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 |
|---|---|---|---|---|---|---|
subg_mx_irr: mx_irreducible rH -> mx_irreducible rG.
Proof. by apply: mxsimple_subg; apply: mxmodule1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | subg_mx_irr | |
subg_mx_abs_irr:
mx_absolutely_irreducible rH -> mx_absolutely_irreducible rG.
Proof.
rewrite /mx_absolutely_irreducible -!sub1mx => /andP[-> /submx_trans-> //].
apply/row_subP=> i; rewrite rowK /= envelop_mx_id //.
by rewrite (subsetP sHG) ?enum_valP.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | subg_mx_abs_irr | |
rfix_eqg: rfix_mx rH = rfix_mx rG. Proof. by []. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rfix_eqg | |
rstabs_eqg: rstabs rH U = rstabs rG U.
Proof. by rewrite rstabs_subg -(eqP eqGH) (setIidPr _) ?rstabs_sub. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstabs_eqg | |
mxmodule_eqg: mxmodule rH U = mxmodule rG U.
Proof. by rewrite /mxmodule rstabs_eqg -(eqP eqGH). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxmodule_eqg | |
mxsimple_eqgM : mxsimple rH M <-> mxsimple rG M.
Proof.
rewrite /mxsimple mxmodule_eqg.
split=> [] [-> -> minM]; split=> // U modU;
by apply: minM; rewrite mxmodule_eqg in modU *.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxsimple_eqg | |
eqg_mx_irr: mx_irreducible rH <-> mx_irreducible rG.
Proof. exact: mxsimple_eqg. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | eqg_mx_irr | |
eqg_mx_abs_irr:
mx_absolutely_irreducible rH = mx_absolutely_irreducible rG.
Proof.
by congr (_ && (_ == _)); rewrite /enveloping_algebra_mx /= -(eqP eqGH).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | eqg_mx_abs_irr | |
rstabs_morphpre: rstabs rGf U = f @*^-1 (rstabs rG U).
Proof. by apply/setP=> x; rewrite !inE andbA. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstabs_morphpre | |
mxmodule_morphpre: G \subset f @* D -> mxmodule rGf U = mxmodule rG U.
Proof. by move=> sGf; rewrite /mxmodule rstabs_morphpre morphpreSK. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxmodule_morphpre | |
rfix_morphpre(H : {set aT}) :
H \subset D -> (rfix_mx rGf H :=: rfix_mx rG (f @* H))%MS.
Proof.
move=> sHD; apply/eqmxP/andP; split.
by apply/rfix_mxP=> _ /morphimP[x _ Hx ->]; rewrite rfix_mx_id.
by apply/rfix_mxP=> x Hx; rewrite rfix_mx_id ?mem_morphim ?(subsetP sHD).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rfix_morphpre | |
morphpre_mx_irr:
G \subset f @* D -> (mx_irreducible rGf <-> mx_irreducible rG).
Proof.
move/mxmodule_morphpre=> modG; split=> /mx_irrP[n_gt0 irrG];
by apply/mx_irrP; split=> // U modU; apply: irrG; rewrite modG in modU *.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | morphpre_mx_irr | |
morphpre_mx_abs_irr:
G \subset f @* D ->
mx_absolutely_irreducible rGf = mx_absolutely_irreducible rG.
Proof.
move=> sGfD; congr (_ && (_ == _)); apply/eqP; rewrite mxrank_leqif_sup //.
apply/row_subP=> i; rewrite rowK.
case/morphimP: (subsetP sGfD _ (enum_valP i)) => x Dx _ def_i.
by rewrite def_i (envelop_mx_id rGf) // !inE Dx -def_i enum_valP.
apply/row_subP=> i; rewrite rowK (envelop_mx_id rG) //.
by case/morphpreP: (enum_valP i).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | morphpre_mx_abs_irr | |
rstabs_morphim: rstabs rG U = G :&: f @*^-1 rstabs rGf U.
