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 |
|---|---|---|---|---|---|---|
cfRepr_standardn (rG : mx_representation algC G n) :
cfRepr (standard_grepr rG)
= \sum_i (standard_irr_coef rG (W i))%:R *: 'chi_i.
Proof.
rewrite cfRepr_dsum (reindex _ (socle_of_Iirr_bij _)).
by apply: eq_bigr => i _; rewrite scaler_nat cfRepr_muln irrRepr.
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_standard | |
cfRepr_injn1 n2 rG1 rG2 :
@cfRepr _ G n1 rG1 = @cfRepr _ G n2 rG2 -> mx_rsim rG1 rG2.
Proof.
move=> eq_repr12; pose c i : algC := (standard_irr_coef _ (W i))%:R.
have [rsim1 rsim2] := (mx_rsim_standard rG1, mx_rsim_standard rG2).
apply: mx_rsim_trans (rsim1) (mx_rsim_sym _).
suffices ->: standard_grepr rG1 = standard... | 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_inj | |
cfRepr_rsimPn1 n2 rG1 rG2 :
reflect (mx_rsim rG1 rG2) (@cfRepr _ G n1 rG1 == @cfRepr _ G n2 rG2).
Proof. by apply: (iffP eqP) => [/cfRepr_inj | /cfRepr_sim]. 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_rsimP | |
irr_reprPxi :
reflect (exists2 rG : representation _ G, mx_irreducible rG & xi = cfRepr rG)
(xi \in irr G).
Proof.
apply: (iffP (irrP xi)) => [[i ->] | [[n rG] irr_rG ->]].
by exists (Representation 'Chi_i); [apply: socle_irr | rewrite irrRepr].
exists (irr_of_socle (irr_comp sG rG)); rewrite -irrRepr irr... | 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_reprP | |
Wedderburn_id_expansioni :
'e_i = #|G|%:R^-1 *: (\sum_(x in G) 'chi_i 1%g * 'chi_i x^-1%g *: aG x).
Proof.
have Rei: ('e_i \in 'R_i)%MS by apply: Wedderburn_id_mem.
have /envelop_mxP[a def_e]: ('e_i \in R_G)%MS; last rewrite -/aG in def_e.
by move: Rei; rewrite genmxE mem_sub_gring => /andP[].
apply: canRL (scalerK... | 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 | Wedderburn_id_expansion | |
character_pred{G : {set gT}} :=
fun phi : 'CF(G) => [forall i, coord (irr G) i phi \in Num.nat].
Arguments character_pred _ _ /. | 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 | character_pred | |
character{G : {set gT}} := [qualify a phi | @character_pred G phi].
Variable G : {group gT}.
Implicit Types (phi chi xi : 'CF(G)) (i : Iirr G). | 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 | character | |
irr_chari : 'chi_i \is a character.
Proof. by apply/forallP=> j; rewrite (tnth_nth 0) coord_free ?irr_free. 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_char | |
cfun1_char: (1 : 'CF(G)) \is a character.
Proof. by rewrite -irr0 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 | cfun1_char | |
cfun0_char: (0 : 'CF(G)) \is a character.
Proof. by apply/forallP=> i; rewrite linear0 rpred0. Qed.
Fact add_char : addr_closed (@character G).
Proof.
split=> [|chi xi /forallP-Nchi /forallP-Nxi]; first exact: cfun0_char.
by apply/forallP=> i; rewrite linearD rpredD /=.
Qed.
HB.instance Definition _ := GRing.isAddClose... | 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 | cfun0_char | |
char_sum_irrP{phi} :
reflect (exists n, phi = \sum_i (n i)%:R *: 'chi_i) (phi \is a character).
Proof.
apply: (iffP idP)=> [/forallP-Nphi | [n ->]]; last first.
by apply: rpred_sum => i _; rewrite scaler_nat rpredMn // irr_char.
do [have [a ->] := cfun_irr_sum phi] in Nphi *; exists (Num.truncn \o a).
apply: eq_big... | 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_sum_irrP | |
char_sum_irrchi :
chi \is a character -> {r | chi = \sum_(i <- r) 'chi_i}.
Proof.
move=> Nchi; apply: sig_eqW; case/char_sum_irrP: Nchi => n {chi}->.
elim/big_rec: _ => [|i _ _ [r ->]]; first by exists nil; rewrite big_nil.
exists (ncons (n i) i r); rewrite scaler_nat.
by elim: {n}(n i) => [|n IHn]; rewrite ?add0r //... | 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_sum_irr | |
Cnat_char1chi : chi \is a character -> chi 1%g \in Num.nat.
