i) Phi i].
Proof.
elim/big_rec2: _ => [|i K psi Pi sK_Ipsi]; first by rewrite setIT inertia0.
by rewrite setICA; apply: subset_trans (setIS _ sK_Ipsi) (inertia_add _ _).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_sum | |
inertia_scalea phi : 'I[phi] \subset 'I[a *: phi].
Proof.
apply/subsetP=> x /setIdP[nHx /eqP Iphi_x].
by rewrite inE nHx linearZ /= Iphi_x.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_scale | |
inertia_scale_nza phi : a != 0 -> 'I[a *: phi] = 'I[phi].
Proof.
move=> nz_a; apply/eqP.
by rewrite eqEsubset -{2}(scalerK nz_a phi) !inertia_scale.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_scale_nz | |
inertia_oppphi : 'I[- phi] = 'I[phi].
Proof. by rewrite -scaleN1r inertia_scale_nz // oppr_eq0 oner_eq0. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_opp | |
inertia1: 'I[1 : 'CF(H)] = 'N(H).
Proof. by apply/setP=> x; rewrite inE rmorph1 eqxx andbT. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia1 | |
Inertia1: H <| G -> 'I_G[1 : 'CF(H)] = G.
Proof. by rewrite inertia1 => /normal_norm/setIidPl. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | Inertia1 | |
inertia_mulphi psi : 'I[phi] :&: 'I[psi] \subset 'I[phi * psi].
Proof.
rewrite !['I[_]]setIdE -setIIr setIS //.
by apply/subsetP=> x /[!(inE, rmorphM)]/= /andP[/eqP-> /eqP->].
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_mul | |
inertia_prodI r (P : pred I) (Phi : I -> 'CF(H)) :
'N(H) :&: \bigcap_(i <- r | P i) 'I[Phi i]
\subset 'I[\prod_(i <- r | P i) Phi i].
Proof.
elim/big_rec2: _ => [|i K psi Pi sK_psi]; first by rewrite inertia1 setIT.
by rewrite setICA; apply: subset_trans (setIS _ sK_psi) (inertia_mul _ _).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_prod | |
inertia_injective(chi : 'CF(H)) :
{in H &, injective chi} -> 'I[chi] = 'C(H).
Proof.
move=> inj_chi; apply/eqP; rewrite eqEsubset cent_sub_inertia andbT.
apply/subsetP=> y Ichi_y; have /setIdP[nHy _] := Ichi_y.
apply/centP=> x Hx; apply/esym/commgP/conjg_fixP.
by apply/inj_chi; rewrite ?memJ_norm ?(inertia_valJ _ Ichi_y).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_injective | |
inertia_irr_primep i :
#|H| = p -> prime p -> i != 0 -> 'I['chi[H]_i] = 'C(H).
Proof. by move=> <- pr_H /(irr_prime_injP pr_H); apply: inertia_injective. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_irr_prime | |
inertia_irr0: 'I['chi[H]_0] = 'N(H).
Proof. by rewrite irr0 inertia1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_irr0 | |
cfConjg_isoy : isometry (cfConjg y : 'CF(H) -> 'CF(H)).
Proof.
move=> phi psi; congr (_ * _).
have [nHy | not_nHy] := boolP (y \in 'N(H)); last by rewrite !cfConjgEout.
rewrite (reindex_astabs 'J y) ?astabsJ //=.
by apply: eq_bigr=> x _; rewrite !cfConjgEJ.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjg_iso | |
cfdot_Res_conjgpsi phi y :
y \in G -> '['Res[H, G] psi, phi ^ y] = '['Res[H] psi, phi].
Proof.
move=> Gy; rewrite -(cfConjg_iso y _ phi); congr '[_, _]; apply/cfunP=> x.
rewrite !cfunElock !genGid; case nHy: (y \in 'N(H)) => //.
by rewrite !(fun_if psi) cfunJ ?memJ_norm ?groupV.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfdot_Res_conjg | |
cfConjg_char(chi : 'CF(H)) y :
chi \is a character -> (chi ^ y)%CF \is a character.