Proof. by rewrite -rstabs_morphpre -(rstabs_subg _ sG_f'fG). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstabs_morphim | |
mxmodule_morphim: mxmodule rG U = mxmodule rGf U.
Proof. by rewrite /mxmodule rstabs_morphim subsetI subxx -sub_morphim_pre. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxmodule_morphim | |
rfix_morphim(H : {set aT}) :
H \subset D -> (rfix_mx rG H :=: rfix_mx rGf (f @* H))%MS.
Proof. exact: rfix_morphpre. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rfix_morphim | |
mxsimple_morphimM : mxsimple rG M <-> mxsimple rGf M.
Proof.
rewrite /mxsimple mxmodule_morphim.
split=> [] [-> -> minM]; split=> // U modU;
by apply: minM; rewrite mxmodule_morphim in modU *.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxsimple_morphim | |
morphim_mx_irr: (mx_irreducible rG <-> mx_irreducible rGf).
Proof. exact: mxsimple_morphim. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | morphim_mx_irr | |
morphim_mx_abs_irr:
mx_absolutely_irreducible rG = mx_absolutely_irreducible rGf.
Proof.
have fG_onto: f @* G \subset restrm sGD f @* G.
by rewrite (morphim_restrm sGD) setIid.
rewrite -(morphpre_mx_abs_irr _ fG_onto); congr (_ && (_ == _)).
by rewrite /enveloping_algebra_mx /= morphpre_restrm (setIidPl _).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | morphim_mx_abs_irr | |
rfix_submod(H : {set gT}) :
H \subset G -> (rfix_mx rU H :=: in_submod U (U :&: rfix_mx rG H))%MS.
Proof.
move=> sHG; apply/eqmxP/andP; split; last first.
apply/rfix_mxP=> x Hx; rewrite -in_submodJ ?capmxSl //.
by rewrite (rfix_mxP H _) ?capmxSr.
rewrite -val_submodS in_submodK ?capmxSl // sub_capmx val_submodP //=.
apply/rfix_mxP=> x Hx.
by rewrite -(val_submodJ Umod) ?(subsetP sHG) ?rfix_mx_id.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rfix_submod | |
rfix_factmod(H : {set gT}) :
H \subset G -> (in_factmod U (rfix_mx rG H) <= rfix_mx rU' H)%MS.
Proof.
move=> sHG; apply/rfix_mxP=> x Hx.
by rewrite -(in_factmodJ Umod) ?(subsetP sHG) ?rfix_mx_id.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rfix_factmod | |
rstab_submodm (W : 'M_(m, \rank U)) :
rstab rU W = rstab rG (val_submod W).
Proof.
apply/setP=> x /[!inE]; apply: andb_id2l => Gx.
by rewrite -(inj_eq val_submod_inj) val_submodJ.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstab_submod | |
rstabs_submodm (W : 'M_(m, \rank U)) :
rstabs rU W = rstabs rG (val_submod W).
Proof.
apply/setP=> x /[!inE]; apply: andb_id2l => Gx.
by rewrite -val_submodS val_submodJ.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstabs_submod | |
val_submod_modulem (W : 'M_(m, \rank U)) :
mxmodule rG (val_submod W) = mxmodule rU W.
Proof. by rewrite /mxmodule rstabs_submod. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_submod_module | |
in_submod_modulem (V : 'M_(m, n)) :
(V <= U)%MS -> mxmodule rU (in_submod U V) = mxmodule rG V.
Proof. by move=> sVU; rewrite -val_submod_module in_submodK. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | in_submod_module | |
rstab_factmodm (W : 'M_(m, n)) :
rstab rG W \subset rstab rU' (in_factmod U W).
Proof.
by apply/subsetP=> x /setIdP[Gx /eqP cUW]; rewrite inE Gx -in_factmodJ //= cUW.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstab_factmod | |
rstabs_factmodm (W : 'M_(m, \rank (cokermx U))) :
rstabs rU' W = rstabs rG (U + val_factmod W)%MS.