Proof.
case/char_sum_irr=> r ->{chi}.
by elim/big_rec: _ => [|i chi _ Nchi1]; rewrite cfunE ?rpredD // Cnat_irr1.
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 | Cnat_char1 | |
char1_ge0chi : chi \is a character -> 0 <= chi 1%g.
Proof. by move/Cnat_char1/natr_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 | char1_ge0 | |
char1_eq0chi : chi \is a character -> (chi 1%g == 0) = (chi == 0).
Proof.
case/char_sum_irr=> r ->; apply/idP/idP=> [|/eqP->]; last by rewrite cfunE.
case: r => [|i r]; rewrite ?big_nil // sum_cfunE big_cons.
rewrite paddr_eq0 ?sumr_ge0 => // [||j _]; rewrite 1?ltW ?irr1_gt0 //.
by rewrite (negbTE (irr1_neq0 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 | char1_eq0 | |
char1_gt0chi : chi \is a character -> (0 < chi 1%g) = (chi != 0).
Proof. by move=> Nchi; rewrite -char1_eq0 // natr_gt0 ?Cnat_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 | char1_gt0 | |
char_reprPphi :
reflect (exists rG : representation algC G, phi = cfRepr rG)
(phi \is a character).
Proof.
apply: (iffP char_sum_irrP) => [[n ->] | [[n rG] ->]]; last first.
exists (fun i => standard_irr_coef rG (socle_of_Iirr i)).
by rewrite -cfRepr_standard (cfRepr_sim (mx_rsim_standard rG)).
exists (... | 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_reprP | |
cfRepr_charn (rG : reprG n) : cfRepr rG \is a character.
Proof. by apply/char_reprP; exists (Representation rG). 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_char | |
cfReg_char: cfReg G \is a character.
Proof. by rewrite -cfReprReg cfRepr_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 | cfReg_char | |
cfRepr_prodn1 n2 (rG1 : reprG n1) (rG2 : reprG n2) :
cfRepr rG1 * cfRepr rG2 = cfRepr (prod_repr rG1 rG2).
Proof. by apply/cfun_inP=> x Gx; rewrite !cfunE /= Gx mxtrace_prod. 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_prod | |
mul_char: mulr_closed (@character G).
Proof.
split=> [|_ _ /char_reprP[rG1 ->] /char_reprP[rG2 ->]]; first exact: cfun1_char.
apply/char_reprP; exists (Representation (prod_repr rG1 rG2)).
by rewrite cfRepr_prod.
Qed.
HB.instance Definition _ := GRing.isMulClosed.Build (classfun G) character_pred
mul_char. | 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 | mul_char | |
cfRepr_mapu n (rG : mx_representation algC G n) :
cfRepr (map_repr u rG) = cfAut u (cfRepr rG).
Proof. by apply/cfun_inP=> x Gx; rewrite !cfunE Gx map_reprE trace_map_mx. 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_map | |
cfAut_charu chi : (cfAut u chi \is a character) = (chi \is a character).
Proof.
without loss /char_reprP[rG ->]: u chi / chi \is a character.
by move=> IHu; apply/idP/idP=> ?; first rewrite -(cfAutK u chi); rewrite IHu.
rewrite cfRepr_char; apply/char_reprP.
by exists (Representation (map_repr u rG)); rewrite cfRepr_... | 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_char | |
cfConjC_charchi : (chi^*%CF \is a character) = (chi \is a character).
Proof. exact: cfAut_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 | cfConjC_char | |
cfAut_char1u (chi : 'CF(G)) :
chi \is a character -> cfAut u chi 1%g = chi 1%g.
Proof. by move/Cnat_char1=> Nchi1; rewrite cfunE /= aut_natr. 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_char1 | |
cfAut_irr1u i : (cfAut u 'chi[G]_i) 1%g = 'chi_i 1%g.
Proof. exact: cfAut_char1 (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 | cfAut_irr1 | |
cfConjC_char1(chi : 'CF(G)) :
chi \is a character -> chi^*%CF 1%g = chi 1%g.
Proof. exact: cfAut_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 | cfConjC_char1 | |
cfConjC_irr1u i : ('chi[G]_i)^*%CF 1%g = 'chi_i 1%g.
