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 |
|---|---|---|---|---|---|---|
cfSdprod_eq1phi : (cfSdprod phi == 1) = (phi == 1).
Proof. exact: rmorph_eq1 cfSdprod_inj. Qed. | Lemma | 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 | cfSdprod_eq1 | |
cfRes_sdprodKphi : K \subset cfker phi -> cfSdprod ('Res[H] phi) = phi.
Proof.
move=> kerK; apply/cfun_inP=> _ /(mem_sdprod defG)[x [y [Kx Hy -> _]]].
by rewrite cfSdprodE // cfResE // cfkerMl ?(subsetP kerK).
Qed. | Lemma | 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 | cfRes_sdprodK | |
sdprod_cfkerphi : K ><| cfker phi = cfker (cfSdprod phi).
Proof.
have [skerH [_ _ nKH tiKH]] := (cfker_sub phi, sdprodP defG).
rewrite unlock cfker_morph ?normal_norm // cfker_isom restrmEsub //=.
rewrite -(sdprod_modl defG) ?sub_cosetpre //=; congr (_ ><| _).
by rewrite quotientK ?(subset_trans skerH) // -group_modr //= setIC tiKH mul1g.
Qed. | Lemma | 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 | sdprod_cfker | |
cforder_sdprodphi : #[cfSdprod phi]%CF = #[phi]%CF.
Proof. exact: cforder_inj_rmorph cfSdprod_inj. Qed. | Lemma | 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 | cforder_sdprod | |
reindex_dprodR idx (op : Monoid.com_law idx) (F : gT -> R) :
\big[op/idx]_(g in G) F g =
\big[op/idx]_(k in K) \big[op/idx]_(h in H) F (k * h)%g.
Proof.
have /mulgmP/misomP[fM /isomP[injf im_f]] := KxH.
rewrite pair_big_dep -im_f morphimEdom big_imset; last exact/injmP.
by apply: eq_big => [][x y]; rewrite ?inE.
Qed. | Lemma | 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 | reindex_dprod | |
cfDprodr:= cfSdprod (dprodWsd KxH). | 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 | cfDprodr | |
cfDprodl:= cfSdprod (dprodWsdC KxH). | 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 | cfDprodl | |
cfDprodphi psi := cfDprodl phi * cfDprodr psi.
HB.instance Definition _ := GRing.LRMorphism.on cfDprodl.
HB.instance Definition _ := GRing.LRMorphism.on cfDprodr. | 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 | cfDprod | |
cfDprodl1phi : cfDprodl phi 1%g = phi 1%g. Proof. exact: cfSdprod1. Qed. | Lemma | 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 | cfDprodl1 | |
cfDprodr1psi : cfDprodr psi 1%g = psi 1%g. Proof. exact: cfSdprod1. Qed. | Lemma | 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 | cfDprodr1 | |
cfDprod1phi psi : cfDprod phi psi 1%g = phi 1%g * psi 1%g.
Proof. by rewrite cfunE /= !cfSdprod1. Qed. | Lemma | 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 | cfDprod1 | |
cfDprodl_eq1phi : (cfDprodl phi == 1) = (phi == 1).
Proof. exact: cfSdprod_eq1. Qed. | Lemma | 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 | cfDprodl_eq1 | |
cfDprodr_eq1psi : (cfDprodr psi == 1) = (psi == 1).
Proof. exact: cfSdprod_eq1. Qed. | Lemma | 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 | cfDprodr_eq1 | |
cfDprod_cfun1rphi : cfDprod phi 1 = cfDprodl phi.
Proof. by rewrite /cfDprod rmorph1 mulr1. Qed. | Lemma | 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 | cfDprod_cfun1r | |
cfDprod_cfun1lpsi : cfDprod 1 psi = cfDprodr psi.
Proof. by rewrite /cfDprod rmorph1 mul1r. Qed. | Lemma | 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 | cfDprod_cfun1l | |
cfDprod_cfun1: cfDprod 1 1 = 1.
Proof. by rewrite cfDprod_cfun1l rmorph1. Qed. | Lemma | 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 | cfDprod_cfun1 | |
cfDprod_splitphi psi : cfDprod phi psi = cfDprod phi 1 * cfDprod 1 psi.
Proof. by rewrite cfDprod_cfun1l cfDprod_cfun1r. Qed.
Let nsKG : K <| G. Proof. by have [] := dprod_normal2 KxH. Qed.
Let nsHG : H <| G. Proof. by have [] := dprod_normal2 KxH. Qed.
Let cKH : H \subset 'C(K). Proof. by have [] := dprodP KxH. Qed.
Let sKG := normal_sub nsKG.
Let sHG := normal_sub nsHG. | Lemma | 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 | cfDprod_split | |
cfDprodlK: cancel cfDprodl 'Res[K]. Proof. exact: cfSdprodK. Qed. | Lemma | 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 | cfDprodlK | |
cfDprodrK: cancel cfDprodr 'Res[H]. Proof. exact: cfSdprodK. Qed. | Lemma | 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 | cfDprodrK | |
cfker_dprodlphi : cfker phi \x H = cfker (cfDprodl phi).
