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 |
|---|---|---|---|---|---|---|
cfker_reprn (rG : mx_representation algC G n) :
cfker (cfRepr rG) = rker rG.
Proof.
apply/esym/setP=> x; rewrite inE mul1mx /=.
case Gx: (x \in G); last by rewrite inE Gx.
apply/eqP/idP=> Kx; last by rewrite max_cfRepr_mx1 // cfker1.
rewrite inE Gx; apply/forallP=> y; rewrite !cfunE !mulrb groupMl //.
by case: ifP => // Gy; rewrite repr_mxM // Kx 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 | cfker_repr | |
cfkerEcharchi :
chi \is a character -> cfker chi = [set x in G | chi x == chi 1%g].
Proof.
move=> Nchi; apply/setP=> x; apply/idP/setIdP=> [Kx | [Gx /eqP chi_x]].
by rewrite (subsetP (cfker_sub chi)) // cfker1.
case/char_reprP: Nchi => rG -> in chi_x *; rewrite inE Gx; apply/forallP=> y.
rewrite !cfunE groupMl // !mulrb; case: ifP => // Gy.
by rewrite repr_mxM // max_cfRepr_mx1 ?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 | cfkerEchar | |
cfker_nzcharEchi :
chi \is a character -> chi != 0 -> cfker chi = [set x | chi x == chi 1%g].
Proof.
move=> Nchi nzchi; apply/setP=> x; rewrite cfkerEchar // !inE andb_idl //.
by apply: contraLR => /cfun0-> //; rewrite eq_sym char1_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 | cfker_nzcharE | |
cfkerEirri : cfker 'chi[G]_i = [set x | 'chi_i x == 'chi_i 1%g].
Proof. by rewrite cfker_nzcharE ?irr_char ?irr_neq0. 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 | cfkerEirr | |
cfker_irr0: cfker 'chi[G]_0 = G.
Proof. by rewrite irr0 cfker_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 | cfker_irr0 | |
cfaithful_reg: cfaithful (cfReg G).
Proof.
apply/subsetP=> x; rewrite cfkerEchar ?cfReg_char // !inE !cfRegE eqxx.
by case/andP=> _; apply: contraLR => /negbTE->; rewrite eq_sym neq0CG.
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 | cfaithful_reg | |
cfkerEchi :
chi \is a character ->
cfker chi = G :&: \bigcap_(i in irr_constt chi) cfker 'chi_i.
Proof.
move=> Nchi; rewrite cfkerEchar //; apply/setP=> x; rewrite !inE.
apply: andb_id2l => Gx; rewrite {1 2}[chi]cfun_sum_constt !sum_cfunE.
apply/eqP/bigcapP=> [Kx i Ci | Kx]; last first.
by apply: eq_bigr => i /Kx Kx_i; rewrite !cfunE cfker1.
rewrite cfkerEirr inE /= -(inj_eq (mulfI Ci)).
have:= (normC_sum_upper _ Kx) i; rewrite !cfunE => -> // {Ci}i _.
have chi_i_ge0: 0 <= '[chi, 'chi_i].
by rewrite natr_ge0 ?Cnat_cfdot_char_irr.
by rewrite !cfunE normrM (normr_idP _) ?ler_wpM2l ?char1_ge_norm ?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 | cfkerE | |
TI_cfker_irr: \bigcap_i cfker 'chi[G]_i = [1].
Proof.
apply/trivgP; apply: subset_trans cfaithful_reg; rewrite cfkerE ?cfReg_char //.
rewrite subsetI (bigcap_min 0) //=; last by rewrite cfker_irr0.
by apply/bigcapsP=> i _; rewrite bigcap_inf.
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 | TI_cfker_irr | |
cfker_constti chi :
chi \is a character -> i \in irr_constt chi ->
cfker chi \subset cfker 'chi[G]_i.
Proof. by move=> Nchi Ci; rewrite cfkerE ?subIset ?(bigcap_min i) ?orbT. 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_constt | |
lin_char_der1: G^`(1)%g \subset cfker xi.
Proof.
rewrite gen_subG /=; apply/subsetP=> _ /imset2P[x y Gx Gy ->].
rewrite cfkerEchar // inE groupR //= !lin_charM ?lin_charV ?in_group //.
by rewrite mulrCA mulKf ?mulVf ?lin_char_neq0 // lin_char1.
