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 |
|---|---|---|---|---|---|---|
cfBigdprodi_lin_chari (phi : 'CF(A i)) :
phi \is a linear_char -> cfBigdprodi defG phi \is a linear_char.
Proof. by move=> Lphi; rewrite cfDprodl_lin_char ?cfRes_lin_char. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfBigdprodi_lin_char | |
cfBigdprodi_lin_charEi (phi : 'CF(A i)) :
P i -> (cfBigdprodi defG phi \is a linear_char) = (phi \is a linear_char).
Proof. by move=> Pi; rewrite qualifE/= cfBigdprodi_charE // cfBigdprodi1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfBigdprodi_lin_charE | |
cfBigdprod_lin_charphi :
(forall i, P i -> phi i \is a linear_char) ->
cfBigdprod defG phi \is a linear_char.
Proof.
by move=> Lphi; apply/rpred_prod=> i /Lphi; apply: cfBigdprodi_lin_char.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfBigdprod_lin_char | |
cfBigdprodi_irri chi :
P i -> (cfBigdprodi defG chi \in irr G) = (chi \in irr (A i)).
Proof. by move=> Pi; rewrite !irrEchar cfBigdprodi_charE ?cfBigdprodi_iso. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfBigdprodi_irr | |
cfBigdprod_irrchi :
(forall i, P i -> chi i \in irr (A i)) -> cfBigdprod defG chi \in irr G.
Proof.
move=> Nchi; rewrite irrEchar cfBigdprod_char => [|i /Nchi/irrWchar] //=.
by rewrite cfdot_bigdprod big1 // => i /Nchi/irrWnorm.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfBigdprod_irr | |
cfBigdprod_eq1phi :
(forall i, P i -> phi i \is a character) ->
(cfBigdprod defG phi == 1) = [forall (i | P i), phi i == 1].
Proof.
move=> Nphi; set Phi := cfBigdprod defG phi.
apply/eqP/eqfun_inP=> [Phi1 i Pi | phi1]; last first.
by apply: big1 => i /phi1->; rewrite rmorph1.
have Phi1_1: Phi 1%g = 1 by rewrite Phi1 cfun1E group1.
have nz_Phi1: Phi 1%g != 0 by rewrite Phi1_1 oner_eq0.
have [_ <-] := cfBigdprodK nz_Phi1 Pi.
rewrite Phi1_1 divr1 -/Phi Phi1 rmorph1.
rewrite prod_cfunE // in Phi1_1; have := natr_prod_eq1 _ Phi1_1 Pi.
rewrite -(cfRes1 (A i)) cfBigdprodiK // => ->; first by rewrite scale1r.
by move=> {i Pi} j /Nphi Nphi_j; rewrite Cnat_char1 ?cfBigdprodi_char.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfBigdprod_eq1 | |
cfBigdprod_Res_linchi :
chi \is a linear_char -> cfBigdprod defG (fun i => 'Res[A i] chi) = chi.
Proof.
move=> Lchi; apply/cfun_inP=> _ /(mem_bigdprod defG)[x [Ax -> _]].
rewrite (lin_char_prod Lchi) ?cfBigdprodE // => [|i Pi]; last first.
by rewrite (subsetP (sAG Pi)) ?Ax.
by apply/eq_bigr=> i Pi; rewrite cfResE ?sAG ?Ax.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfBigdprod_Res_lin | |
cfBigdprodKlinphi :
(forall i, P i -> phi i \is a linear_char) ->
forall i, P i -> 'Res (cfBigdprod defG phi) = phi i.
Proof.
move=> Lphi i Pi; have Lpsi := cfBigdprod_lin_char Lphi.
have [_ <-] := cfBigdprodK (lin_char_neq0 Lpsi (group1 G)) Pi.
by rewrite !lin_char1 ?Lphi // divr1 scale1r.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfBigdprodKlin | |
cfBigdprodKabelianIphi (phi := fun i => 'chi_(Iphi i)) :
abelian G -> forall i, P i -> 'Res (cfBigdprod defG phi) = 'chi_(Iphi i).
Proof.
move=> /(abelianS _) cGG.
by apply: cfBigdprodKlin => i /sAG/cGG/char_abelianP->.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfBigdprodKabelian | |
conjC_charAutu (chi : 'CF(G)) x :
chi \is a character -> (u (chi x))^* = u (chi x)^*.
Proof.
have [Gx | /cfun0->] := boolP (x \in G); last by rewrite !rmorph0.
case/char_reprP=> rG ->; have [e [_ [en1 _] [-> _] _]] := repr_rsim_diag rG Gx.
by rewrite !rmorph_sum; apply: eq_bigr => i _; apply: aut_unity_rootC (en1 i).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | conjC_charAut | |
conjC_irrAutu i x : (u ('chi[G]_i x))^* = u ('chi_i x)^*.
Proof. exact: conjC_charAut (irr_char i). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | conjC_irrAut | |
cfdot_aut_charu (phi chi : 'CF(G)) :
chi \is a character -> '[cfAut u phi, cfAut u chi] = u '[phi, chi].
Proof. by move/conjC_charAut=> Nchi; apply: cfdot_cfAut => _ /mapP[x _ ->]. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfdot_aut_char | |
cfdot_aut_irru phi i :
'[cfAut u phi, cfAut u 'chi[G]_i] = u '[phi, 'chi_i].
Proof. exact: cfdot_aut_char (irr_char i). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfdot_aut_irr | |
cfAut_irru chi : (cfAut u chi \in irr G) = (chi \in irr G).
Proof.
rewrite !irrEchar cfAut_char; apply/andb_id2l=> /cfdot_aut_char->.
exact: fmorph_eq1.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfAut_irr | |
cfConjC_irri : (('chi_i)^*)%CF \in irr G.
Proof. by rewrite cfAut_irr mem_irr. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfConjC_irr | |
irr_aut_closedu : cfAut_closed u (irr G).