Proof.
by rewrite dprodC -sdprod_cfker dprodEsd // centsC (centsS (cfker_sub _)).
Qed. | Lemma | 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 | cfker_dprodl | |
cfker_dprodrpsi : K \x cfker psi = cfker (cfDprodr psi).
Proof. by rewrite -sdprod_cfker dprodEsd // (subset_trans (cfker_sub _)). Qed. | Lemma | 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 | cfker_dprodr | |
cfDprodElphi : {in K & H, forall k h, cfDprodl phi (k * h)%g = phi k}.
Proof. by move=> k h Kk Hh /=; rewrite -(centsP cKH) // cfSdprodE. Qed. | Lemma | 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 | cfDprodEl | |
cfDprodErpsi : {in K & H, forall k h, cfDprodr psi (k * h)%g = psi h}.
Proof. exact: cfSdprodE. Qed. | Lemma | 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 | cfDprodEr | |
cfDprodEphi psi :
{in K & H, forall h k, cfDprod phi psi (h * k)%g = phi h * psi k}.
Proof. by move=> k h Kk Hh /=; rewrite cfunE cfDprodEl ?cfDprodEr. Qed. | Lemma | 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 | cfDprodE | |
cfDprod_Reslphi psi : 'Res[K] (cfDprod phi psi) = psi 1%g *: phi.
Proof.
by apply/cfun_inP=> x Kx; rewrite cfunE cfResE // -{1}[x]mulg1 mulrC cfDprodE.
Qed. | Lemma | 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 | cfDprod_Resl | |
cfDprod_Resrphi psi : 'Res[H] (cfDprod phi psi) = phi 1%g *: psi.
Proof.
by apply/cfun_inP=> y Hy; rewrite cfunE cfResE // -{1}[y]mul1g cfDprodE.
Qed. | Lemma | 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 | cfDprod_Resr | |
cfDprodKl(psi : 'CF(H)) : psi 1%g = 1 -> cancel (cfDprod^~ psi) 'Res.
Proof. by move=> psi1 phi; rewrite cfDprod_Resl psi1 scale1r. Qed. | Lemma | 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 | cfDprodKl | |
cfDprodKr(phi : 'CF(K)) : phi 1%g = 1 -> cancel (cfDprod phi) 'Res.
Proof. by move=> phi1 psi; rewrite cfDprod_Resr phi1 scale1r. Qed. | Lemma | 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 | cfDprodKr | |
cfker_dprodphi psi :
cfker phi <*> cfker psi \subset cfker (cfDprod phi psi).
Proof.
rewrite -genM_join gen_subG; apply/subsetP=> _ /mulsgP[x y kKx kHy ->] /=.
have [[Kx _] [Hy _]] := (setIdP kKx, setIdP kHy).
have Gxy: (x * y)%g \in G by rewrite -(dprodW KxH) mem_mulg.
rewrite inE Gxy; apply/forallP=> g.
have [Gg | G'g] := boolP (g \in G); last by rewrite !cfun0 1?groupMl.
have{g Gg} [k [h [Kk Hh -> _]]] := mem_dprod KxH Gg.
rewrite mulgA -(mulgA x) (centsP cKH y) // mulgA -mulgA !cfDprodE ?groupM //.
by rewrite !cfkerMl.
Qed. | Lemma | 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 | cfker_dprod | |
cfdot_dprodphi1 phi2 psi1 psi2 :
'[cfDprod phi1 psi1, cfDprod phi2 psi2] = '[phi1, phi2] * '[psi1, psi2].
Proof.
rewrite !cfdotE mulrCA -mulrA mulrCA mulrA -invfM -natrM (dprod_card KxH).
congr (_ * _); rewrite big_distrl reindex_dprod /=; apply: eq_bigr => k Kk.
rewrite big_distrr; apply: eq_bigr => h Hh /=.
by rewrite mulrCA -mulrA -rmorphM mulrCA mulrA !cfDprodE.
Qed. | Lemma | 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 | cfdot_dprod | |
cfDprodl_iso: isometry cfDprodl.
Proof.
by move=> phi1 phi2; rewrite -!cfDprod_cfun1r cfdot_dprod cfnorm1 mulr1.
Qed. | Lemma | 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 | cfDprodl_iso | |
cfDprodr_iso: isometry cfDprodr.
Proof.
by move=> psi1 psi2; rewrite -!cfDprod_cfun1l cfdot_dprod cfnorm1 mul1r.
Qed. | Lemma | 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 | cfDprodr_iso | |
cforder_dprodlphi : #[cfDprodl phi]%CF = #[phi]%CF.
Proof. exact: cforder_sdprod. Qed. | Lemma | 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 | cforder_dprodl | |
cforder_dprodrpsi : #[cfDprodr psi]%CF = #[psi]%CF.
Proof. exact: cforder_sdprod. Qed. | Lemma | 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 | cforder_dprodr | |
cfDprodC(gT : finGroupType) (G K H : {group gT})
(KxH : K \x H = G) (HxK : H \x K = G) chi psi :
cfDprod KxH chi psi = cfDprod HxK psi chi.