Proof. exact: cfAut_irr1. 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_irr1 | |
linear_char_pred{B : {set gT}} :=
fun phi : 'CF(B) => (phi \is a character) && (phi 1%g == 1).
Arguments linear_char_pred _ _ /. | 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 | linear_char_pred | |
linear_char{B : {set gT}} :=
[qualify a phi | @linear_char_pred B 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 | linear_char | |
lin_char1: xi 1%g = 1.
Proof. by case/andP: CFxi => _ /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 | lin_char1 | |
lin_charW: xi \is a character.
Proof. by case/andP: CFxi. 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_charW | |
cfun1_lin_char: (1 : 'CF(G)) \is a linear_char.
Proof. by rewrite qualifE/= cfun1_char /= cfun11. 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 | cfun1_lin_char | |
lin_charM: {in G &, {morph xi : x y / (x * y)%g >-> x * y}}.
Proof.
move=> x y Gx Gy; case/andP: CFxi => /char_reprP[[n rG] -> /=].
rewrite cfRepr1 pnatr_eq1 => /eqP n1; rewrite {n}n1 in rG *.
rewrite !cfunE Gx Gy groupM //= !mulr1n repr_mxM //.
by rewrite [rG x]mx11_scalar [rG y]mx11_scalar -scalar_mxM !mxtrace_scalar... | 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_charM | |
lin_char_prodI r (P : pred I) (x : I -> gT) :
(forall i, P i -> x i \in G) ->
xi (\prod_(i <- r | P i) x i)%g = \prod_(i <- r | P i) xi (x i).
Proof.
move=> Gx; elim/(big_load (fun y => y \in G)): _.
elim/big_rec2: _ => [|i a y Pi [Gy <-]]; first by rewrite lin_char1.
by rewrite groupM ?lin_charM ?Gx.
Qed.
Let xi... | 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_prod | |
lin_char_neq0x : x \in G -> xi x != 0.
Proof.
by move/xiMV/(congr1 (predC1 0)); rewrite /= oner_eq0 mulf_eq0 => /norP[].
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_neq0 | |
lin_charVx : x \in G -> xi x^-1%g = (xi x)^-1.
Proof. by move=> Gx; rewrite -[_^-1]mulr1 -(xiMV Gx) mulKf ?lin_char_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 | lin_charV | |
lin_charXx n : x \in G -> xi (x ^+ n)%g = xi x ^+ n.
Proof.
move=> Gx; elim: n => [|n IHn]; first exact: lin_char1.
by rewrite expgS exprS lin_charM ?groupX ?IHn.
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_charX | |
lin_char_unity_rootx : x \in G -> xi x ^+ #[x] = 1.
Proof. by move=> Gx; rewrite -lin_charX // expg_order 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_unity_root | |
normC_lin_charx : x \in G -> `|xi x| = 1.
Proof.
move=> Gx; apply/eqP; rewrite -(@pexpr_eq1 _ _ #[x]) //.
by rewrite -normrX // lin_char_unity_root ?normr1.
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 | normC_lin_char | |
lin_charV_conjx : x \in G -> xi x^-1%g = (xi x)^*.
Proof.
move=> Gx; rewrite lin_charV // invC_norm mulrC normC_lin_char //.
by rewrite expr1n divr1.
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_charV_conj | |
lin_char_irr: xi \in irr G.
Proof.
case/andP: CFxi => /char_reprP[rG ->]; rewrite cfRepr1 pnatr_eq1 => /eqP n1.
by apply/irr_reprP; exists rG => //; apply/mx_abs_irrW/linear_mx_abs_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 | lin_char_irr | |
mul_conjC_lin_char: xi * xi^*%CF = 1.
Proof.
apply/cfun_inP=> x Gx.
by rewrite !cfunE cfun1E Gx -normCK normC_lin_char ?expr1n.
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 | mul_conjC_lin_char | |
lin_char_unitr: xi \in GRing.unit.
Proof. by apply/unitrPr; exists xi^*%CF; apply: mul_conjC_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 | lin_char_unitr | |
invr_lin_char: xi^-1 = xi^*%CF.
Proof. by rewrite -[_^-1]mulr1 -mul_conjC_lin_char mulKr ?lin_char_unitr. 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 | invr_lin_char | |
fful_lin_char_inj: cfaithful xi -> {in G &, injective xi}.