Proof. by move=> chi; rewrite /= cfAut_irr. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr_aut_closed | |
aut_Iirru i := cfIirr (cfAut u 'chi[G]_i). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | aut_Iirr | |
aut_IirrEu i : 'chi_(aut_Iirr u i) = cfAut u 'chi_i.
Proof. by rewrite cfIirrE ?cfAut_irr ?mem_irr. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | aut_IirrE | |
conjC_Iirr:= aut_Iirr Num.conj. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | conjC_Iirr | |
conjC_IirrEi : 'chi[G]_(conjC_Iirr i) = ('chi_i)^*%CF.
Proof. exact: aut_IirrE. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | conjC_IirrE | |
conjC_IirrK: involutive conjC_Iirr.
Proof. by move=> i; apply: irr_inj; rewrite !conjC_IirrE cfConjCK. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | conjC_IirrK | |
aut_Iirr0u : aut_Iirr u 0 = 0 :> Iirr G.
Proof. by apply/irr_inj; rewrite aut_IirrE irr0 cfAut_cfun1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | aut_Iirr0 | |
conjC_Iirr0: conjC_Iirr 0 = 0 :> Iirr G.
Proof. exact: aut_Iirr0. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | conjC_Iirr0 | |
aut_Iirr_eq0u i : (aut_Iirr u i == 0) = (i == 0).
Proof. by rewrite -!irr_eq1 aut_IirrE cfAut_eq1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | aut_Iirr_eq0 | |
conjC_Iirr_eq0i : (conjC_Iirr i == 0 :> Iirr G) = (i == 0).
Proof. exact: aut_Iirr_eq0. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | conjC_Iirr_eq0 | |
aut_Iirr_inju : injective (aut_Iirr u).
Proof.
by move=> i j eq_ij; apply/irr_inj/(cfAut_inj u); rewrite -!aut_IirrE eq_ij.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | aut_Iirr_inj | |
cfQuo_charG H (chi : 'CF(G)) :
chi \is a character -> (chi / H)%CF \is a character.
Proof.
move=> Nchi; without loss kerH: / H \subset cfker chi.
move/contraNF=> IHchi; apply/wlog_neg=> N'chiH.
suffices ->: (chi / H)%CF = (chi 1%g)%:A.
by rewrite rpredZ_nat ?Cnat_char1 ?rpred1.
by apply/cfunP=> x; rewrite cfunE cfun1E mulr_natr cfunElock IHchi.
without loss nsHG: G chi Nchi kerH / H <| G.
move=> IHchi; have nsHN := normalSG (subset_trans kerH (cfker_sub chi)).
rewrite cfQuoInorm//; apply/cfRes_char/IHchi => //; first exact: cfRes_char.
by apply: sub_cfker_Res => //; apply: normal_sub.
have [rG Dchi] := char_reprP Nchi; rewrite Dchi cfker_repr in kerH.
apply/char_reprP; exists (Representation (quo_repr kerH (normal_norm nsHG))).
apply/cfun_inP=> _ /morphimP[x nHx Gx ->]; rewrite Dchi cfQuoE ?cfker_repr //=.
by rewrite !cfunE Gx quo_repr_coset ?mem_quotient.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfQuo_char | |
cfQuo_lin_charG H (chi : 'CF(G)) :
chi \is a linear_char -> (chi / H)%CF \is a linear_char.
Proof. by case/andP=> Nchi; rewrite qualifE/= cfQuo_char ?cfQuo1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfQuo_lin_char | |
cfMod_charG H (chi : 'CF(G / H)) :
chi \is a character -> (chi %% H)%CF \is a character.
Proof. exact: cfMorph_char. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfMod_char | |
cfMod_lin_charG H (chi : 'CF(G / H)) :
chi \is a linear_char -> (chi %% H)%CF \is a linear_char.
Proof. exact: cfMorph_lin_char. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfMod_lin_char | |
cfMod_charEG H (chi : 'CF(G / H)) :
H <| G -> (chi %% H \is a character)%CF = (chi \is a character).
Proof. by case/andP=> _; apply: cfMorph_charE. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfMod_charE | |
cfMod_lin_charEG H (chi : 'CF(G / H)) :
H <| G -> (chi %% H \is a linear_char)%CF = (chi \is a linear_char).
Proof. by case/andP=> _; apply: cfMorph_lin_charE. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfMod_lin_charE | |
cfQuo_charEG H (chi : 'CF(G)) :
H <| G -> H \subset cfker chi ->
(chi / H \is a character)%CF = (chi \is a character).
Proof. by move=> nsHG kerH; rewrite -cfMod_charE ?cfQuoK. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfQuo_charE | |
cfQuo_lin_charEG H (chi : 'CF(G)) :
H <| G -> H \subset cfker chi ->
(chi / H \is a linear_char)%CF = (chi \is a linear_char).
Proof. by move=> nsHG kerH; rewrite -cfMod_lin_charE ?cfQuoK. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfQuo_lin_charE | |
cfMod_irrG H chi :
H <| G -> (chi %% H \in irr G)%CF = (chi \in irr (G / H)).
Proof. by case/andP=> _; apply: cfMorph_irr. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfMod_irr | |
mod_IirrG H i := cfIirr ('chi[G / H]_i %% H)%CF. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | mod_Iirr | |
mod_Iirr0G H : mod_Iirr (0 : Iirr (G / H)) = 0.
Proof. exact: morph_Iirr0. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | mod_Iirr0 | |
mod_IirrEG H i : H <| G -> 'chi_(mod_Iirr i) = ('chi[G / H]_i %% H)%CF.
Proof. by move=> nsHG; rewrite cfIirrE ?cfMod_irr ?mem_irr. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | mod_IirrE | |
mod_Iirr_eq0G H i :
H <| G -> (mod_Iirr i == 0) = (i == 0 :> Iirr (G / H)).