Proof.
rewrite /cfDprod mulrC.
by congr (_ * _); congr (cfSdprod _ _); apply: eq_irrelevance.
Qed. | Lemma | 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 | cfDprodC | |
cfBigdprodii := cfDprodl (cfBigdprodi_subproof i) \o 'Res[_, A i].
HB.instance Definition _ i := GRing.LRMorphism.on (@cfBigdprodi i). | 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 | cfBigdprodi | |
cfBigdprodi1i (phi : 'CF(A i)) : cfBigdprodi phi 1%g = phi 1%g.
Proof. by rewrite cfDprodl1 cfRes1. Qed. | Lemma | 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 | cfBigdprodi1 | |
cfBigdprodi_eq1i (phi : 'CF(A i)) :
P i -> (cfBigdprodi phi == 1) = (phi == 1).
Proof. by move=> Pi; rewrite cfSdprod_eq1 Pi cfRes_id. Qed. | Lemma | 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 | cfBigdprodi_eq1 | |
cfBigdprodiKi : P i -> cancel (@cfBigdprodi i) 'Res[A i].
Proof.
move=> Pi phi; have:= cfDprodlK (cfBigdprodi_subproof i) ('Res phi).
by rewrite -[cfDprodl _ _]/(cfBigdprodi phi) Pi cfRes_id.
Qed. | Lemma | 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 | cfBigdprodiK | |
cfBigdprodi_inji : P i -> injective (@cfBigdprodi i).
Proof. by move/cfBigdprodiK; apply: can_inj. Qed. | Lemma | 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 | cfBigdprodi_inj | |
cfBigdprodEii (phi : 'CF(A i)) x :
P i -> (forall j, P j -> x j \in A j) ->
cfBigdprodi phi (\prod_(j | P j) x j)%g = phi (x i).
Proof.
have [r big_r [Ur mem_r] _] := big_enumP P => Pi AxP.
have:= bigdprodWcp defG; rewrite -!big_r => defGr.
have{AxP} [r_i Axr]: i \in r /\ {in r, forall j, x j \in A j}.
by split=> [|j]; rewrite mem_r // => /AxP.
rewrite (perm_bigcprod defGr Axr (perm_to_rem r_i)) big_cons.
rewrite cfDprodEl ?Pi ?cfRes_id ?Axr // big_seq group_prod // => j.
rewrite mem_rem_uniq // => /andP[i'j /= r_j].
by apply/mem_gen/bigcupP; exists j; [rewrite -mem_r r_j | apply: Axr].
Qed. | Lemma | 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 | cfBigdprodEi | |
cfBigdprodi_isoi : P i -> isometry (@cfBigdprodi i).
Proof. by move=> Pi phi psi; rewrite cfDprodl_iso Pi !cfRes_id. Qed. | Lemma | 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 | cfBigdprodi_iso | |
cfBigdprod(phi : forall i, 'CF(A i)) :=
\prod_(i | P i) cfBigdprodi (phi i). | 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 | cfBigdprod | |
cfBigdprodEphi x :
(forall i, P i -> x i \in A i) ->
cfBigdprod phi (\prod_(i | P i) x i)%g = \prod_(i | P i) phi i (x i).
Proof.
move=> Ax; rewrite prod_cfunE; last by rewrite -(bigdprodW defG) mem_prodg.
by apply: eq_bigr => i Pi; rewrite cfBigdprodEi.
Qed. | Lemma | 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 | cfBigdprodE | |
cfBigdprod1phi : cfBigdprod phi 1%g = \prod_(i | P i) phi i 1%g.
Proof. by rewrite prod_cfunE //; apply/eq_bigr=> i _; apply: cfBigdprodi1. Qed. | Lemma | 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 | cfBigdprod1 | |
cfBigdprodKphi (Phi := cfBigdprod phi) i (a := phi i 1%g / Phi 1%g) :
Phi 1%g != 0 -> P i -> a != 0 /\ a *: 'Res[A i] Phi = phi i.
Proof.
move=> nzPhi Pi; split.
rewrite mulf_neq0 ?invr_eq0 // (contraNneq _ nzPhi) // => phi_i0.
by rewrite cfBigdprod1 (bigD1 i) //= phi_i0 mul0r.
apply/cfun_inP=> x Aix; rewrite cfunE cfResE ?sAG // mulrAC.
have {1}->: x = (\prod_(j | P j) (if j == i then x else 1))%g.
rewrite -big_mkcondr (big_pred1 i) ?eqxx // => j /=.
by apply: andb_idl => /eqP->.
rewrite cfBigdprodE => [|j _]; last by case: eqP => // ->.
apply: canLR (mulfK nzPhi) _; rewrite cfBigdprod1 !(bigD1 i Pi) /= eqxx.
by rewrite mulrCA !mulrA; congr (_ * _); apply: eq_bigr => j /andP[_ /negPf->].
Qed. | Lemma | 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 | cfBigdprodK | |
cfdot_bigdprodphi psi :
'[cfBigdprod phi, cfBigdprod psi] = \prod_(i | P i) '[phi i, psi i].