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_char_der1 | |
cforder_lin_char: #[xi]%CF = exponent (G / cfker xi)%g.
Proof.
apply/eqP; rewrite eqn_dvd; apply/andP; split.
apply/dvdn_cforderP=> x Gx; rewrite -lin_charX // -cfQuoEker ?groupX //.
rewrite morphX ?(subsetP (cfker_norm xi)) //= expg_exponent ?mem_quotient //.
by rewrite cfQuo1 ?cfker_normal ?lin_char1.
have abGbar: abelian (G / cfker xi) := sub_der1_abelian lin_char_der1.
have [_ /morphimP[x Nx Gx ->] ->] := exponent_witness (abelian_nil abGbar).
rewrite order_dvdn -morphX //= coset_id cfkerEchar // !inE groupX //=.
by rewrite lin_charX ?lin_char1 // (dvdn_cforderP _ _ _).
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 | cforder_lin_char | |
cforder_lin_char_dvdG: #[xi]%CF %| #|G|.
Proof.
by rewrite cforder_lin_char (dvdn_trans (exponent_dvdn _)) ?dvdn_morphim.
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 | cforder_lin_char_dvdG | |
cforder_lin_char_gt0: (0 < #[xi]%CF)%N.
Proof. by rewrite cforder_lin_char exponent_gt0. 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 | cforder_lin_char_gt0 | |
cfRepr_subn (rG : mx_representation algC G n) (sHG : H \subset G) :
cfRepr (subg_repr rG sHG) = 'Res[H] (cfRepr rG).
Proof.
by apply/cfun_inP => x Hx; rewrite cfResE // !cfunE Hx (subsetP sHG).
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 | cfRepr_sub | |
cfRes_charchi : chi \is a character -> 'Res[H, G] chi \is a character.
Proof.
have [sHG | not_sHG] := boolP (H \subset G).
by case/char_reprP=> rG ->; rewrite -(cfRepr_sub rG sHG) cfRepr_char.
by move/Cnat_char1=> Nchi1; rewrite cfResEout // rpredZ_nat ?rpred1.
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 | cfRes_char | |
cfRes_eq0phi : phi \is a character -> ('Res[H, G] phi == 0) = (phi == 0).
Proof. by move=> Nchi; rewrite -!char1_eq0 ?cfRes_char // cfRes1. 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 | cfRes_eq0 | |
cfRes_lin_charchi :
chi \is a linear_char -> 'Res[H, G] chi \is a linear_char.
Proof. by case/andP=> Nchi; rewrite qualifE/= cfRes_char ?cfRes1. 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 | cfRes_lin_char | |
Res_irr_neq0i : 'Res[H, G] 'chi_i != 0.
Proof. by rewrite cfRes_eq0 ?irr_neq0 ?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 | Res_irr_neq0 | |
cfRes_lin_lin(chi : 'CF(G)) :
chi \is a character -> 'Res[H] chi \is a linear_char -> chi \is a linear_char.
Proof. by rewrite !qualifE/= !qualifE/= cfRes1 => -> /andP[]. 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 | cfRes_lin_lin | |
cfRes_irr_irrchi :
chi \is a character -> 'Res[H] chi \in irr H -> chi \in irr G.
Proof.
have [sHG /char_reprP[rG ->] | not_sHG Nchi] := boolP (H \subset G).
rewrite -(cfRepr_sub _ sHG) => /irr_reprP[rH irrH def_rH]; apply/irr_reprP.
suffices /subg_mx_irr: mx_irreducible (subg_repr rG sHG) by exists rG.
by apply: mx_rsim_irr irrH; apply/cfRepr_rsimP/eqP.
rewrite cfResEout // => /irrP[j Dchi_j]; apply/lin_char_irr/cfRes_lin_lin=> //.
suffices j0: j = 0 by rewrite cfResEout // Dchi_j j0 irr0 rpred1.
apply: contraNeq (irr1_neq0 j) => nz_j.
have:= xcfun_id j 0; rewrite -Dchi_j cfunE xcfunZl -irr0 xcfun_id eqxx => ->.
by rewrite (negPf nz_j).