Proof. by case/andP=> _ /morph_Iirr_eq0->. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | mod_Iirr_eq0 | |
cfQuo_irrG H chi :
H <| G -> H \subset cfker chi ->
((chi / H)%CF \in irr (G / H)) = (chi \in irr G).
Proof. by move=> nsHG kerH; rewrite -cfMod_irr ?cfQuoK. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfQuo_irr | |
quo_IirrG H i := cfIirr ('chi[G]_i / H)%CF. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | quo_Iirr | |
quo_Iirr0G H : quo_Iirr H (0 : Iirr G) = 0.
Proof. by rewrite /quo_Iirr irr0 cfQuo_cfun1 -irr0 irrK. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | quo_Iirr0 | |
quo_IirrEG H i :
H <| G -> H \subset cfker 'chi[G]_i -> 'chi_(quo_Iirr H i) = ('chi_i / H)%CF.
Proof. by move=> nsHG kerH; rewrite cfIirrE ?cfQuo_irr ?mem_irr. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | quo_IirrE | |
quo_Iirr_eq0G H i :
H <| G -> H \subset cfker 'chi[G]_i -> (quo_Iirr H i == 0) = (i == 0).
Proof. by move=> nsHG kerH; rewrite -!irr_eq1 quo_IirrE ?cfQuo_eq1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | quo_Iirr_eq0 | |
mod_IirrKG H : H <| G -> cancel (@mod_Iirr G H) (@quo_Iirr G H).
Proof.
move=> nsHG i; apply: irr_inj.
by rewrite quo_IirrE ?mod_IirrE ?cfker_mod // cfModK.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | mod_IirrK | |
quo_IirrKG H i :
H <| G -> H \subset cfker 'chi[G]_i -> mod_Iirr (quo_Iirr H i) = i.
Proof.
by move=> nsHG kerH; apply: irr_inj; rewrite mod_IirrE ?quo_IirrE ?cfQuoK.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | quo_IirrK | |
quo_IirrKeqG H :
H <| G ->
forall i, (mod_Iirr (quo_Iirr H i) == i) = (H \subset cfker 'chi[G]_i).
Proof.
move=> nsHG i; apply/eqP/idP=> [<- | ]; last exact: quo_IirrK.
by rewrite mod_IirrE ?cfker_mod.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | quo_IirrKeq | |
mod_Iirr_bijH G :
H <| G -> {on [pred i | H \subset cfker 'chi_i], bijective (@mod_Iirr G H)}.
Proof.
by exists (quo_Iirr H) => [i _ | i]; [apply: mod_IirrK | apply: quo_IirrK].
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | mod_Iirr_bij | |
sum_norm_irr_quoH G x :
x \in G -> H <| G ->
\sum_i `|'chi[G / H]_i (coset H x)| ^+ 2
= \sum_(i | H \subset cfker 'chi_i) `|'chi[G]_i x| ^+ 2.
Proof.
move=> Gx nsHG; rewrite (reindex _ (mod_Iirr_bij nsHG)) /=.
by apply/esym/eq_big=> [i | i _]; rewrite mod_IirrE ?cfker_mod ?cfModE.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | sum_norm_irr_quo | |
cap_cfker_normalG H :
H <| G -> \bigcap_(i | H \subset cfker 'chi[G]_i) (cfker 'chi_i) = H.
Proof.
move=> nsHG; have [sHG nHG] := andP nsHG; set lhs := \bigcap_(i | _) _.
have nHlhs: lhs \subset 'N(H) by rewrite (bigcap_min 0) ?cfker_irr0.
apply/esym/eqP; rewrite eqEsubset (introT bigcapsP) //= -quotient_sub1 //.
rewrite -(TI_cfker_irr (G / H)); apply/bigcapsP=> i _.
rewrite sub_quotient_pre // (bigcap_min (mod_Iirr i)) ?mod_IirrE ?cfker_mod //.
by rewrite cfker_morph ?subsetIr.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cap_cfker_normal | |
cfker_reg_quoG H : H <| G -> cfker (cfReg (G / H)%g %% H) = H.
Proof.
move=> nsHG; have [sHG nHG] := andP nsHG.
apply/setP=> x; rewrite cfkerEchar ?cfMod_char ?cfReg_char //.
rewrite -[in RHS in _ = RHS](setIidPr sHG) !inE; apply: andb_id2l => Gx.
rewrite !cfModE // !cfRegE // morph1 eqxx.
rewrite (sameP eqP (kerP _ (subsetP nHG x Gx))) ker_coset.
by rewrite -!mulrnA eqr_nat eqn_pmul2l ?cardG_gt0 // (can_eq oddb) eqb_id.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfker_reg_quo | |
lin_irr_der1G i :
('chi_i \is a linear_char) = (G^`(1)%g \subset cfker 'chi[G]_i).
Proof.
apply/idP/idP=> [|sG'K]; first exact: lin_char_der1.
have nsG'G: G^`(1) <| G := der_normal 1 G.
rewrite qualifE/= irr_char -[i](quo_IirrK nsG'G) // mod_IirrE //=.
by rewrite cfModE // morph1 lin_char1 //; apply/char_abelianP/der_abelian.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | lin_irr_der1 | |
subGcfkerG i : (G \subset cfker 'chi[G]_i) = (i == 0).
Proof.
rewrite -irr_eq1; apply/idP/eqP=> [chiG1 | ->]; last by rewrite cfker_cfun1.
apply/cfun_inP=> x Gx; rewrite cfun1E Gx cfker1 ?(subsetP chiG1) ?lin_char1 //.
by rewrite lin_irr_der1 (subset_trans (der_sub 1 G)).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | subGcfker | |
irr_prime_injPG i :
prime #|G| -> reflect {in G &, injective 'chi[G]_i} (i != 0).