Proof.
apply: canLR (mulKf (neq0CG G)) _; rewrite -(bigdprod_card defG).
rewrite (big_morph _ (@natrM _) (erefl _)) -big_split /=.
rewrite (eq_bigr _ (fun i _ => mulVKf (neq0CG _) _)) (big_distr_big_dep 1%g) /=.
set F := pfamily _ _ _; pose h (f : {ffun I -> gT}) := (\prod_(i | P i) f i)%g.
pose is_hK x f := forall f1, (f1 \in F) && (h f1 == x) = (f == f1).
have /fin_all_exists[h1 Dh1] x: exists f, x \in G -> is_hK x f.
case Gx: (x \in G); last by exists [ffun _ => x].
have [f [Af fK Uf]] := mem_bigdprod defG Gx.
exists [ffun i => if P i then f i else 1%g] => _ f1.
apply/andP/eqP=> [[/pfamilyP[Pf1 Af1] /eqP Dx] | <-].
by apply/ffunP=> i; rewrite ffunE; case: ifPn => [/Uf-> | /(supportP Pf1)].
split; last by rewrite fK; apply/eqP/eq_bigr=> i Pi; rewrite ffunE Pi.
by apply/familyP=> i; rewrite ffunE !unfold_in; case: ifP => //= /Af.
rewrite (reindex_onto h h1) /= => [|x /Dh1/(_ (h1 x))]; last first.
by rewrite eqxx => /andP[_ /eqP].
apply/eq_big => [f | f /andP[/Dh1<- /andP[/pfamilyP[_ Af] _]]]; last first.
by rewrite !cfBigdprodE // rmorph_prod -big_split /=.
apply/idP/idP=> [/andP[/Dh1<-] | Ff]; first by rewrite eqxx andbT.
have /pfamilyP[_ Af] := Ff; suffices Ghf: h f \in G by rewrite -Dh1 ?Ghf ?Ff /=.
by apply/group_prod=> i Pi; rewrite (subsetP (sAG Pi)) ?Af.
Qed. | Lemma | 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 | cfdot_bigdprod | |
cfMorph_isoaT rT (G D : {group aT}) (f : {morphism D >-> rT}) :
G \subset D -> isometry (cfMorph : 'CF(f @* G) -> 'CF(G)).
Proof.
move=> sGD phi psi; rewrite !cfdotE card_morphim (setIidPr sGD).
rewrite -(LagrangeI G ('ker f)) /= mulnC natrM invfM -mulrA.
congr (_ * _); apply: (canLR (mulKf (neq0CG _))).
rewrite mulr_sumr (partition_big_imset f) /= -morphimEsub //.
apply: eq_bigr => _ /morphimP[x Dx Gx ->].
rewrite -(card_rcoset _ x) mulr_natl -sumr_const.
apply/eq_big => [y | y /andP[Gy /eqP <-]]; last by rewrite !cfMorphE.
rewrite mem_rcoset inE groupMr ?groupV // -mem_rcoset.
by apply: andb_id2l => /(subsetP sGD) Dy; apply: sameP eqP (rcoset_kerP f _ _).
Qed. | Lemma | 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 | cfMorph_iso | |
cfIsom_isorT G (R : {group rT}) (f : {morphism G >-> rT}) :
forall isoG : isom G R f, isometry (cfIsom isoG).
Proof.
move=> isoG phi psi; rewrite unlock cfMorph_iso //; set G1 := _ @* R.
by rewrite -(isom_im (isom_sym isoG)) -/G1 in phi psi *; rewrite !cfRes_id.
Qed. | Lemma | 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 | cfIsom_iso | |
cfMod_isoH G : H <| G -> isometry (@cfMod _ G H).
Proof. by case/andP=> _; apply: cfMorph_iso. Qed. | Lemma | 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 | cfMod_iso | |
cfQuo_isoH G :
H <| G -> {in [pred phi | H \subset cfker phi] &, isometry (@cfQuo _ G H)}.
Proof.
by move=> nsHG phi psi sHkphi sHkpsi; rewrite -(cfMod_iso nsHG) !cfQuoK.
Qed. | Lemma | 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 | cfQuo_iso | |
cfnorm_quoH G phi :
H <| G -> H \subset cfker phi -> '[phi / H] = '[phi]_G.
Proof. by move=> nsHG sHker; apply: cfQuo_iso. Qed. | Lemma | 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 | cfnorm_quo | |
cfSdprod_isoK H G (defG : K ><| H = G) : isometry (cfSdprod defG).
Proof.
move=> phi psi; have [/andP[_ nKG] _ _ _ _] := sdprod_context defG.
by rewrite [cfSdprod _]locked_withE cfMorph_iso ?cfIsom_iso.
Qed. | Lemma | 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 | cfSdprod_iso | |
ffun_cfInd(phi : 'CF(A)) :=
[ffun x => if H \subset G then #|A|%:R^-1 * (\sum_(y in G) phi (x ^ y))
else #|G|%:R * '[phi, 1] *+ (x == 1%g)].