Proof.
move=> fful_phi x y Gx Gy xi_xy; apply/eqP; rewrite eq_mulgV1 -in_set1.
rewrite (subsetP fful_phi) // inE groupM ?groupV //=; apply/forallP=> z.
have [Gz | G'z] := boolP (z \in G); last by rewrite !cfun0 ?groupMl ?groupV.
by rewrite -mulgA lin_charM ?xi_... | 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 | fful_lin_char_inj | |
cfAut_lin_charu (xi : 'CF(G)) :
(cfAut u xi \is a linear_char) = (xi \is a linear_char).
Proof. by rewrite qualifE/= cfAut_char; apply/andb_id2l=> /cfAut_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 | cfAut_lin_char | |
cfConjC_lin_char(xi : 'CF(G)) :
(xi^*%CF \is a linear_char) = (xi \is a linear_char).
Proof. exact: cfAut_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 | cfConjC_lin_char | |
card_Iirr_abelian: abelian G -> #|Iirr G| = #|G|.
Proof. by rewrite card_ord NirrE card_classes_abelian => /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 | card_Iirr_abelian | |
card_Iirr_cyclic: cyclic G -> #|Iirr G| = #|G|.
Proof. by move/cyclic_abelian/card_Iirr_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 | card_Iirr_cyclic | |
char_abelianP:
reflect (forall i : Iirr G, 'chi_i \is a linear_char) (abelian G).
Proof.
apply: (iffP idP) => [cGG i | CF_G].
rewrite qualifE/= irr_char /= irr1_degree.
by rewrite irr_degree_abelian //; last apply: groupC.
rewrite card_classes_abelian -NirrE -eqC_nat -irr_sum_square //.
rewrite -{1}[Nirr G]card_o... | 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_abelianP | |
irr_repr_lin_char(i : Iirr G) x :
x \in G -> 'chi_i \is a linear_char ->
irr_repr (socle_of_Iirr i) x = ('chi_i x)%:M.
Proof.
move=> Gx CFi; rewrite -irrRepr cfunE Gx.
move: (_ x); rewrite -[irr_degree _](@natrK algC) -irr1_degree lin_char1 //.
by rewrite (natrK 1) => A; rewrite trace_mx11 -mx11_scalar.
Qed.
Fact... | 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_repr_lin_char | |
irr_cyclic_lini : cyclic G -> 'chi[G]_i \is a linear_char.
Proof. by move/cyclic_abelian/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 | irr_cyclic_lin | |
irr_prime_lini : prime #|G| -> 'chi[G]_i \is a linear_char.
Proof. by move/prime_cyclic/irr_cyclic_lin. 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_lin | |
repr_rsim_diag(G : {group gT}) f (rG : mx_representation algC G f) x :
x \in G -> let chi := cfRepr rG in
exists e,
[/\ exists2 B, B \in unitmx & rG x = invmx B *m diag_mx e *m B, | 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 | repr_rsim_diag | |
char_inv(chi : 'CF(G)) x : chi \is a character -> chi x^-1%g = (chi x)^*.
Proof.
case Gx: (x \in G); last by rewrite !cfun0 ?rmorph0 ?groupV ?Gx.
by case/char_reprP=> rG ->; have [e [_ _ _]] := repr_rsim_diag rG 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_inv | |
irr_invi x : 'chi[G]_i x^-1%g = ('chi_i x)^*.
Proof. exact/char_inv/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_inv | |
generalized_orthogonality_relationy (i j : Iirr G) :
#|G|%:R^-1 * (\sum_(x in G) 'chi_i (x * y)%g * 'chi_j x^-1%g)
= (i == j)%:R * ('chi_i y / 'chi_i 1%g).
Proof.
pose W := @socle_of_Iirr _ G; pose e k := Wedderburn_id (W k).
pose aG := regular_repr algC G.
have [Gy | notGy] := boolP (y \in G); last first.
rewr... | Theorem | 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 | generalized_orthogonality_relation | |
irr_classi := enum_val (cast_ord (NirrE 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 | irr_class | |
class_IirrxG :=
cast_ord (esym (NirrE G)) (enum_rank_in (classes1 G) xG).
Local Notation c := irr_class.
Local Notation g i := (repr (c i)).
Local Notation iC := class_Iirr. | 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 | class_Iirr | |
character_table:= \matrix_(i, j) 'chi[G]_i (g j).
Local Notation X := character_table. | 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 | character_table | |
irr_classPi : c i \in classes G.