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 | cfRes_irr_irr | |
Res_Iirr(A B : {set gT}) i := cfIirr ('Res[B, A] 'chi_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 | Res_Iirr | |
Res_Iirr0: Res_Iirr H (0 : Iirr G) = 0.
Proof. by rewrite /Res_Iirr irr0 rmorph1 -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 | Res_Iirr0 | |
lin_Res_IirrEi : 'chi[G]_i 1%g = 1 -> 'chi_(Res_Iirr H i) = 'Res 'chi_i.
Proof.
move=> chi1; rewrite cfIirrE ?lin_char_irr ?cfRes_lin_char //.
by rewrite qualifE/= irr_char /= chi1.
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_Res_IirrE | |
constt_Ind_Resi j :
i \in irr_constt ('Ind[G] 'chi_j) = (j \in irr_constt ('Res[H] 'chi_i)).
Proof. by rewrite !irr_consttE cfdotC conjC_eq0 -cfdot_Res_l. 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 | constt_Ind_Res | |
cfdot_Res_ge_constti j psi :
psi \is a character -> j \in irr_constt psi ->
'['Res[H, G] 'chi_j, 'chi_i] <= '['Res[H] psi, 'chi_i].
Proof.
move=> {psi} _ /constt_charP[// | psi Npsi ->].
rewrite linearD cfdotDl addrC -subr_ge0 addrK natr_ge0 //=.
by rewrite Cnat_cfdot_char_irr // cfRes_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 | cfdot_Res_ge_constt | |
constt_Res_transj psi :
psi \is a character -> j \in irr_constt psi ->
{subset irr_constt ('Res[H, G] 'chi_j) <= irr_constt ('Res[H] psi)}.
Proof.
move=> Npsi Cj i; apply: contraNneq; rewrite eq_le => {1}<-.
rewrite cfdot_Res_ge_constt ?natr_ge0 ?Cnat_cfdot_char_irr //.
by rewrite 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 | constt_Res_trans | |
cfRepr_morphimn (rfG : mx_representation algC (f @* G) n) sGD :
cfRepr (morphim_repr rfG sGD) = cfMorph (cfRepr rfG).
Proof.
apply/cfun_inP=> x Gx; have Dx: x \in D := subsetP sGD x Gx.
by rewrite cfMorphE // !cfunE ?mem_morphim ?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 | cfRepr_morphim | |
cfMorph_charchi : chi \is a character -> cfMorph chi \is a character.
Proof.
have [sGD /char_reprP[rfG ->] | outGD Nchi] := boolP (G \subset D); last first.
by rewrite cfMorphEout // rpredZ_nat ?rpred1 ?Cnat_char1.
apply/char_reprP; exists (Representation (morphim_repr rfG sGD)).
by rewrite cfRepr_morphim.
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 | cfMorph_char | |
cfMorph_lin_charchi :
chi \is a linear_char -> cfMorph chi \is a linear_char.
Proof. by case/andP=> Nchi; rewrite qualifE/= cfMorph1 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 | cfMorph_lin_char | |
cfMorph_charEchi :
G \subset D -> (cfMorph chi \is a character) = (chi \is a character).
Proof.
move=> sGD; apply/idP/idP=> [/char_reprP[[n rG] /=Dfchi] | /cfMorph_char//].
pose H := 'ker_G f; have kerH: H \subset rker rG.
by rewrite -cfker_repr -Dfchi cfker_morph // setIS // ker_sub_pre.
have nHG: G \subset 'N(H) by rewrite normsI // (subset_trans sGD) ?ker_norm.
have [h injh im_h] := first_isom_loc f sGD; rewrite -/H in h injh im_h.
have DfG: invm injh @*^-1 (G / H) == (f @* G)%g by rewrite morphpre_invm im_h.
pose rfG := eqg_repr (morphpre_repr _ (quo_repr kerH nHG)) DfG.
apply/char_reprP; exists (Representation rfG).
apply/cfun_inP=> _ /morphimP[x Dx Gx ->]; rewrite -cfMorphE // Dfchi !cfunE Gx.
pose xH := coset H x; have GxH: xH \in (G / H)%g by apply: mem_quotient.
suffices Dfx: f x = h xH by rewrite mem_morphim //= Dfx invmE ?quo_repr_coset.
by apply/set1_inj; rewrite -?morphim_set1 ?im_h ?(subsetP nHG) ?sub1set.