Fact cfInd_subproof phi : is_class_fun G (ffun_cfInd phi).
Proof.
apply: intro_class_fun => [x y Gx Gy | x H'x]; last first.
case: subsetP => [sHG | _]; last by rewrite (negPf (group1_contra H'x)).
rewrite big1 ?mulr0 // => y Gy; rewrite cfun0gen ?(contra _ H'x) //= => /sHG.
by rewrite memJ_norm ?(subsetP (normG _)).
rewrite conjg_eq1 (reindex_inj (mulgI y^-1)%g); congr (if _ then _ * _ else _).
by apply: eq_big => [z | z Gz]; rewrite ?groupMl ?groupV // -conjgM mulKVg.
Qed. | 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 | ffun_cfInd | |
cfIndphi := Cfun 1 (cfInd_subproof phi). | 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 | cfInd | |
cfInd_is_linear: linear cfInd.
Proof.
move=> c phi psi; apply/cfunP=> x; rewrite !cfunElock; case: ifP => _.
rewrite mulrCA -mulrDr [c * _]mulr_sumr -big_split /=.
by congr (_ * _); apply: eq_bigr => y _; rewrite !cfunE.
rewrite mulrnAr -mulrnDl !(mulrCA c) -!mulrDr [c * _]mulr_sumr -big_split /=.
by congr (_ * (_ * _) *+ _); apply: eq_bigr => y; rewrite !cfunE mulrA mulrDl.
Qed.
HB.instance Definition _ := GRing.isSemilinear.Build algC _ _ _ cfInd
(GRing.semilinear_linear cfInd_is_linear). | Lemma | 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 | cfInd_is_linear | |
cfIndE(G H : {group gT}) phi x :
H \subset G -> 'Ind[G, H] phi x = #|H|%:R^-1 * (\sum_(y in G) phi (x ^ y)).
Proof. by rewrite cfunElock !genGid => ->. Qed.
Variables G K H : {group gT}.
Implicit Types (phi : 'CF(H)) (psi : 'CF(G)). | Lemma | 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 | cfIndE | |
cfIndEoutphi :
~~ (H \subset G) -> 'Ind[G] phi = (#|G|%:R * '[phi, 1]) *: '1_1%G.
Proof.
move/negPf=> not_sHG; apply/cfunP=> x; rewrite cfunE cfuniE ?normal1 // inE.
by rewrite mulr_natr cfunElock !genGid not_sHG.
Qed. | Lemma | 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 | cfIndEout | |
cfIndEsdprod(phi : 'CF(K)) x :
K ><| H = G -> 'Ind[G] phi x = \sum_(w in H) phi (x ^ w)%g.
Proof.
move=> defG; have [/andP[sKG _] _ mulKH nKH _] := sdprod_context defG.
rewrite cfIndE //; apply: canLR (mulKf (neq0CG _)) _; rewrite -mulKH mulr_sumr.
rewrite (set_partition_big _ (rcosets_partition_mul H K)) ?big_imset /=.
apply: eq_bigr => y Hy; rewrite rcosetE norm_rlcoset ?(subsetP nKH) //.
rewrite -lcosetE mulr_natl big_imset /=; last exact: in2W (mulgI _).
by rewrite -sumr_const; apply: eq_bigr => z Kz; rewrite conjgM cfunJ.
have [{}nKH /isomP[injf _]] := sdprod_isom defG.
apply: can_in_inj (fun Ky => invm injf (coset K (repr Ky))) _ => y Hy.
by rewrite rcosetE -val_coset ?(subsetP nKH) // coset_reprK invmE.
Qed. | Lemma | 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 | cfIndEsdprod | |
cfInd_onA phi :
H \subset G -> phi \in 'CF(H, A) -> 'Ind[G] phi \in 'CF(G, class_support A G).
Proof.
move=> sHG Af; apply/cfun_onP=> g AG'g; rewrite cfIndE ?big1 ?mulr0 // => h Gh.
apply: (cfun_on0 Af); apply: contra AG'g => Agh.
by rewrite -[g](conjgK h) memJ_class_support // groupV.
Qed. | Lemma | 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 | cfInd_on | |
cfInd_idphi : 'Ind[H] phi = phi.
Proof.
apply/cfun_inP=> x Hx; rewrite cfIndE // (eq_bigr _ (cfunJ phi x)) sumr_const.
by rewrite -[phi x *+ _]mulr_natl mulKf ?neq0CG.
Qed. | Lemma | 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 | cfInd_id | |
cfInd_normalphi : H <| G -> 'Ind[G] phi \in 'CF(G, H).
Proof.
case/andP=> sHG nHG; apply: (cfun_onS (class_support_sub_norm (subxx _) nHG)).
by rewrite cfInd_on ?cfun_onG.
Qed. | Lemma | 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 | cfInd_normal | |
cfInd1phi : H \subset G -> 'Ind[G] phi 1%g = #|G : H|%:R * phi 1%g.