Proof.
apply/setP=> x /[!inE]; apply: andb_id2l => Gx.
rewrite addsmxMr addsmx_sub (submx_trans (mxmoduleP Umod x Gx)) ?addsmxSl //.
rewrite -val_factmodS val_factmodJ //= val_factmodS; apply/idP/idP=> nWx.
rewrite (submx_trans (addsmxSr U _)) // -(in_factmodsK (addsmxSl U _)) //.
by rewrite addsmxS // val_factmodS in_factmod_addsK.
rewrite in_factmodE (submx_trans (submxMr _ nWx)) // -in_factmodE.
by rewrite in_factmod_addsK val_factmodK.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstabs_factmod | |
val_factmod_modulem (W : 'M_(m, \rank (cokermx U))) :
mxmodule rG (U + val_factmod W)%MS = mxmodule rU' W.
Proof. by rewrite /mxmodule rstabs_factmod. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_factmod_module | |
in_factmod_modulem (V : 'M_(m, n)) :
mxmodule rU' (in_factmod U V) = mxmodule rG (U + V)%MS.
Proof.
rewrite -(eqmx_module _ (in_factmodsK (addsmxSl U V))).
by rewrite val_factmod_module (eqmx_module _ (in_factmod_addsK _ _)).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | in_factmod_module | |
rker_submod: rker rU = rstab rG U.
Proof. by rewrite /rker rstab_submod; apply: eqmx_rstab (val_submod1 U). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rker_submod | |
rstab_norm: G \subset 'N(rstab rG U).
Proof. by rewrite -rker_submod rker_norm. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstab_norm | |
rstab_normal: rstab rG U <| G.
Proof. by rewrite -rker_submod rker_normal. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstab_normal | |
submod_mx_faithful: mx_faithful rU -> mx_faithful rG.
Proof. by apply: subset_trans; rewrite rker_submod rstabS ?submx1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | submod_mx_faithful | |
rker_factmod: rker rG \subset rker rU'.
Proof.
apply/subsetP=> x /rkerP[Gx cVx].
by rewrite inE Gx /= /factmod_mx cVx mul1mx mulmx1 val_factmodK.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rker_factmod | |
factmod_mx_faithful: mx_faithful rU' -> mx_faithful rG.
Proof. exact: subset_trans rker_factmod. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | factmod_mx_faithful | |
submod_mx_irr: mx_irreducible rU <-> mxsimple rG U.
Proof.
split=> [] [_ nzU simU].
rewrite -mxrank_eq0 mxrank1 mxrank_eq0 in nzU; split=> // V modV sVU nzV.
rewrite -(in_submodK sVU) -val_submod1 val_submodS.
rewrite -(genmxE (in_submod U V)) simU ?genmxE ?submx1 //=.
by rewrite (eqmx_module _ (genmxE _)) in_submod_module.
by rewrite -submx0 genmxE -val_submodS in_submodK // linear0 eqmx0 submx0.
apply/mx_irrP; rewrite lt0n mxrank_eq0; split=> // V modV.
rewrite -(inj_eq val_submod_inj) linear0 -(eqmx_eq0 (genmxE _)) => nzV.
rewrite -sub1mx -val_submodS val_submod1 -(genmxE (val_submod V)).
rewrite simU ?genmxE ?val_submodP //=.
by rewrite (eqmx_module _ (genmxE _)) val_submod_module.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | submod_mx_irr | |
rfix_conj(H : {set gT}) :
(rfix_mx rGB H :=: B *m rfix_mx rG H *m invmx B)%MS.