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 | cfMorph_charE | |
cfMorph_lin_charEchi :
G \subset D -> (cfMorph chi \is a linear_char) = (chi \is a linear_char).
Proof. by rewrite qualifE/= cfMorph1 => /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 | cfMorph_lin_charE | |
cfMorph_irrchi :
G \subset D -> (cfMorph chi \in irr G) = (chi \in irr (f @* G)).
Proof. by move=> sGD; rewrite !irrEchar cfMorph_charE // cfMorph_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 | cfMorph_irr | |
morph_Iirri := cfIirr (cfMorph 'chi[f @* 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 | morph_Iirr | |
morph_Iirr0: morph_Iirr 0 = 0.
Proof. by rewrite /morph_Iirr irr0 rmorph1 -irr0 irrK. Qed.
Hypothesis sGD : G \subset D. | 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 | morph_Iirr0 | |
morph_IirrEi : 'chi_(morph_Iirr i) = cfMorph 'chi_i.
Proof. by rewrite cfIirrE ?cfMorph_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 | morph_IirrE | |
morph_Iirr_inj: injective morph_Iirr.
Proof.
by move=> i j eq_ij; apply/irr_inj/cfMorph_inj; rewrite // -!morph_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 | morph_Iirr_inj | |
morph_Iirr_eq0i : (morph_Iirr i == 0) = (i == 0).
Proof. by rewrite -!irr_eq1 morph_IirrE cfMorph_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 | morph_Iirr_eq0 | |
cfIsom_charchi :
(cfIsom isoGR chi \is a character) = (chi \is a character).
Proof.
rewrite [cfIsom _]locked_withE cfMorph_charE //.
by rewrite (isom_im (isom_sym _)) cfRes_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 | cfIsom_char | |
cfIsom_lin_charchi :
(cfIsom isoGR chi \is a linear_char) = (chi \is a linear_char).
Proof. by rewrite qualifE/= cfIsom_char cfIsom1. 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 | cfIsom_lin_char | |
cfIsom_irrchi : (cfIsom isoGR chi \in irr R) = (chi \in irr G).
Proof. by rewrite !irrEchar cfIsom_char cfIsom_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 | cfIsom_irr | |
isom_Iirri := cfIirr (cfIsom isoGR 'chi_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 | isom_Iirr | |
isom_IirrEi : 'chi_(isom_Iirr i) = cfIsom isoGR 'chi_i.
Proof. by rewrite cfIirrE ?cfIsom_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 | isom_IirrE | |
isom_Iirr_inj: injective isom_Iirr.
Proof.
by move=> i j eqij; apply/irr_inj/(cfIsom_inj isoGR); rewrite -!isom_IirrE eqij.
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 | isom_Iirr_inj | |
isom_Iirr_eq0i : (isom_Iirr i == 0) = (i == 0).
Proof. by rewrite -!irr_eq1 isom_IirrE cfIsom_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 | isom_Iirr_eq0 | |
isom_Iirr0: isom_Iirr 0 = 0.
Proof. by apply/eqP; rewrite isom_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 | isom_Iirr0 | |
isom_IirrK: cancel (isom_Iirr isoGR) (isom_Iirr (isom_sym isoGR)).
Proof. by move=> i; apply: irr_inj; rewrite !isom_IirrE cfIsomK. 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 | isom_IirrK | |
isom_IirrKV: cancel (isom_Iirr (isom_sym isoGR)) (isom_Iirr isoGR).
Proof. by move=> i; apply: irr_inj; rewrite !isom_IirrE cfIsomKV. 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 | isom_IirrKV | |
cfSdprod_charchi :
(cfSdprod defG chi \is a character) = (chi \is a character).
Proof. by rewrite unlock cfMorph_charE // cfIsom_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 | cfSdprod_char | |
cfSdprod_lin_charchi :
(cfSdprod defG chi \is a linear_char) = (chi \is a linear_char).
Proof. by rewrite qualifE/= cfSdprod_char cfSdprod1. 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 | cfSdprod_lin_char | |
cfSdprod_irrchi : (cfSdprod defG chi \in irr G) = (chi \in irr H).