Proof.
move=> sHG; rewrite cfIndE // natf_indexg // -mulrA mulrCA; congr (_ * _).
by rewrite mulr_natl -sumr_const; apply: eq_bigr => x; rewrite conj1g.
Qed. | Lemma | 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 | cfInd1 | |
cfInd_cfun1: H <| G -> 'Ind[G, H] 1 = #|G : H|%:R *: '1_H.
Proof.
move=> nsHG; have [sHG nHG] := andP nsHG; rewrite natf_indexg // mulrC.
apply/cfunP=> x; rewrite cfIndE ?cfunE ?cfuniE // -mulrA; congr (_ * _).
rewrite mulr_natl -sumr_const; apply: eq_bigr => y Gy.
by rewrite cfun1E -{1}(normsP nHG y Gy) memJ_conjg.
Qed. | Lemma | 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 | cfInd_cfun1 | |
cfnorm_Ind_cfun1: H <| G -> '['Ind[G, H] 1] = #|G : H|%:R.
Proof.
move=> nsHG; rewrite cfInd_cfun1 // cfnormZ normr_nat cfdot_cfuni // setIid.
by rewrite expr2 {2}natf_indexg ?normal_sub // !mulrA divfK ?mulfK ?neq0CG.
Qed. | Lemma | 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 | cfnorm_Ind_cfun1 | |
cfIndIndphi :
K \subset G -> H \subset K -> 'Ind[G] ('Ind[K] phi) = 'Ind[G] phi.
Proof.
move=> sKG sHK; apply/cfun_inP=> x Gx; rewrite !cfIndE ?(subset_trans sHK) //.
apply: canLR (mulKf (neq0CG K)) _; rewrite mulr_sumr mulr_natl.
transitivity (\sum_(y in G) \sum_(z in K) #|H|%:R^-1 * phi ((x ^ y) ^ z)).
by apply: eq_bigr => y Gy; rewrite cfIndE // -mulr_sumr.
symmetry; rewrite exchange_big /= -sumr_const; apply: eq_bigr => z Kz.
rewrite (reindex_inj (mulIg z)).
by apply: eq_big => [y | y _]; rewrite ?conjgM // groupMr // (subsetP sKG).
Qed. | Lemma | 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 | cfIndInd | |
Frobenius_reciprocityphi psi : '[phi, 'Res[H] psi] = '['Ind[G] phi, psi].
Proof.
have [sHG | not_sHG] := boolP (H \subset G); last first.
rewrite cfResEout // cfIndEout // cfdotZr cfdotZl mulrAC; congr (_ * _).
rewrite (cfdotEl _ (cfuni_on _ _)) mulVKf ?neq0CG // big_set1.
by rewrite cfuniE ?normal1 ?set11 ?mul1r.
transitivity (#|H|%:R^-1 * \sum_(x in G) phi x * (psi x)^* ).
rewrite (big_setID H) /= (setIidPr sHG) addrC big1 ?add0r; last first.
by move=> x /setDP[_ /cfun0->]; rewrite mul0r.
by congr (_ * _); apply: eq_bigr => x Hx; rewrite cfResE.
set h' := _^-1; apply: canRL (mulKf (neq0CG G)) _.
transitivity (h' * \sum_(y in G) \sum_(x in G) phi (x ^ y) * (psi (x ^ y))^* ).
rewrite mulrCA mulr_natl -sumr_const; congr (_ * _); apply: eq_bigr => y Gy.
by rewrite (reindex_acts 'J _ Gy) ?astabsJ ?normG.
rewrite exchange_big mulr_sumr; apply: eq_bigr => x _; rewrite cfIndE //=.
by rewrite -mulrA mulr_suml; congr (_ * _); apply: eq_bigr => y /(cfunJ psi)->.
Qed. | Lemma | 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 | Frobenius_reciprocity | |
cfdot_Res_r:= Frobenius_reciprocity. | 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 | cfdot_Res_r | |
cfdot_Res_lpsi phi : '['Res[H] psi, phi] = '[psi, 'Ind[G] phi].
Proof. by rewrite cfdotC cfdot_Res_r -cfdotC. Qed. | Lemma | 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 | cfdot_Res_l | |
cfIndMphi psi: H \subset G ->
'Ind[G] (phi * ('Res[H] psi)) = 'Ind[G] phi * psi.
Proof.
move=> HsG; apply/cfun_inP=> x Gx; rewrite !cfIndE // !cfunE !cfIndE // -mulrA.
congr (_ * _); rewrite mulr_suml; apply: eq_bigr=> i iG; rewrite !cfunE.
case: (boolP (x ^ i \in H)) => xJi; last by rewrite cfun0gen ?mul0r ?genGid.
by rewrite !cfResE //; congr (_ * _); rewrite cfunJgen ?genGid.
Qed. | Lemma | 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 | cfIndM | |
cfIndMorph(f : {morphism D >-> rT}) (phi : 'CF(f @* H)) :
'ker f \subset H -> H \subset G -> G \subset D ->
'Ind[G] (cfMorph phi) = cfMorph ('Ind[f @* G] phi).