Proof.
apply/eqmxP/andP; split.
rewrite -mulmxA (eqmxMfull (_ *m _)) ?row_full_unit //.
rewrite -[rfix_mx rGB H](mulmxK uB) submxMr //; apply/rfix_mxP=> x Hx.
apply: (canRL (mulmxKV uB)); rewrite -(rconj_mxJ _ uB) mulmxK //.
by rewrite rfix_mx_id.
apply/rfix_mxP=> x Gx; rewrite -3!mulmxA; congr (_ *m _).
by rewrite !mulmxA mulmxKV // rfix_mx_id.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rfix_conj | |
rstabs_conjm (U : 'M_(m, n)) : rstabs rGB U = rstabs rG (U *m B).
Proof.
apply/setP=> x; rewrite !inE rconj_mxE !mulmxA.
by rewrite -{2}[U](mulmxK uB) submxMfree // row_free_unit unitmx_inv.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstabs_conj | |
mxmodule_conjm (U : 'M_(m, n)) : mxmodule rGB U = mxmodule rG (U *m B).
Proof. by rewrite /mxmodule rstabs_conj. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxmodule_conj | |
conj_mx_irr: mx_irreducible rGB <-> mx_irreducible rG.
Proof.
have Bfree: row_free B by rewrite row_free_unit.
split => /mx_irrP[n_gt0 irrG]; apply/mx_irrP; split=> // U.
rewrite -[U](mulmxKV uB) -mxmodule_conj -mxrank_eq0 /row_full mxrankMfree //.
by rewrite mxrank_eq0; apply: irrG.
rewrite -mxrank_eq0 /row_full -(mxrankMfree _ Bfree) mxmodule_conj mxrank_eq0.
exact: irrG.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | conj_mx_irr | |
quo_mx_quotient: (E_ rGH :=: E_ rG)%MS.
Proof.
apply/eqmxP/andP; split; apply/row_subP=> i.
rewrite rowK; case/morphimP: (enum_valP i) => x _ Gx ->{i}.
rewrite quo_repr_coset // (eq_row_sub (enum_rank_in Gx x)) // rowK.
by rewrite enum_rankK_in.
rewrite rowK -(quo_mx_coset krH nHG) ?enum_valP //; set Hx := coset H _.
have GHx: Hx \in (G / H)%g by rewrite mem_quotient ?enum_valP.
by rewrite (eq_row_sub (enum_rank_in GHx Hx)) // rowK enum_rankK_in.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | quo_mx_quotient | |
rfix_quo(K : {group gT}) :
K \subset G -> (rfix_mx rGH (K / H)%g :=: rfix_mx rG K)%MS.
Proof.
move=> sKG; apply/eqmxP/andP; (split; apply/rfix_mxP) => [x Kx | Hx].
have Gx := subsetP sKG x Kx; rewrite -(quo_mx_coset krH nHG) // rfix_mx_id //.
by rewrite mem_morphim ?(subsetP nHG).
case/morphimP=> x _ Kx ->; have Gx := subsetP sKG x Kx.
by rewrite quo_repr_coset ?rfix_mx_id.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rfix_quo | |
rstabs_quom (U : 'M_(m, n)) : rstabs rGH U = (rstabs rG U / H)%g.
Proof.
apply/setP=> Hx /[!inE]; apply/andP/idP=> [[]|] /morphimP[x Nx Gx ->{Hx}].
by rewrite quo_repr_coset // => nUx; rewrite mem_morphim // inE Gx.
by case/setIdP: Gx => Gx nUx; rewrite quo_repr_coset ?mem_morphim.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rstabs_quo | |
mxmodule_quom (U : 'M_(m, n)) : mxmodule rGH U = mxmodule rG U.
Proof.
rewrite /mxmodule rstabs_quo quotientSGK // ?(subset_trans krH) //.
by apply/subsetP=> x /[!inE]/andP[-> /[1!mul1mx]/eqP->/=]; rewrite mulmx1.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxmodule_quo | |
quo_mx_irr: mx_irreducible rGH <-> mx_irreducible rG.