Proof. by rewrite !irrEchar cfSdprod_char cfSdprod_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 | cfSdprod_irr | |
sdprod_Iirrj := cfIirr (cfSdprod defG 'chi_j). | 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 | sdprod_Iirr | |
sdprod_IirrEj : 'chi_(sdprod_Iirr j) = cfSdprod defG 'chi_j.
Proof. by rewrite cfIirrE ?cfSdprod_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 | sdprod_IirrE | |
sdprod_IirrK: cancel sdprod_Iirr (Res_Iirr H).
Proof. by move=> j; rewrite /Res_Iirr sdprod_IirrE cfSdprodK 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 | sdprod_IirrK | |
sdprod_Iirr_inj: injective sdprod_Iirr.
Proof. exact: can_inj sdprod_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 | sdprod_Iirr_inj | |
sdprod_Iirr_eq0i : (sdprod_Iirr i == 0) = (i == 0).
Proof. by rewrite -!irr_eq1 sdprod_IirrE cfSdprod_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 | sdprod_Iirr_eq0 | |
sdprod_Iirr0: sdprod_Iirr 0 = 0.
Proof. by apply/eqP; rewrite sdprod_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 | sdprod_Iirr0 | |
Res_sdprod_irrphi :
K \subset cfker phi -> phi \in irr G -> 'Res phi \in irr H.
Proof.
move=> kerK /irrP[i Dphi]; rewrite irrEchar -(cfSdprod_iso defG).
by rewrite cfRes_sdprodK // Dphi cfnorm_irr 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 | Res_sdprod_irr | |
sdprod_Res_IirrEi :
K \subset cfker 'chi[G]_i -> 'chi_(Res_Iirr H i) = 'Res 'chi_i.
Proof. by move=> kerK; rewrite cfIirrE ?Res_sdprod_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 | sdprod_Res_IirrE | |
sdprod_Res_IirrKi :
K \subset cfker 'chi_i -> sdprod_Iirr (Res_Iirr H i) = i.
Proof.
by move=> kerK; rewrite /sdprod_Iirr sdprod_Res_IirrE ?cfRes_sdprodK ?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 | sdprod_Res_IirrK | |
cfDprodKl_abelianj : abelian H -> cancel ((cfDprod KxH)^~ 'chi_j) 'Res.
Proof. by move=> cHH; apply: cfDprodKl; apply/lin_char1/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 | cfDprodKl_abelian | |
cfDprodKr_abeliani : abelian K -> cancel (cfDprod KxH 'chi_i) 'Res.
Proof. by move=> cKK; apply: cfDprodKr; apply/lin_char1/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 | cfDprodKr_abelian | |
cfDprodl_charphi :
(cfDprodl KxH phi \is a character) = (phi \is a character).
Proof. exact: cfSdprod_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 | cfDprodl_char | |
cfDprodr_charpsi :
(cfDprodr KxH psi \is a character) = (psi \is a character).
Proof. exact: cfSdprod_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 | cfDprodr_char | |
cfDprod_charphi psi :
phi \is a character -> psi \is a character ->
cfDprod KxH phi psi \is a character.
Proof. by move=> Nphi Npsi; rewrite rpredM ?cfDprodl_char ?cfDprodr_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 | cfDprod_char | |
cfDprod_eq1phi psi :
phi \is a character -> psi \is a character ->
(cfDprod KxH phi psi == 1) = (phi == 1) && (psi == 1).
Proof.
move=> /Cnat_char1 Nphi /Cnat_char1 Npsi.
apply/eqP/andP=> [phi_psi_1 | [/eqP-> /eqP->]]; last by rewrite cfDprod_cfun1.
have /andP[/eqP phi1 /eqP psi1]: (phi 1%g == 1) && (psi 1%g == 1).
by rewrite -natr_mul_eq1 // -(cfDprod1 KxH) phi_psi_1 cfun11.
rewrite -[phi](cfDprodKl KxH psi1) -{2}[psi](cfDprodKr KxH phi1) phi_psi_1.
by rewrite !rmorph1.
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 | cfDprod_eq1 | |
cfDprodl_lin_charphi :
(cfDprodl KxH phi \is a linear_char) = (phi \is a linear_char).