Proof.
move=> sKH sHG sGD; have [sHD inD] := (subset_trans sHG sGD, subsetP sGD).
apply/cfun_inP=> /= x Gx; have [Dx sKG] := (inD x Gx, subset_trans sKH sHG).
rewrite cfMorphE ?cfIndE ?morphimS // (partition_big_imset f) -morphimEsub //=.
rewrite card_morphim (setIidPr sHD) natf_indexg // invfM invrK -mulrA.
congr (_ * _); rewrite mulr_sumr; apply: eq_bigr => _ /morphimP[y Dy Gy ->].
rewrite -(card_rcoset _ y) mulr_natl -sumr_const.
apply: eq_big => [z | z /andP[Gz /eqP <-]].
have [Gz | G'z] := boolP (z \in G).
by rewrite (sameP eqP (rcoset_kerP _ _ _)) ?inD.
by case: rcosetP G'z => // [[t Kt ->]]; rewrite groupM // (subsetP sKG).
have [Dz Dxz] := (inD z Gz, inD (x ^ z) (groupJ Gx Gz)); rewrite -morphJ //.
have [Hxz | notHxz] := boolP (x ^ z \in H); first by rewrite cfMorphE.
by rewrite !cfun0 // -sub1set -morphim_set1 // morphimSGK ?sub1set.
Qed.
Variables (g : {morphism G >-> rT}) (h : {morphism H >-> rT}).
Hypotheses (isoG : isom G R g) (isoH : isom H S h) (eq_hg : {in H, h =1 g}).
Hypothesis sHG : H \subset G. | Lemma | 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 | cfIndMorph | |
cfResIsomphi : 'Res[S] (cfIsom isoG phi) = cfIsom isoH ('Res[H] phi).
Proof.
have [[injg defR] [injh defS]] := (isomP isoG, isomP isoH).
rewrite !morphimEdom in defS defR; apply/cfun_inP=> s.
rewrite -{1}defS => /imsetP[x Hx ->] {s}; have Gx := subsetP sHG x Hx.
rewrite {1}eq_hg ?(cfResE, cfIsomE) // -defS -?eq_hg ?imset_f // -defR.
by rewrite (eq_in_imset eq_hg) imsetS.
Qed. | Lemma | 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 | cfResIsom | |
cfIndIsomphi : 'Ind[R] (cfIsom isoH phi) = cfIsom isoG ('Ind[G] phi).
Proof.
have [[injg defR] [_ defS]] := (isomP isoG, isomP isoH).
rewrite morphimEdom (eq_in_imset eq_hg) -morphimEsub // in defS.
apply/cfun_inP=> s; rewrite -{1}defR => /morphimP[x _ Gx ->]{s}.
rewrite cfIsomE ?cfIndE // -defR -{1}defS ?morphimS ?card_injm // morphimEdom.
congr (_ * _); rewrite big_imset //=; last exact/injmP.
apply: eq_bigr => y Gy; rewrite -morphJ //.
have [Hxy | H'xy] := boolP (x ^ y \in H); first by rewrite -eq_hg ?cfIsomE.
by rewrite !cfun0 -?defS // -sub1set -morphim_set1 ?injmSK ?sub1set // groupJ.
Qed. | Lemma | 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 | cfIndIsom | |
cfAutZ_natn phi : (n%:R *: phi)^u = n%:R *: phi^u.
Proof. exact: raddfZnat. Qed. | Lemma | 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 | cfAutZ_nat | |
cfAutZ_Cnatz phi : z \in Num.nat -> (z *: phi)^u = z *: phi^u.
Proof. exact: raddfZ_nat. Qed. | Lemma | 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 | cfAutZ_Cnat | |
cfAutZ_Cintz phi : z \in Num.int -> (z *: phi)^u = z *: phi^u.
Proof. exact: raddfZ_int. Qed. | Lemma | 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 | cfAutZ_Cint | |
cfAutK: cancel (@cfAut gT G u) (cfAut (algC_invaut u)).
Proof. by move=> phi; apply/cfunP=> x; rewrite !cfunE /= algC_autK. Qed. | Lemma | 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 | cfAutK | |
cfAutVK: cancel (cfAut (algC_invaut u)) (@cfAut gT G u).
Proof. by move=> phi; apply/cfunP=> x; rewrite !cfunE /= algC_invautK. Qed. | Lemma | 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 | cfAutVK | |
cfAut_inj: injective (@cfAut gT G u).
Proof. exact: can_inj cfAutK. Qed. | Lemma | 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 | cfAut_inj | |
cfAut_eq1phi : (cfAut u phi == 1) = (phi == 1).
Proof. by rewrite rmorph_eq1 //; apply: cfAut_inj. Qed. | Lemma | 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 | cfAut_eq1 | |
support_cfAutphi : support phi^u =i support phi.
Proof. by move=> x; rewrite !inE cfunE fmorph_eq0. Qed. | Lemma | 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 | support_cfAut | |
map_cfAut_freeS : cfAut_closed u S -> free S -> free (map (cfAut u) S).