Proof.
move=> pr_G; apply: (iffP idP) => [nz_i | inj_chi].
apply: fful_lin_char_inj (irr_prime_lin i pr_G) _.
by rewrite cfaithfulE -(setIidPr (cfker_sub _)) prime_TIg // subGcfker.
have /trivgPn[x Gx ntx]: G :!=: 1%g by rewrite -cardG_gt1 prime_gt1.
apply: contraNneq ntx => i0; apply/eqP/inj_chi=> //.
by rewrite i0 irr0 !cfun1E Gx group1.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr_prime_injP | |
cap_cfker_lin_irrG :
\bigcap_(i | 'chi[G]_i \is a linear_char) (cfker 'chi_i) = G^`(1)%g.
Proof.
rewrite -(cap_cfker_normal (der_normal 1 G)).
by apply: eq_bigl => i; rewrite lin_irr_der1.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cap_cfker_lin_irr | |
card_lin_irrG :
#|[pred i | 'chi[G]_i \is a linear_char]| = #|G : G^`(1)%g|.
Proof.
have nsG'G := der_normal 1 G; rewrite (eq_card (@lin_irr_der1 G)).
rewrite -(on_card_preimset (mod_Iirr_bij nsG'G)).
rewrite -card_quotient ?normal_norm //.
move: (der_abelian 0 G); rewrite card_classes_abelian; move/eqP<-.
rewrite -NirrE -[RHS]card_ord.
by apply: eq_card => i; rewrite !inE mod_IirrE ?cfker_mod. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | card_lin_irr | |
solvable_has_lin_charG :
G :!=: 1%g -> solvable G ->
exists2 i, 'chi[G]_i \is a linear_char & 'chi_i != 1.
Proof.
move=> ntG solG.
suff /subsetPn[i]: ~~ ([pred i | 'chi[G]_i \is a linear_char] \subset pred1 0).
by rewrite !inE -(inj_eq irr_inj) irr0; exists i.
rewrite (contra (@subset_leq_card _ _ _)) // -ltnNge card1 card_lin_irr.
by rewrite indexg_gt1 proper_subn // (sol_der1_proper solG).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | solvable_has_lin_char | |
lin_char_groupG :
{linG : finGroupType & {cF : linG -> 'CF(G) |
[/\ injective cF, #|linG| = #|G : G^`(1)|,
forall u, cF u \is a linear_char
& forall phi, phi \is a linear_char -> exists u, phi = cF u]
& [/\ cF 1%g = 1%R,
{morph cF : u v / (u * v)%g >-> (u * v)%R},
forall k, {morph cF : u / (u^+ k)%g >-> u ^+ k},
{morph cF: u / u^-1%g >-> u^-1%CF}
& {mono cF: u / #[u]%g >-> #[u]%CF} ]}}.
Proof.
pose linT := {i : Iirr G | 'chi_i \is a linear_char}.
pose cF (u : linT) := 'chi_(sval u).
have cFlin u: cF u \is a linear_char := svalP u.
have cFinj: injective cF := inj_comp irr_inj val_inj.
have inT xi : xi \is a linear_char -> {u | cF u = xi}.
move=> lin_xi; have /irrP/sig_eqW[i Dxi] := lin_char_irr lin_xi.
by apply: (exist _ (Sub i _)) => //; rewrite -Dxi.
have [one cFone] := inT 1 (rpred1 _).
pose inv u := sval (inT _ (rpredVr (cFlin u))).
pose mul u v := sval (inT _ (rpredM (cFlin u) (cFlin v))).
have cFmul u v: cF (mul u v) = cF u * cF v := svalP (inT _ _).
have cFinv u: cF (inv u) = (cF u)^-1 := svalP (inT _ _).
have mulA: associative mul by move=> u v w; apply: cFinj; rewrite !cFmul mulrA.
have mul1: left_id one mul by move=> u; apply: cFinj; rewrite cFmul cFone mul1r.
have mulV: left_inverse one inv mul.
by move=> u; apply: cFinj; rewrite cFmul cFinv cFone mulVr ?lin_char_unitr.
pose imA := Finite_isGroup.Build linT mulA mul1 mulV.
pose linG : finGroupType := HB.pack linT imA.
have cFexp k: {
... | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | lin_char_group | |
cfExp_prime_transitiveG (i j : Iirr G) :
prime #|G| -> i != 0 -> j != 0 ->
exists2 k, coprime k #['chi_i]%CF & 'chi_j = 'chi_i ^+ k.
Proof.
set p := #|G| => pr_p nz_i nz_j; have cycG := prime_cyclic pr_p.
have [L [h [injh oL Lh h_ontoL]] [h1 hM hX _ o_h]] := lin_char_group G.
rewrite (derG1P (cyclic_abelian cycG)) indexg1 -/p in oL.
have /fin_all_exists[h' h'K] := h_ontoL _ (irr_cyclic_lin _ cycG).
have o_h' k: k != 0 -> #[h' k] = p.
rewrite -cforder_irr_eq1 h'K -o_h => nt_h'k.
by apply/prime_nt_dvdP=> //; rewrite cforder_lin_char_dvdG.
have{oL} genL k: k != 0 -> generator [set: L] (h' k).
move=> /o_h' o_h'k; rewrite /generator eq_sym eqEcard subsetT /=.
by rewrite cardsT oL -o_h'k.
have [/(_ =P <[_]>)-> gen_j] := (genL i nz_i, genL j nz_j).
have /cycleP[k Dj] := cycle_generator gen_j.
by rewrite !h'K Dj o_h hX generator_coprime coprime_sym in gen_j *; exists k.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfExp_prime_transitive | |
card_subcent1_cosetG H x :
x \in G -> H <| G -> (#|'C_(G / H)[coset H x]| <= #|'C_G[x]|)%N.