Proof. exact: enum_valP. 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_classP | |
repr_irr_classKi : g i ^: G = c i.
Proof. by case/repr_classesP: (irr_classP 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 | repr_irr_classK | |
irr_classK: cancel c iC.
Proof. by move=> i; rewrite /iC enum_valK_in cast_ordK. 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_classK | |
class_IirrK: {in classes G, cancel iC c}.
Proof. by move=> xG GxG; rewrite /c cast_ordKV enum_rankK_in. 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 | class_IirrK | |
reindex_irr_classR idx (op : @Monoid.com_law R idx) F :
\big[op/idx]_(xG in classes G) F xG = \big[op/idx]_i F (c i).
Proof.
rewrite (reindex c); first by apply: eq_bigl => i; apply: enum_valP.
by exists iC; [apply: in1W; apply: irr_classK | apply: class_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 | reindex_irr_class | |
character_table_unit: X \in unitmx.
Proof. by case/mulmx1_unit: XX'_1. Qed.
Let uX := character_table_unit. | 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 | character_table_unit | |
second_orthogonality_relationx y :
y \in G ->
\sum_i 'chi[G]_i x * ('chi_i y)^* = #|'C_G[x]|%:R *+ (x \in y ^: G).
Proof.
move=> Gy; pose i_x := iC (x ^: G); pose i_y := iC (y ^: G).
have [Gx | notGx] := boolP (x \in G); last first.
rewrite (contraNF (subsetP _ x) notGx) ?class_subG ?big1 // => i _.
by rewrit... | Theorem | 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 | second_orthogonality_relation | |
eq_irr_mem_classPx y :
y \in G -> reflect (forall i, 'chi[G]_i x = 'chi_i y) (x \in y ^: G).
Proof.
move=> Gy; apply: (iffP idP) => [/imsetP[z Gz ->] i | xGy]; first exact: cfunJ.
have Gx: x \in G.
congr is_true: Gy; apply/eqP; rewrite -(can_eq oddb) -eqC_nat -!cfun1E.
by rewrite -irr0 xGy.
congr is_true: (class_... | 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 | eq_irr_mem_classP | |
card_afix_irr_classes(ito : action A (Iirr G)) (cto : action A _) a :
a \in A -> [acts A, on classes G | cto] ->
(forall i x y, x \in G -> y \in cto (x ^: G) a ->
'chi_i x = 'chi_(ito i a) y) ->
#|'Fix_ito[a]| = #|'Fix_(classes G | cto)[a]|.
Proof.
move=> Aa actsAG stabAchi; apply/eqP; rewrite -eqC_nat;... | 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_afix_irr_classes | |
cfdot_irri j : '['chi_i, 'chi_j]_G = (i == j)%:R.
Proof.
rewrite -first_orthogonality_relation; congr (_ * _).
by apply: eq_bigr => x Gx; rewrite irr_inv.
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_irr | |
cfnorm_irri : '['chi[G]_i] = 1.
Proof. by rewrite cfdot_irr eqxx. 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_irr | |
irr_orthonormal: orthonormal (irr G).
Proof.
apply/orthonormalP; split; first exact: free_uniq (irr_free G).
move=> _ _ /irrP[i ->] /irrP[j ->].
by rewrite cfdot_irr (inj_eq irr_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 | irr_orthonormal | |
coord_cfdotphi i : coord (irr G) i phi = '[phi, 'chi_i].
Proof.
rewrite {2}(coord_basis (irr_basis G) (memvf phi)).
rewrite cfdot_suml (bigD1 i) // cfdotZl /= -tnth_nth cfdot_irr eqxx mulr1.
rewrite big1 ?addr0 // => j neq_ji; rewrite cfdotZl /= -tnth_nth cfdot_irr.
by rewrite (negbTE neq_ji) mulr0.
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 | coord_cfdot | |
cfun_sum_cfdotphi : phi = \sum_i '[phi, 'chi_i]_G *: 'chi_i.
Proof.
rewrite {1}(coord_basis (irr_basis G) (memvf phi)).
by apply: eq_bigr => i _; rewrite coord_cfdot -tnth_nth.
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 | cfun_sum_cfdot | |
cfdot_sum_irrphi psi :
'[phi, psi]_G = \sum_i '[phi, 'chi_i] * '[psi, 'chi_i]^*.
Proof.
rewrite {1}[phi]cfun_sum_cfdot cfdot_suml; apply: eq_bigr => i _.
by rewrite cfdotZl -cfdotC.