Proof.
have [nHy Nchi | /cfConjgEout-> //] := boolP (y \in 'N(H)).
by rewrite cfConjgEin cfIsom_char.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjg_char | |
cfConjg_lin_char(chi : 'CF(H)) y :
chi \is a linear_char -> (chi ^ y)%CF \is a linear_char.
Proof. by case/andP=> Nchi chi1; rewrite qualifE/= cfConjg1 cfConjg_char. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjg_lin_char | |
cfConjg_irry chi : chi \in irr H -> (chi ^ y)%CF \in irr H.
Proof. by rewrite !irrEchar cfConjg_iso => /andP[/cfConjg_char->]. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjg_irr | |
conjg_Iirri y := cfIirr ('chi[H]_i ^ y)%CF. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | conjg_Iirr | |
conjg_IirrEi y : 'chi_(conjg_Iirr i y) = ('chi_i ^ y)%CF.
Proof. by rewrite cfIirrE ?cfConjg_irr ?mem_irr. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | conjg_IirrE | |
conjg_IirrKy : cancel (conjg_Iirr^~ y) (conjg_Iirr^~ y^-1%g).
Proof. by move=> i; apply/irr_inj; rewrite !conjg_IirrE cfConjgK. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | conjg_IirrK | |
conjg_IirrKVy : cancel (conjg_Iirr^~ y^-1%g) (conjg_Iirr^~ y).
Proof. by rewrite -{2}[y]invgK; apply: conjg_IirrK. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | conjg_IirrKV | |
conjg_Iirr_injy : injective (conjg_Iirr^~ y).
Proof. exact: can_inj (conjg_IirrK y). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | conjg_Iirr_inj | |
conjg_Iirr_eq0i y : (conjg_Iirr i y == 0) = (i == 0).
Proof. by rewrite -!irr_eq1 conjg_IirrE cfConjg_eq1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | conjg_Iirr_eq0 | |
conjg_Iirr0x : conjg_Iirr 0 x = 0.
Proof. by apply/eqP; rewrite conjg_Iirr_eq0. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | conjg_Iirr0 | |
cfdot_irr_conjgi 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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfdot_irr_conjg | |
cfclass(A : {set gT}) (phi : 'CF(A)) (B : {set gT}) :=
[seq (phi ^ repr Tx)%CF | Tx in rcosets 'I_B[phi] B].
Local Notation "phi ^: G" := (cfclass phi G) : cfun_scope. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfclass | |
size_cfclassi : size ('chi[H]_i ^: G)%CF = #|G : 'I_G['chi_i]|.
Proof. by rewrite size_map -cardE. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | size_cfclass | |
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_A[phi] :* y); first by rewrite -rcosetE imset_f.
case: repr_rcosetP => z /setIP[_ /setIdP[nHz /eqP Tz]].
by rewrite cfConjgMnorm ?Tz.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfclassP | |
cfclassInormphi : (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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfclassInorm | |
cfclass_reflphi : phi \in (phi ^: G)%CF.
Proof. by apply/cfclassP; exists 1%g => //; rewrite cfConjgJ1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfclass_refl | |
cfclass_transrphi 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 -?cfConjgMnorm ?groupM ?nHN.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfclass_transr | |
cfclass_symphi psi : (psi \in phi ^: G)%CF = (phi \in psi ^: G)%CF.
Proof. by apply/idP/idP=> /cfclass_transr <-; apply: cfclass_refl. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfclass_sym | |
cfclass_uniqphi : 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 rewrite !rcosetM (rcoset_id Iphi_v) (rcoset_id Iphi_u).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfclass_uniq | |
cfclass_invariantphi : 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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfclass_invariant | |
cfclass1: H <| G -> (1 ^: G)%CF = [:: 1 : 'CF(H)].