Proof. exact: cfSdprod_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 | cfDprodl_lin_char | |
cfDprodr_lin_charpsi :
(cfDprodr KxH psi \is a linear_char) = (psi \is a linear_char).
Proof. exact: cfSdprod_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 | cfDprodr_lin_char | |
cfDprod_lin_charphi psi :
phi \is a linear_char -> psi \is a linear_char ->
cfDprod KxH phi psi \is a linear_char.
Proof. by move=> Nphi Npsi; rewrite rpredM ?cfSdprod_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 | cfDprod_lin_char | |
cfDprodl_irrchi : (cfDprodl KxH chi \in irr G) = (chi \in irr K).
Proof. exact: cfSdprod_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 | cfDprodl_irr | |
cfDprodr_irrchi : (cfDprodr KxH chi \in irr G) = (chi \in irr H).
Proof. exact: cfSdprod_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 | cfDprodr_irr | |
dprodl_Iirri := cfIirr (cfDprodl KxH 'chi_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 | dprodl_Iirr | |
dprodl_IirrEi : 'chi_(dprodl_Iirr i) = cfDprodl KxH 'chi_i.
Proof. exact: sdprod_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 | dprodl_IirrE | |
dprodl_IirrK: cancel dprodl_Iirr (Res_Iirr K).
Proof. exact: sdprod_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 | dprodl_IirrK | |
dprodl_Iirr_eq0i : (dprodl_Iirr i == 0) = (i == 0).
Proof. exact: sdprod_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 | dprodl_Iirr_eq0 | |
dprodl_Iirr0: dprodl_Iirr 0 = 0.
Proof. exact: sdprod_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 | dprodl_Iirr0 | |
dprodr_Iirrj := cfIirr (cfDprodr KxH 'chi_j). | 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 | dprodr_Iirr | |
dprodr_IirrEj : 'chi_(dprodr_Iirr j) = cfDprodr KxH 'chi_j.
Proof. exact: sdprod_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 | dprodr_IirrE | |
dprodr_IirrK: cancel dprodr_Iirr (Res_Iirr H).
Proof. exact: sdprod_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 | dprodr_IirrK | |
dprodr_Iirr_eq0j : (dprodr_Iirr j == 0) = (j == 0).
Proof. exact: sdprod_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 | dprodr_Iirr_eq0 | |
dprodr_Iirr0: dprodr_Iirr 0 = 0.
Proof. exact: sdprod_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 | dprodr_Iirr0 | |
cfDprod_irri j : cfDprod KxH 'chi_i 'chi_j \in irr G.
Proof.
rewrite irrEchar cfDprod_char ?irr_char //=.
by rewrite cfdot_dprod !cfdot_irr !eqxx mul1r.
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 | cfDprod_irr | |
dprod_Iirrij := cfIirr (cfDprod KxH 'chi_ij.1 'chi_ij.2). | 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 | dprod_Iirr | |
dprod_IirrEi j : 'chi_(dprod_Iirr (i, j)) = cfDprod KxH 'chi_i 'chi_j.
Proof. by rewrite cfIirrE ?cfDprod_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 | dprod_IirrE | |
dprod_IirrEli : 'chi_(dprod_Iirr (i, 0)) = cfDprodl KxH 'chi_i.
Proof. by rewrite dprod_IirrE /cfDprod irr0 rmorph1 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 | dprod_IirrEl | |
dprod_IirrErj : 'chi_(dprod_Iirr (0, j)) = cfDprodr KxH 'chi_j.
Proof. by rewrite dprod_IirrE /cfDprod irr0 rmorph1 mul1r. 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 | dprod_IirrEr | |
dprod_Iirr_inj: injective dprod_Iirr.
Proof.
move=> [i1 j1] [i2 j2] /eqP; rewrite -[_ == _]oddb -(@natrK algC (_ == _)).
rewrite -cfdot_irr !dprod_IirrE cfdot_dprod !cfdot_irr -natrM mulnb.
by rewrite natrK oddb -xpair_eqE => /eqP.
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 | dprod_Iirr_inj | |
dprod_Iirr0: dprod_Iirr (0, 0) = 0.