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_sum_irr | |
Cnat_cfdot_char_irri phi :
phi \is a character -> '[phi, 'chi_i]_G \in Num.nat.
Proof. by move/forallP/(_ i); rewrite coord_cfdot. 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 | Cnat_cfdot_char_irr | |
cfdot_char_rphi chi :
chi \is a character -> '[phi, chi]_G = \sum_i '[phi, 'chi_i] * '[chi, 'chi_i].
Proof.
move=> Nchi; rewrite cfdot_sum_irr; apply: eq_bigr => i _; congr (_ * _).
by rewrite conj_natr ?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 | cfdot_char_r | |
Cnat_cfdot_charchi xi :
chi \is a character -> xi \is a character -> '[chi, xi]_G \in Num.nat.
Proof.
move=> Nchi Nxi; rewrite cfdot_char_r ?rpred_sum // => i _.
by rewrite rpredM ?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 | Cnat_cfdot_char | |
cfdotC_charchi xi :
chi \is a character-> xi \is a character -> '[chi, xi]_G = '[xi, chi].
Proof. by move=> Nchi Nxi; rewrite cfdotC conj_natr ?Cnat_cfdot_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 | cfdotC_char | |
irrEcharchi : (chi \in irr G) = (chi \is a character) && ('[chi] == 1).
Proof.
apply/irrP/andP=> [[i ->] | [Nchi]]; first by rewrite irr_char cfnorm_irr.
rewrite cfdot_sum_irr => /eqP/natr_sum_eq1[i _| i [_ ci1 cj0]].
by rewrite rpredM // ?conj_natr ?Cnat_cfdot_char_irr.
exists i; rewrite [chi]cfun_sum_cfdot (bigD1 i... | 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 | irrEchar | |
irrWcharchi : chi \in irr G -> chi \is a character.
Proof. by rewrite irrEchar => /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 | irrWchar | |
irrWnormchi : chi \in irr G -> '[chi] = 1.
Proof. by rewrite irrEchar => /andP[_ /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 | irrWnorm | |
mul_lin_irrxi chi :
xi \is a linear_char -> chi \in irr G -> xi * chi \in irr G.
Proof.
move=> Lxi; rewrite !irrEchar => /andP[Nphi /eqP <-].
rewrite rpredM // ?lin_charW //=; apply/eqP; congr (_ * _).
apply: eq_bigr=> x Gx; rewrite !cfunE rmorphM/= mulrACA -(lin_charV_conj Lxi)//.
by rewrite -lin_charM ?groupV // mu... | 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 | mul_lin_irr | |
eq_scaled_irra b i j :
(a *: 'chi[G]_i == b *: 'chi_j) = (a == b) && ((a == 0) || (i == j)).
Proof.
apply/eqP/andP=> [|[/eqP-> /pred2P[]-> //]]; last by rewrite !scale0r.
move/(congr1 (cfdotr 'chi__)) => /= eq_ai_bj.
move: {eq_ai_bj}(eq_ai_bj i) (esym (eq_ai_bj j)); rewrite !cfdotZl !cfdot_irr.
by rewrite !mulr_natr ... | 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 | eq_scaled_irr | |
eq_signed_irr(s t : bool) i j :
((-1) ^+ s *: 'chi[G]_i == (-1) ^+ t *: 'chi_j) = (s == t) && (i == j).
Proof. by rewrite eq_scaled_irr signr_eq0 (inj_eq signr_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 | eq_signed_irr | |
eq_scale_irra (i j : Iirr G) :
(a *: 'chi_i == a *: 'chi_j) = (a == 0) || (i == j).
Proof. by rewrite eq_scaled_irr eqxx. 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 | eq_scale_irr | |
eq_addZ_irra b (i j r t : Iirr G) :
(a *: 'chi_i + b *: 'chi_j == a *: 'chi_r + b *: 'chi_t)
= [|| [&& (a == 0) || (i == r) & (b == 0) || (j == t)],
[&& i == t, j == r & a == b] | [&& i == j, r == t & a == - b]].
Proof.
rewrite -!eq_scale_irr; apply/eqP/idP; last first.
case/orP; first by case/andP=> /e... | 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 | eq_addZ_irr | |
eq_subZnat_irr(a b : nat) (i j r t : Iirr G) :
(a%:R *: 'chi_i - b%:R *: 'chi_j == a%:R *: 'chi_r - b%:R *: 'chi_t)
= [|| a == 0 | i == r] && [|| b == 0 | j == t]
|| [&& i == j, r == t & a == b].