Proof.
split; case/mx_irrP=> n_gt0 irrG; apply/mx_irrP; split=> // U modU;
by apply: irrG; rewrite mxmodule_quo in modU *.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | quo_mx_irr | |
group_splitting_fieldgT (G : {group gT}) :=
forall n (rG : mx_representation F G n),
mx_irreducible rG -> mx_absolutely_irreducible rG. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | group_splitting_field | |
group_closure_fieldgT :=
forall G : {group gT}, group_splitting_field G. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | group_closure_field | |
quotient_splitting_fieldgT (G : {group gT}) (H : {set gT}) :
G \subset 'N(H) -> group_splitting_field G -> group_splitting_field (G / H).
Proof.
move=> nHG splitG n rGH irrGH.
by rewrite -(morphim_mx_abs_irr _ nHG) splitG //; apply/morphim_mx_irr.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | quotient_splitting_field | |
coset_splitting_fieldgT (H : {set gT}) :
group_closure_field gT -> group_closure_field (coset_of H).
Proof.
move=> split_gT Gbar; have ->: Gbar = (coset H @*^-1 Gbar / H)%G.
by apply: val_inj; rewrite /= /quotient morphpreK ?sub_im_coset.
by apply: quotient_splitting_field; [apply: subsetIl | apply: split_gT].
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | coset_splitting_field | |
mx_faithful_irr_center_cyclicn (rG : mx_representation F G n) :
mx_faithful rG -> mx_irreducible rG -> cyclic 'Z(G).
Proof.
case: n rG => [|n] rG injG irrG; first by case/mx_irrP: irrG.
move/trivgP: injG => KrG1; pose rZ := subg_repr rG (center_sub _).
apply: (div_ring_mul_group_cyclic (repr_mx1 rZ)) (repr_mxM rZ) _ _; last first.
exact: center_abelian.
move=> x; rewrite -[[set _]]KrG1 !inE mul1mx -subr_eq0 andbC; set U := _ - _.
do 2![case/andP]=> Gx cGx; rewrite Gx /=; apply: (mx_Schur irrG).
apply/centgmxP=> y Gy; rewrite mulmxBl mulmxBr mulmx1 mul1mx.
by rewrite -!repr_mxM // (centP cGx).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_faithful_irr_center_cyclic | |
mx_faithful_irr_abelian_cyclicn (rG : mx_representation F G n) :
mx_faithful rG -> mx_irreducible rG -> abelian G -> cyclic G.
Proof.
move=> injG irrG cGG; rewrite -(setIidPl cGG).
exact: mx_faithful_irr_center_cyclic injG irrG.
Qed.
Hypothesis splitG : group_splitting_field G. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_faithful_irr_abelian_cyclic | |
mx_irr_abelian_linearn (rG : mx_representation F G n) :
mx_irreducible rG -> abelian G -> n = 1.
Proof.
by move=> irrG cGG; apply/eqP; rewrite -(abelian_abs_irr rG) ?splitG.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_irr_abelian_linear | |
mxsimple_abelian_linearn (rG : mx_representation F G n) M :
abelian G -> mxsimple rG M -> \rank M = 1.
Proof.
move=> cGG simM; have [modM _ _] := simM.
by move/(submod_mx_irr modM)/mx_irr_abelian_linear: simM => ->.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxsimple_abelian_linear | |
linear_mxsimplen (rG : mx_representation F G n) (M : 'M_n) :
mxmodule rG M -> \rank M = 1 -> mxsimple rG M.
Proof.
move=> modM rM1; apply/(submod_mx_irr modM).
by apply: mx_abs_irrW; rewrite linear_mx_abs_irr.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | linear_mxsimple | |
center_kquo_cyclic: mx_irreducible rG -> cyclic 'Z(G / rker rG)%g.
Proof.
move=> irrG; apply: mx_faithful_irr_center_cyclic (kquo_mx_faithful rG) _.
exact/quo_mx_irr.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | center_kquo_cyclic | |
der1_sub_rker:
group_splitting_field G -> mx_irreducible rG ->
(G^`(1) \subset rker rG)%g = (n == 1)%N.