Proof. by move/normal_norm=> nHG; rewrite cfclass_invariant ?inertia1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfclass1 | |
cfclass_Iirr(A : {set gT}) i := conjg_Iirr i @: A. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfclass_Iirr | |
cfclass_IirrEi 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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfclass_IirrE | |
eq_cfclass_IirrEi 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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | eq_cfclass_IirrE | |
im_cfclass_Iirri :
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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | im_cfclass_Iirr | |
card_cfclass_Iirri : 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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | card_cfclass_Iirr | |
reindex_cfclassR 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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | reindex_cfclass | |
cfResIndj:
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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfResInd | |
Clifford_Res_sum_cfclassi 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[_ chiHk].
have{chiHk chiHj}: '['Res[H] ('Ind[G] 'chi_j), 'chi_k] != 0.
rewrite !inE !cfdot_Res_l in chiHj chiHk *.
apply: contraNneq chiHk; rewrite cfdot_sum_irr => /psumr_eq0P/(_ i isT)/eqP.
rewrite -cfdotC cfdotC mulf_eq0 conjC_eq0 (negbTE chiHj) /= => -> // i1.
by rewrite -cfdotC natr_ge0 // rpredM ?Cnat_cfdot_char ?cfInd_char ?irr_char.
rewrite cfResInd // cfdotZl mulf_eq0 cfdot_suml => /norP[_].
apply: contraR => chiGk'j; rewrite big1 // => x Gx; apply: contraNeq chiGk'j.
rewrite -conjg_IirrE cfdot_irr pnatr_eq0; case: (_ =P k) => // <- _.
by rewrite conjg_IirrE; apply/cfclassP; exists x.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | Clifford_Res_sum_cfclass | |
cfRes_Ind_invariantpsi :
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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfRes_Ind_invariant | |
dvdn_constt_Res1_irr1i 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; congr (_ * _).
rewrite mulr_natl -[size _]card_ord big_tnth -sumr_const; apply: eq_bigr => k _.
by have /cfclassP[y Gy ->]:= mem_tnth k (in_tuple _); rewrite cfConjg1.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | dvdn_constt_Res1_irr1 | |
cfclass_Indphi 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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfclass_Ind | |
cfConjgRes_normphi 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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgRes_norm | |
cfConjgResphi 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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgRes | |
sub_inertia_Resphi :
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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | sub_inertia_Res | |
cfConjgInd_normphi 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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgInd_norm | |
cfConjgIndphi 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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgInd | |
sub_inertia_Indphi :
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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | sub_inertia_Ind | |
inertia_id: 'I_T['chi_i] = T. Proof. by rewrite -setIA setIid. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_id | |
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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfclass_inertia | |
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 morphJ ?morphV ?groupV // (subsetP sHD).
by rewrite (subsetP (morphim_norm _ _)) ?mem_morphim.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgMorph | |
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 apply/cfun_inP=> _ /morphimP[x Dx Hx ->]; rewrite -!cfMorphE ?Iphi_y.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_morph_pre | |
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.
Variables (R S : {group rT}).
Variables (g : {morphism G >-> rT}) (h : {morphism H >-> rT}).
Hypotheses (isoG : isom G R g) (isoH : isom H S h).
Hypotheses (eq_hg : {in H, h =1 g}) (sHG : H \subset G). | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_morph_im | |
cfConjgIsomphi 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; last by rewrite -defS inE -morphimJ ?(normP nHy).
by rewrite -morphV -?morphJ -?eq_hg ?cfIsomE ?cfConjgE ?memJ_norm ?groupV.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgIsom | |
inertia_isomphi : '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 // injmSK ?sub1set //= inE.
apply/eqP/eqP=> [Iphi_y | -> //].
by apply/cfun_inP=> x Hx; rewrite -!(cfIsomE isoH) ?Iphi_y.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | inertia_isom | |
cfConjgMod_normH 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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgMod_norm | |
cfConjgModG 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 | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div tuple finfun bigop prime order",
"From mathcomp Require Import ssralg ssrnum finset fingroup morphism perm",
"From mathcomp Require Import aut... | character/inertia.v | cfConjgMod |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.