Proof.
set Su := map _ S => sSuS freeS; have uniqS := free_uniq freeS.
have uniqSu: uniq Su by rewrite (map_inj_uniq cfAut_inj).
have{} sSuS: {subset Su <= S} by move=> _ /mapP[phi Sphi ->]; apply: sSuS.
have [|_ eqSuS] := uniq_min_size uniqSu sSuS; first by rewrite size_map.
by rewrite (perm_free (uniq_perm uniqSu uniqS eqSuS)).
Qed. | Lemma | 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 | map_cfAut_free | |
cfAut_onA phi : (phi^u \in 'CF(G, A)) = (phi \in 'CF(G, A)).
Proof. by rewrite !cfun_onE (eq_subset (support_cfAut phi)). Qed. | Lemma | 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 | cfAut_on | |
cfker_autphi : cfker phi^u = cfker phi.
Proof.
apply/setP=> x /[!inE]; apply: andb_id2l => Gx.
by apply/forallP/forallP=> Kx y;
have:= Kx y; rewrite !cfunE (inj_eq (fmorph_inj u)).
Qed. | Lemma | 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 | cfker_aut | |
cfAut_cfuniA : ('1_A)^u = '1_A :> 'CF(G).
Proof. by apply/cfunP=> x; rewrite !cfunElock rmorph_nat. Qed. | Lemma | 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 | cfAut_cfuni | |
cforder_autphi : #[phi^u]%CF = #[phi]%CF.
Proof. exact: cforder_inj_rmorph cfAut_inj. Qed. | Lemma | 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 | cforder_aut | |
cfAutResphi : ('Res[H] phi)^u = 'Res phi^u.
Proof. by apply/cfunP=> x; rewrite !cfunElock rmorphMn. Qed. | Lemma | 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 | cfAutRes | |
cfAutMorph(psi : 'CF(f @* H)) : (cfMorph psi)^u = cfMorph psi^u.
Proof. by apply/cfun_inP=> x Hx; rewrite !cfunElock Hx. Qed. | Lemma | 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 | cfAutMorph | |
cfAutIsom(isoGR : isom G R f) phi :
(cfIsom isoGR phi)^u = cfIsom isoGR phi^u.
Proof.
apply/cfun_inP=> y; have [_ {1}<-] := isomP isoGR => /morphimP[x _ Gx ->{y}].
by rewrite !(cfunE, cfIsomE).
Qed. | Lemma | 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 | cfAutIsom | |
cfAutQuophi : (phi / H)^u = (phi^u / H)%CF.
Proof. by apply/cfunP=> Hx; rewrite !cfunElock cfker_aut rmorphMn. Qed. | Lemma | 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 | cfAutQuo | |
cfAutMod(psi : 'CF(G / H)) : (psi %% H)^u = (psi^u %% H)%CF.
Proof. by apply/cfunP=> x; rewrite !cfunElock rmorphMn. Qed. | Lemma | 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 | cfAutMod | |
cfAutInd(psi : 'CF(H)) : ('Ind[G] psi)^u = 'Ind psi^u.
Proof.
have [sHG | not_sHG] := boolP (H \subset G).
apply/cfunP=> x; rewrite !(cfunE, cfIndE) // rmorphM /= fmorphV rmorph_nat.
by congr (_ * _); rewrite rmorph_sum; apply: eq_bigr => y; rewrite !cfunE.
rewrite !cfIndEout // linearZ /= cfAut_cfuni rmorphM rmorph_nat /=.
rewrite -cfdot_cfAut ?rmorph1 // => _ /imageP[x Hx ->].
by rewrite cfun1E Hx !rmorph1.
Qed.
Hypothesis KxH : K \x H = G. | Lemma | 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 | cfAutInd | |
cfAutDprodl(phi : 'CF(K)) : (cfDprodl KxH phi)^u = cfDprodl KxH phi^u.
Proof.
apply/cfun_inP=> _ /(mem_dprod KxH)[x [y [Kx Hy -> _]]].
by rewrite !(cfunE, cfDprodEl).
Qed. | Lemma | 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 | cfAutDprodl | |
cfAutDprodr(psi : 'CF(H)) : (cfDprodr KxH psi)^u = cfDprodr KxH psi^u.
Proof.
apply/cfun_inP=> _ /(mem_dprod KxH)[x [y [Kx Hy -> _]]].
by rewrite !(cfunE, cfDprodEr).
Qed. | Lemma | 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 | cfAutDprodr | |
cfAutDprod(phi : 'CF(K)) (psi : 'CF(H)) :
(cfDprod KxH phi psi)^u = cfDprod KxH phi^u psi^u.
Proof. by rewrite rmorphM /= cfAutDprodl cfAutDprodr. Qed. | Lemma | 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 | cfAutDprod | |
conj_cfRes:= cfAutRes 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 | conj_cfRes | |
cfker_conjC:= cfker_aut 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 | cfker_conjC | |
conj_cfQuo:= cfAutQuo 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 | conj_cfQuo | |
conj_cfMod:= cfAutMod 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 | conj_cfMod | |
conj_cfInd:= cfAutInd 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 | conj_cfInd |
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