Proof.
move=> splitG irrG; apply/idP/idP; last by move/eqP; apply: rker_linear.
move/sub_der1_abelian; move/(abelian_abs_irr (kquo_repr rG))=> <-.
by apply: (quotient_splitting_field (rker_norm _) splitG); apply/quo_mx_irr.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | der1_sub_rker | |
mx_rsimn1 (rG1 : reprG n1) n2 (rG2 : reprG n2) : Prop :=
MxReprSim B of n1 = n2 & row_free B
& forall x, x \in G -> rG1 x *m B = B *m rG2 x. | Variant | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_rsim | |
mxrank_rsimn1 n2 (rG1 : reprG n1) (rG2 : reprG n2) :
mx_rsim rG1 rG2 -> n1 = n2.
Proof. by case. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxrank_rsim | |
mx_rsim_refln (rG : reprG n) : mx_rsim rG rG.
Proof.
exists 1%:M => // [|x _]; first by rewrite row_free_unit unitmx1.
by rewrite mulmx1 mul1mx.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_rsim_refl | |
mx_rsim_symn1 n2 (rG1 : reprG n1) (rG2 : reprG n2) :
mx_rsim rG1 rG2 -> mx_rsim rG2 rG1.
Proof.
case=> B def_n1; rewrite def_n1 in rG1 B *.
rewrite row_free_unit => injB homB; exists (invmx B) => // [|x Gx].
by rewrite row_free_unit unitmx_inv.
by apply: canRL (mulKmx injB) _; rewrite mulmxA -homB ?mulmxK.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_rsim_sym | |
mx_rsim_transn1 n2 n3
(rG1 : reprG n1) (rG2 : reprG n2) (rG3 : reprG n3) :
mx_rsim rG1 rG2 -> mx_rsim rG2 rG3 -> mx_rsim rG1 rG3.
Proof.
case=> [B1 defn1 freeB1 homB1] [B2 defn2 freeB2 homB2].
exists (B1 *m B2); rewrite /row_free ?mxrankMfree 1?defn1 // => x Gx.
by rewrite mulmxA homB1 // -!mulmxA homB2.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_rsim_trans | |
mx_rsim_defn1 n2 (rG1 : reprG n1) (rG2 : reprG n2) :
mx_rsim rG1 rG2 ->
exists B, exists2 B', B' *m B = 1%:M &
forall x, x \in G -> rG1 x = B *m rG2 x *m B'.
Proof.
case=> B def_n1; rewrite def_n1 in rG1 B *; rewrite row_free_unit => injB homB.
by exists B, (invmx B) => [|x Gx]; rewrite ?mulVmx // -homB // mulmxK.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_rsim_def | |
mx_rsim_ison (rG : reprG n) (U V : 'M_n)
(modU : mxmodule rG U) (modV : mxmodule rG V) :
mx_rsim (submod_repr modU) (submod_repr modV) <-> mx_iso rG U V.
Proof.
split=> [[B eqrUV injB homB] | [f injf homf defV]].
have: \rank (U *m val_submod (in_submod U 1%:M *m B)) = \rank U.
do 2!rewrite mulmxA mxrankMfree ?row_base_free //.
by rewrite -(eqmxMr _ (val_submod1 U)) -in_submodE val_submodK mxrank1.
case/complete_unitmx => f injf defUf; exists f => //.
apply/hom_mxP=> x Gx; rewrite -defUf -2!mulmxA -(val_submodJ modV) //.
rewrite -(mulmxA _ B) -homB // val_submodE 3!(mulmxA U) (mulmxA _ _ B).
rewrite -in_submodE -in_submodJ //.
have [u ->] := submxP (mxmoduleP modU x Gx).
by rewrite in_submodE -mulmxA -defUf !mulmxA !mulmx1.