Proof.
move=> Gx nsHG; rewrite -leC_nat.
move: (second_orthogonality_relation x Gx); rewrite mulrb class_refl => <-.
have GHx: coset H x \in (G / H)%g by apply: mem_quotient.
move: (second_orthogonality_relation (coset H x) GHx).
rewrite mulrb class_refl => <-.
rewrite -2!(eq_bigr _ (fun _ _ => normCK _)) sum_norm_irr_quo // -subr_ge0.
rewrite (bigID (fun i => H \subset cfker 'chi[G]_i)) //= [X in X + _]addrC addrK.
by apply: sumr_ge0 => i _; rewrite normCK mul_conjC_ge0.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | card_subcent1_coset | |
det_repr_mxx : 'M_1 := (\det (rG x))%:M.
Fact det_is_repr : mx_repr G det_repr_mx.
Proof.
split=> [|g h Gg Gh]; first by rewrite /det_repr_mx repr_mx1 det1.
by rewrite /det_repr_mx repr_mxM // det_mulmx !mulmxE scalar_mxM.
Qed. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | det_repr_mx | |
det_repr:= MxRepresentation det_is_repr. | Canonical | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | det_repr | |
detRepr:= cfRepr det_repr. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | detRepr | |
detRepr_lin_char: detRepr \is a linear_char.
Proof.
by rewrite qualifE/= cfRepr_char cfunE group1 repr_mx1 mxtrace1 mulr1n /=.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | detRepr_lin_char | |
cfDet(gT : finGroupType) (G : {group gT}) phi :=
\prod_i detRepr 'Chi_i ^+ Num.truncn '[phi, 'chi[G]_i]. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDet | |
cfDet_unlockable:= Unlockable cfDet.unlock. | Canonical | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDet_unlockable | |
cfDet_lin_charphi : cfDet phi \is a linear_char.
Proof.
rewrite unlock; apply: rpred_prod => i _; apply: rpredX.
exact: detRepr_lin_char.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDet_lin_char | |
cfDetD:
{in character &, {morph cfDet : phi psi / phi + psi >-> phi * psi}}.
Proof.
move=> phi psi Nphi Npsi; rewrite unlock /= -big_split; apply: eq_bigr => i _ /=.
by rewrite -exprD cfdotDl truncnD ?nnegrE ?natr_ge0 // Cnat_cfdot_char_irr.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDetD | |
cfDet0: cfDet 0 = 1.
Proof. by rewrite unlock big1 // => i _; rewrite cfdot0l truncn0. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDet0 | |
cfDetMnk :
{in character, {morph cfDet : phi / phi *+ k >-> phi ^+ k}}.
Proof.
move=> phi Nphi; elim: k => [|k IHk]; rewrite ?cfDet0 // mulrS exprS -{}IHk.
by rewrite cfDetD ?rpredMn.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDetMn | |
cfDetReprn rG : cfDet (cfRepr rG) = @detRepr _ _ n rG.
Proof.
transitivity (\prod_W detRepr (socle_repr W) ^+ standard_irr_coef rG W).
rewrite (reindex _ (socle_of_Iirr_bij _)) unlock /=.
apply: eq_bigr => i _; congr (_ ^+ _).
rewrite (cfRepr_sim (mx_rsim_standard rG)) cfRepr_standard.
rewrite cfdot_suml (bigD1 i) ?big1 //= => [|j i'j]; last first.
by rewrite cfdotZl cfdot_irr (negPf i'j) mulr0.
by rewrite cfdotZl cfnorm_irr mulr1 addr0 natrK.
apply/cfun_inP=> x Gx; rewrite prod_cfunE //.
transitivity (detRepr (standard_grepr rG) x); last first.
rewrite !cfunE Gx !trace_mx11 !mxE eqxx !mulrb.
case: (standard_grepr rG) (mx_rsim_standard rG) => /= n1 rG1 [B Dn1].
rewrite -{n1}Dn1 in rG1 B *; rewrite row_free_unit => uB rG_B.
by rewrite -[rG x](mulmxK uB) rG_B // !det_mulmx mulrC -!det_mulmx mulKmx.
rewrite /standard_grepr; elim/big_rec2: _ => [|W y _ _ ->].
by rewrite cfunE trace_mx11 mxE Gx det1.
rewrite !cfunE Gx /= !{1}trace_mx11 !{1}mxE det_ublock; congr (_ * _).
rewrite exp_cfunE //; elim: (standard_irr_coef rG W) => /= [|k IHk].
by rewrite /muln_grepr big_ord0 det1.
rewrite exprS /muln_grepr big_ord_recl det_ublock -IHk; congr (_ * _).
by rewrite cfunE trace_mx11 mxE Gx.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDetRepr | |
cfDet_idxi : xi \is a linear_char -> cfDet xi = xi.
Proof.
move=> lin_xi; have /irrP[i Dxi] := lin_char_irr lin_xi.
apply/cfun_inP=> x Gx; rewrite Dxi -irrRepr cfDetRepr !cfunE trace_mx11 mxE.
move: lin_xi (_ x) => /andP[_]; rewrite Dxi irr1_degree pnatr_eq1 => /eqP-> X.
by rewrite {1}[X]mx11_scalar det_scalar1 trace_mx11.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDet_id | |
cfDet_orderphi := #[cfDet phi]%CF. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDet_order | |
cfDet_order_linxi :
xi \is a linear_char -> cfDet_order xi = #[xi]%CF.
Proof. by rewrite /cfDet_order => /cfDet_id->. Qed. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDet_order_lin | |
cfDet_order_dvdGphi : cfDet_order phi %| #|G|.
Proof. by rewrite cforder_lin_char_dvdG ?cfDet_lin_char. Qed. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDet_order_dvdG | |
cfDetResgT (G H : {group gT}) phi :
phi \is a character -> cfDet ('Res[H, G] phi) = 'Res (cfDet phi).