Proof.
rewrite -!scaleNr eq_addZ_irr oppr_eq0 opprK -addr_eq0 -natrD eqr_nat.
by rewrite !pnatr_eq0 addn_eq0; ca... | 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 | eq_subZnat_irr | |
char1_ge_norm(chi : 'CF(G)) x :
chi \is a character -> `|chi x| <= chi 1%g.
Proof.
case/char_reprP=> rG ->; case Gx: (x \in G); last first.
by rewrite cfunE cfRepr1 Gx normr0 ler0n.
by have [e [_ _ []]] := repr_rsim_diag rG 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 | char1_ge_norm | |
max_cfRepr_norm_scalarn (rG : mx_representation algC G n) x :
x \in G -> `|cfRepr rG x| = cfRepr rG 1%g ->
exists2 c, `|c| = 1 & rG x = c%:M.
Proof.
move=> Gx; have [e [[B uB def_x] [_ e1] [-> _] _]] := repr_rsim_diag rG Gx.
rewrite cfRepr1 -[n in n%:R]card_ord -sumr_const -(eq_bigr _ (in1W e1)).
case/normC_sum... | 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 | max_cfRepr_norm_scalar | |
max_cfRepr_mx1n (rG : mx_representation algC G n) x :
x \in G -> cfRepr rG x = cfRepr rG 1%g -> rG x = 1%:M.
Proof.
move=> Gx kerGx; have [|c _ def_x] := @max_cfRepr_norm_scalar n rG x Gx.
by rewrite kerGx cfRepr1 normr_nat.
move/eqP: kerGx; rewrite cfRepr1 cfunE Gx {rG}def_x mxtrace_scalar.
case: n => [_|n]; firs... | 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 | max_cfRepr_mx1 | |
irr_constt(B : {set gT}) phi := [pred i | '[phi, 'chi_i]_B != 0]. | 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 | irr_constt | |
irr_consttEi phi : (i \in irr_constt phi) = ('[phi, 'chi_i]_G != 0).
Proof. by []. 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_consttE | |
constt_charP(i : Iirr G) chi :
chi \is a character ->
reflect (exists2 chi', chi' \is a character & chi = 'chi_i + chi')
(i \in irr_constt chi).
Proof.
move=> Nchi; apply: (iffP idP) => [i_in_chi| [chi' Nchi' ->]]; last first.
rewrite inE /= cfdotDl cfdot_irr eqxx -(eqP (Cnat_cfdot_char_irr i Nchi')).... | 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_charP | |
cfun_sum_constt(phi : 'CF(G)) :
phi = \sum_(i in irr_constt phi) '[phi, 'chi_i] *: 'chi_i.
Proof.
rewrite {1}[phi]cfun_sum_cfdot (bigID [pred i | '[phi, 'chi_i] == 0]) /=.
by rewrite big1 ?add0r // => i /eqP->; rewrite scale0r.
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 | cfun_sum_constt | |
neq0_has_constt(phi : 'CF(G)) :
phi != 0 -> exists i, i \in irr_constt phi.
Proof.
move=> nz_phi; apply/existsP; apply: contra nz_phi => /pred0P phi0.
by rewrite [phi]cfun_sum_constt big_pred0.
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 | neq0_has_constt | |
constt_irri : irr_constt 'chi[G]_i =i pred1 i.
Proof.
by move=> j; rewrite !inE cfdot_irr pnatr_eq0 (eq_sym j); case: (i == 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 | constt_irr | |
char1_ge_constt(i : Iirr G) chi :
chi \is a character -> i \in irr_constt chi -> 'chi_i 1%g <= chi 1%g.
Proof.
move=> {chi} _ /constt_charP[// | chi Nchi ->].
by rewrite cfunE addrC -subr_ge0 addrK 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 | char1_ge_constt | |
constt_ortho_char(phi psi : 'CF(G)) i j :
phi \is a character -> psi \is a character ->
i \in irr_constt phi -> j \in irr_constt psi ->
'[phi, psi] = 0 -> '['chi_i, 'chi_j] = 0.
Proof.
move=> _ _ /constt_charP[//|phi1 Nphi1 ->] /constt_charP[//|psi1 Npsi1 ->].
rewrite cfdot_irr; case: eqP => // -> /eqP/idPn... | 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_ortho_char |
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