Proof. by apply/irr_inj; rewrite dprod_IirrE !irr0 cfDprod_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 | dprod_Iirr0 | |
dprod_Iirr0lj : dprod_Iirr (0, j) = dprodr_Iirr j.
Proof.
by apply/irr_inj; rewrite dprod_IirrE irr0 dprodr_IirrE cfDprod_cfun1l.
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 | dprod_Iirr0l | |
dprod_Iirr0ri : dprod_Iirr (i, 0) = dprodl_Iirr i.
Proof.
by apply/irr_inj; rewrite dprod_IirrE irr0 dprodl_IirrE cfDprod_cfun1r.
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 | dprod_Iirr0r | |
dprod_Iirr_eq0i j : (dprod_Iirr (i, j) == 0) = (i == 0) && (j == 0).
Proof. by rewrite -xpair_eqE -(inj_eq dprod_Iirr_inj) dprod_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 | dprod_Iirr_eq0 | |
cfdot_dprod_irri1 i2 j1 j2 :
'['chi_(dprod_Iirr (i1, j1)), 'chi_(dprod_Iirr (i2, j2))]
= ((i1 == i2) && (j1 == j2))%:R.
Proof. by rewrite cfdot_irr (inj_eq dprod_Iirr_inj). 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_dprod_irr | |
dprod_Iirr_ontok : k \in codom dprod_Iirr.
Proof.
set D := codom _; have Df: dprod_Iirr _ \in D := codom_f dprod_Iirr _.
have: 'chi_k 1%g ^+ 2 != 0 by rewrite mulf_neq0 ?irr1_neq0.
apply: contraR => notDk; move/eqP: (irr_sum_square G).
rewrite (bigID [in D]) (reindex _ (bij_on_codom dprod_Iirr_inj (0, 0))) /=.
have ->: #|G|%:R = \sum_i \sum_j 'chi_(dprod_Iirr (i, j)) 1%g ^+ 2.
rewrite -(dprod_card KxH) natrM.
do 2![rewrite -irr_sum_square (mulr_suml, mulr_sumr); apply: eq_bigr => ? _].
by rewrite dprod_IirrE -exprMn -{3}(mulg1 1%g) cfDprodE.
rewrite (eq_bigl _ _ Df) pair_bigA addrC -subr_eq0 addrK.
by move/eqP/psumr_eq0P=> -> //= i _; rewrite irr1_degree -natrX ler0n.
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 | dprod_Iirr_onto | |
inv_dprod_Iirri := iinv (dprod_Iirr_onto 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 | inv_dprod_Iirr | |
dprod_IirrK: cancel dprod_Iirr inv_dprod_Iirr.
Proof. by move=> p; apply: (iinv_f dprod_Iirr_inj). 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 | dprod_IirrK | |
inv_dprod_IirrK: cancel inv_dprod_Iirr dprod_Iirr.
Proof. by move=> i; apply: f_iinv. 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 | inv_dprod_IirrK | |
inv_dprod_Iirr0: inv_dprod_Iirr 0 = (0, 0).
Proof. by apply/(canLR dprod_IirrK); rewrite dprod_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 | inv_dprod_Iirr0 | |
dprod_IirrC(gT : finGroupType) (G K H : {group gT})
(KxH : K \x H = G) (HxK : H \x K = G) i j :
dprod_Iirr KxH (i, j) = dprod_Iirr HxK (j, i).
Proof. by apply: irr_inj; rewrite !dprod_IirrE; apply: cfDprodC. 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 | dprod_IirrC | |
cfBigdprodi_chari (phi : 'CF(A i)) :
phi \is a character -> cfBigdprodi defG phi \is a character.
Proof. by move=> Nphi; rewrite cfDprodl_char cfRes_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_char | |
cfBigdprodi_charEi (phi : 'CF(A i)) :
P i -> (cfBigdprodi defG phi \is a character) = (phi \is a character).
Proof. by move=> Pi; rewrite cfDprodl_char Pi cfRes_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 | cfBigdprodi_charE | |
cfBigdprod_charphi :
(forall i, P i -> phi i \is a character) ->
cfBigdprod defG phi \is a character.
Proof.
by move=> Nphi; apply: rpred_prod => i /Nphi; apply: 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_char |
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