Proof.
move=> Nphi; have [sGH | not_sHG] := boolP (H \subset G); last first.
have /natrP[n Dphi1] := Cnat_char1 Nphi.
rewrite !cfResEout // Dphi1 lin_char1 ?cfDet_lin_char // scale1r.
by rewrite scaler_nat cfDetMn ?cfDet_id ?rpred1 // expr1n.
have [rG ->] := char_reprP Nphi; rewrite !(=^~ cfRepr_sub, cfDetRepr) //.
apply: cfRepr_sim; exists 1%:M; rewrite ?row_free_unit ?unitmx1 // => x Hx.
by rewrite mulmx1 mul1mx.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDetRes | |
cfDetMorphaT rT (D G : {group aT}) (f : {morphism D >-> rT})
(phi : 'CF(f @* G)) :
phi \is a character -> cfDet (cfMorph phi) = cfMorph (cfDet phi).
Proof.
move=> Nphi; have [sGD | not_sGD] := boolP (G \subset D); last first.
have /natrP[n Dphi1] := Cnat_char1 Nphi.
rewrite !cfMorphEout // Dphi1 lin_char1 ?cfDet_lin_char // scale1r.
by rewrite scaler_nat cfDetMn ?cfDet_id ?rpred1 // expr1n.
have [rG ->] := char_reprP Nphi; rewrite !(=^~ cfRepr_morphim, cfDetRepr) //.
apply: cfRepr_sim; exists 1%:M; rewrite ?row_free_unit ?unitmx1 // => x Hx.
by rewrite mulmx1 mul1mx.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDetMorph | |
cfDetIsomaT rT (G : {group aT}) (R : {group rT})
(f : {morphism G >-> rT}) (isoGR : isom G R f) phi :
cfDet (cfIsom isoGR phi) = cfIsom isoGR (cfDet phi).
Proof.
rewrite unlock rmorph_prod (reindex (isom_Iirr isoGR)); last first.
by exists (isom_Iirr (isom_sym isoGR)) => i; rewrite ?isom_IirrK ?isom_IirrKV.
apply: eq_bigr=> i; rewrite -!cfDetRepr !irrRepr isom_IirrE rmorphXn cfIsom_iso.
by rewrite /= ![in cfIsom _]unlock cfDetMorph ?cfRes_char ?cfDetRes ?irr_char.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDetIsom | |
cfDet_mul_lingT (G : {group gT}) (lambda phi : 'CF(G)) :
lambda \is a linear_char -> phi \is a character ->
cfDet (lambda * phi) = lambda ^+ Num.truncn (phi 1%g) * cfDet phi.
Proof.
case/andP=> /char_reprP[[n1 rG1] ->] /= n1_1 /char_reprP[[n2 rG2] ->] /=.
do [rewrite !cfRepr1 pnatr_eq1 natrK; move/eqP] in n1_1 *.
rewrite {n1}n1_1 in rG1 *; rewrite cfRepr_prod cfDetRepr.
apply/cfun_inP=> x Gx; rewrite !cfunE cfDetRepr cfunE Gx !mulrb !trace_mx11.
rewrite !mxE prod_repr_lin ?mulrb //=; case: _ / (esym _); rewrite detZ.
congr (_ * _); case: {rG2}n2 => [|n2]; first by rewrite cfun1E Gx.
by rewrite expS_cfunE //= cfunE Gx trace_mx11.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfDet_mul_lin | |
cfcenter(gT : finGroupType) (G : {set gT}) (phi : 'CF(G)) :=
if phi \is a character then [set g in G | `|phi g| == phi 1%g] else cfker phi. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfcenter | |
cfcenter_reprn (rG : mx_representation algC G n) :
'Z(cfRepr rG)%CF = rcenter rG.
Proof.
rewrite /cfcenter /rcenter cfRepr_char /=.
apply/setP=> x /[!inE]; apply/andb_id2l=> Gx.
apply/eqP/is_scalar_mxP=> [|[c rG_c]].
by case/max_cfRepr_norm_scalar=> // c; exists c.
rewrite -(sqrCK (char1_ge0 (cfRepr_char rG))) normC_def; congr (sqrtC _).
rewrite expr2 -{2}(mulgV x) -char_inv ?cfRepr_char ?cfunE ?groupM ?groupV //.
rewrite Gx group1 repr_mx1 repr_mxM ?repr_mxV ?groupV // !mulrb rG_c.
by rewrite invmx_scalar -scalar_mxM !mxtrace_scalar mulrnAr mulrnAl mulr_natl.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfcenter_repr | |
cfcenter_groupf := Group (cfcenter_group_set f). | Canonical | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfcenter_group | |
char_cfcenterEchi x :
chi \is a character -> x \in G ->
(x \in ('Z(chi))%CF) = (`|chi x| == chi 1%g).
Proof. by move=> Nchi Gx; rewrite /cfcenter Nchi inE Gx. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | char_cfcenterE | |
irr_cfcenterEi x :
x \in G -> (x \in 'Z('chi[G]_i)%CF) = (`|'chi_i x| == 'chi_i 1%g).
Proof. by move/char_cfcenterE->; rewrite ?irr_char. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr_cfcenterE | |
cfcenter_subphi : ('Z(phi))%CF \subset G.
Proof. by rewrite /cfcenter /cfker !setIdE -fun_if subsetIl. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfcenter_sub | |
cfker_center_normalphi : cfker phi <| 'Z(phi)%CF.
Proof.
apply: normalS (cfcenter_sub phi) (cfker_normal phi).
rewrite /= /cfcenter; case: ifP => // Hphi; rewrite cfkerEchar //.
apply/subsetP=> x /[!inE] /andP[-> /eqP->] /=.
by rewrite ger0_norm ?char1_ge0.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfker_center_normal | |
cfcenter_normalphi : 'Z(phi)%CF <| G.
Proof.
have [[rG ->] | /negbTE notNphi] := altP (@char_reprP _ _ phi).
by rewrite cfcenter_repr rcenter_normal.
by rewrite /cfcenter notNphi cfker_normal.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfcenter_normal | |
cfcenter_Reschi :
exists2 chi1, chi1 \is a linear_char & 'Res['Z(chi)%CF] chi = chi 1%g *: chi1.
Proof.
have [[rG ->] | /negbTE notNphi] := altP (@char_reprP _ _ chi); last first.
exists 1; first exact: cfun1_lin_char.
rewrite /cfcenter notNphi; apply/cfun_inP=> x Kx.
by rewrite cfunE cfun1E Kx mulr1 cfResE ?cfker_sub // cfker1.
rewrite cfcenter_repr -(cfRepr_sub _ (normal_sub (rcenter_normal _))).
case: rG => [[|n] rG] /=; rewrite cfRepr1.
exists 1; first exact: cfun1_lin_char.
by apply/cfun_inP=> x Zx; rewrite scale0r !cfunE flatmx0 raddf0 Zx.
pose rZmx x := ((rG x 0 0)%:M : 'M_(1,1)).
have rZmxP: mx_repr [group of rcenter rG] rZmx.
split=> [|x y]; first by rewrite /rZmx repr_mx1 mxE eqxx.
move=> /setIdP[Gx /is_scalar_mxP[a rGx]] /setIdP[Gy /is_scalar_mxP[b rGy]].
by rewrite /rZmx repr_mxM // rGx rGy -!scalar_mxM !mxE.
exists (cfRepr (MxRepresentation rZmxP)).
by rewrite qualifE/= cfRepr_char cfRepr1 eqxx.
apply/cfun_inP=> x Zx; rewrite !cfunE Zx /= /rZmx mulr_natl.
by case/setIdP: Zx => Gx /is_scalar_mxP[a ->]; rewrite mxE !mxtrace_scalar.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfcenter_Res | |
cfcenter_cyclicchi : cyclic ('Z(chi)%CF / cfker chi)%g.
Proof.
case Nchi: (chi \is a character); last first.
by rewrite /cfcenter Nchi trivg_quotient cyclic1.
have [-> | nz_chi] := eqVneq chi 0.
rewrite quotientS1 ?cyclic1 //= /cfcenter cfkerEchar ?cfun0_char //.
by apply/subsetP=> x /setIdP[Gx _]; rewrite inE Gx /= !cfunE.
have [xi Lxi def_chi] := cfcenter_Res chi.
set Z := ('Z(_))%CF in xi Lxi def_chi *.
have sZG: Z \subset G by apply: cfcenter_sub.
have ->: cfker chi = cfker xi.
rewrite -(setIidPr (normal_sub (cfker_center_normal _))) -/Z.
rewrite !cfkerEchar // ?lin_charW //= -/Z.
apply/setP=> x /[!inE]; apply: andb_id2l => Zx.
rewrite (subsetP sZG) //= -!(cfResE chi sZG) ?group1 // def_chi !cfunE.
by rewrite (inj_eq (mulfI _)) ?char1_eq0.
have: abelian (Z / cfker xi) by rewrite sub_der1_abelian ?lin_char_der1.
have /irr_reprP[rG irrG ->] := lin_char_irr Lxi; rewrite cfker_repr.
apply: mx_faithful_irr_abelian_cyclic (kquo_mx_faithful rG) _.
exact/quo_mx_irr.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfcenter_cyclic | |
cfcenter_subset_centerchi :
('Z(chi)%CF / cfker chi)%g \subset 'Z(G / cfker chi)%g.
Proof.
case Nchi: (chi \is a character); last first.
by rewrite /cfcenter Nchi trivg_quotient sub1G.
rewrite subsetI quotientS ?cfcenter_sub // quotient_cents2r //=.
case/char_reprP: Nchi => rG ->{chi}; rewrite cfker_repr cfcenter_repr gen_subG.
apply/subsetP=> _ /imset2P[x y /setIdP[Gx /is_scalar_mxP[c rGx]] Gy ->].
rewrite inE groupR //= !repr_mxM ?groupM ?groupV // rGx -(scalar_mxC c) -rGx.
by rewrite !mulmxA !repr_mxKV.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfcenter_subset_center | |
cfcenter_eq_center(i : Iirr G) :
('Z('chi_i)%CF / cfker 'chi_i)%g = 'Z(G / cfker 'chi_i)%g.
Proof.
apply/eqP; rewrite eqEsubset; rewrite cfcenter_subset_center ?irr_char //.
apply/subsetP=> _ /setIP[/morphimP[x /= _ Gx ->] cGx]; rewrite mem_quotient //=.
rewrite -irrRepr cfker_repr cfcenter_repr inE Gx in cGx *.
apply: mx_abs_irr_cent_scalar 'Chi_i _ _ _; first exact/groupC/socle_irr.
have nKG: G \subset 'N(rker 'Chi_i) by apply: rker_norm. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfcenter_eq_center | |
cap_cfcenter_irr: \bigcap_i 'Z('chi[G]_i)%CF = 'Z(G).
Proof.
apply/esym/eqP; rewrite eqEsubset (introT bigcapsP) /= => [|i _]; last first.
rewrite -(quotientSGK _ (normal_sub (cfker_center_normal _))).
by rewrite cfcenter_eq_center morphim_center.
by rewrite subIset // normal_norm // cfker_normal.
set Z := \bigcap_i _.
have sZG: Z \subset G by rewrite (bigcap_min 0) ?cfcenter_sub.
rewrite subsetI sZG (sameP commG1P trivgP) -(TI_cfker_irr G).
apply/bigcapsP=> i _; have nKiG := normal_norm (cfker_normal 'chi_i).
rewrite -quotient_cents2 ?(subset_trans sZG) //.
rewrite (subset_trans (quotientS _ (bigcap_inf i _))) //.
by rewrite cfcenter_eq_center subsetIr.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cap_cfcenter_irr | |
cfnorm_Res_leifH phi :
H \subset G ->
'['Res[H] phi] <= #|G : H|%:R * '[phi] ?= iff (phi \in 'CF(G, H)).
Proof.
move=> sHG; rewrite cfun_onE mulrCA natf_indexg // -mulrA mulKf ?neq0CG //.
rewrite (big_setID H) (setIidPr sHG) /= addrC.
rewrite (mono_leif (ler_pM2l _)) ?invr_gt0 ?gt0CG // -leifBLR -sumrB.
rewrite big1 => [|x Hx]; last by rewrite !cfResE ?subrr.
have ->: (support phi \subset H) = (G :\: H \subset [set x | phi x == 0]).
rewrite subDset setUC -subDset; apply: eq_subset => x.
by rewrite !inE (andb_idr (contraR _)) // => /cfun0->.
rewrite (sameP subsetP forall_inP); apply: leif_0_sum => x _.
by rewrite !inE /<?=%R mul_conjC_ge0 eq_sym mul_conjC_eq0.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfnorm_Res_leif | |
irr1_bound(i : Iirr G) :
('chi_i 1%g) ^+ 2 <= #|G : 'Z('chi_i)%CF|%:R
?= iff ('chi_i \in 'CF(G, 'Z('chi_i)%CF)).
Proof.
congr (_ <= _ ?= iff _): (cfnorm_Res_leif 'chi_i (cfcenter_sub 'chi_i)).
have [xi Lxi ->] := cfcenter_Res 'chi_i.
have /irrP[j ->] := lin_char_irr Lxi; rewrite cfdotZl cfdotZr cfdot_irr eqxx.
by rewrite mulr1 irr1_degree conjC_nat.
by rewrite cfdot_irr eqxx mulr1.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr1_bound | |
irr1_abelian_bound(i : Iirr G) :
abelian (G / 'Z('chi_i)%CF) -> ('chi_i 1%g) ^+ 2 = #|G : 'Z('chi_i)%CF|%:R.
Proof.
move=> AbGc; apply/eqP; rewrite irr1_bound cfun_onE; apply/subsetP=> x nz_chi_x.
have Gx: x \in G by apply: contraR nz_chi_x => /cfun0->.
have nKx := subsetP (normal_norm (cfker_normal 'chi_i)) _ Gx.
rewrite -(quotientGK (cfker_center_normal _)) inE nKx inE /=.
rewrite cfcenter_eq_center inE mem_quotient //=.
apply/centP=> _ /morphimP[y nKy Gy ->]; apply/commgP; rewrite -morphR //=.
set z := [~ x, y]; rewrite coset_id //.
have: z \in 'Z('chi_i)%CF.
apply: subsetP (mem_commg Gx Gy).
by rewrite der1_min // normal_norm ?cfcenter_normal.
rewrite -irrRepr cfker_repr cfcenter_repr !inE in nz_chi_x *.
case/andP=> Gz /is_scalar_mxP[c Chi_z]; rewrite Gz Chi_z mul1mx /=.
apply/eqP; congr _%:M; apply: (mulIf nz_chi_x); rewrite mul1r.
rewrite -{2}(cfunJ _ x Gy) conjg_mulR -/z !cfunE Gx groupM // !{1}mulrb.
by rewrite repr_mxM // Chi_z mul_mx_scalar mxtraceZ.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr1_abelian_bound | |
irr_faithful_centeri : cfaithful 'chi[G]_i -> cyclic 'Z(G).
Proof.
rewrite (isog_cyclic (isog_center (quotient1_isog G))) /=.
by move/trivgP <-; rewrite -cfcenter_eq_center cfcenter_cyclic.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr_faithful_center | |
cfcenter_fful_irri : cfaithful 'chi[G]_i -> 'Z('chi_i)%CF = 'Z(G).
Proof.
move/trivgP=> Ki1; have:= cfcenter_eq_center i; rewrite {}Ki1.
have inj1: 'injm (@coset gT 1%g) by rewrite ker_coset.
by rewrite -injm_center; first apply: injm_morphim_inj; rewrite ?norms1.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfcenter_fful_irr | |
pgroup_cyclic_faithful(p : nat) :
p.-group G -> cyclic 'Z(G) -> exists i, cfaithful 'chi[G]_i.
Proof.
pose Z := 'Ohm_1('Z(G)) => pG cycZG; have nilG := pgroup_nil pG.
have [-> | ntG] := eqsVneq G [1]; first by exists 0; apply: cfker_sub.
have{pG} [[p_pr _ _] pZ] := (pgroup_pdiv pG ntG, pgroupS (center_sub G) pG).
have ntZ: 'Z(G) != [1] by rewrite center_nil_eq1.
have{pZ} oZ: #|Z| = p by apply: Ohm1_cyclic_pgroup_prime.
apply/existsP; apply: contraR ntZ => /existsPn-not_ffulG.
rewrite -Ohm1_eq1 -subG1 /= -/Z -(TI_cfker_irr G); apply/bigcapsP=> i _.
rewrite prime_meetG ?oZ // setIC meet_Ohm1 // meet_center_nil ?cfker_normal //.
by rewrite -subG1 not_ffulG.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | pgroup_cyclic_faithful | |
cfInd_charchi : chi \is a character -> 'Ind[G] chi \is a character.
Proof.
move=> Nchi; apply/forallP=> i; rewrite coord_cfdot -Frobenius_reciprocity //.
by rewrite Cnat_cfdot_char ?cfRes_char ?irr_char.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfInd_char | |
cfInd_eq0chi :
H \subset G -> chi \is a character -> ('Ind[G] chi == 0) = (chi == 0).
Proof.
move=> sHG Nchi; rewrite -!(char1_eq0) ?cfInd_char // cfInd1 //.
by rewrite (mulrI_eq0 _ (mulfI _)) ?neq0CiG.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfInd_eq0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.