apply/eqmxP; rewrite -mxrank_leqif_eq.
by rewrite mxrankMfree ?eqrUV ?row_free_unit.
by rewrite -defUf mulmxA val_submodP.
have eqrUV: \rank U = \rank V by rewrite -defV mxrankMfree ?row_free_unit.
exists (in_submod V (val_submod 1%:M *m f)) => // [|x Gx].
rewrite /row_free {6}eqrUV -[_ == _]sub1mx -val_submodS.
rewrite in_submodK; last by rewrite -defV submxMr ?val_submodP.
by rewrite val_submod1 -defV submxMr ?val_submod1.
rewrite -in_submodJ; last by rewrite -defV submxMr ?val_submodP.
rewrite -(hom_mxP (submx_trans (val_submodP _) homf)) // -(val_submodJ modU) //.
by rewrite mul1mx 2!(mulmxA ((submod_repr _) x)) -val_submodE.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_rsim_iso | |
mx_rsim_irrn1 n2 (rG1 : reprG n1) (rG2 : reprG n2) :
mx_rsim rG1 rG2 -> mx_irreducible rG1 -> mx_irreducible rG2.
Proof.
case/mx_rsim_sym=> f def_n2; rewrite {n2}def_n2 in f rG2 * => injf homf.
case/mx_irrP=> n1_gt0 minG; apply/mx_irrP; split=> // U modU nzU.
rewrite /row_full -(mxrankMfree _ injf) -genmxE.
apply: minG; last by rewrite -mxrank_eq0 genmxE mxrankMfree // mxrank_eq0.
rewrite (eqmx_module _ (genmxE _)); apply/mxmoduleP=> x Gx.
by rewrite -mulmxA -homf // mulmxA submxMr // (mxmoduleP modU).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_rsim_irr | |
mx_rsim_abs_irrn1 n2 (rG1 : reprG n1) (rG2 : reprG n2) :
mx_rsim rG1 rG2 ->
mx_absolutely_irreducible rG1 = mx_absolutely_irreducible rG2.
Proof.
case=> f def_n2; rewrite -{n2}def_n2 in f rG2 *.
rewrite row_free_unit => injf homf; congr (_ && (_ == _)).
pose Eg (g : 'M[F]_n1) := lin_mx (mulmxr (invmx g) \o mulmx g).
have free_Ef: row_free (Eg f).
apply/row_freeP; exists (Eg (invmx f)); apply/row_matrixP=> i.
rewrite rowE row1 mulmxA mul_rV_lin mx_rV_lin /=.
by rewrite invmxK !{1}mulmxA mulmxKV // -mulmxA mulKmx // vec_mxK.
symmetry; rewrite -(mxrankMfree _ free_Ef); congr (\rank _).
apply/row_matrixP=> i; rewrite row_mul !rowK mul_vec_lin /=.
by rewrite -homf ?enum_valP // mulmxK.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_rsim_abs_irr | |
rker_mx_rsimn1 n2 (rG1 : reprG n1) (rG2 : reprG n2) :
mx_rsim rG1 rG2 -> rker rG1 = rker rG2.
Proof.
case=> f def_n2; rewrite -{n2}def_n2 in f rG2 *.
rewrite row_free_unit => injf homf.
apply/setP=> x; rewrite !inE !mul1mx; apply: andb_id2l => Gx.
by rewrite -(can_eq (mulmxK injf)) homf // -scalar_mxC (can_eq (mulKmx injf)).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rker_mx_rsim | |
mx_rsim_faithfuln1 n2 (rG1 : reprG n1) (rG2 : reprG n2) :
mx_rsim rG1 rG2 -> mx_faithful rG1 = mx_faithful rG2.
Proof. by move=> simG12; rewrite /mx_faithful (rker_mx_rsim simG12). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_rsim_faithful | |
mx_rsim_factmodn (rG : reprG n) U V
(modU : mxmodule rG U) (modV : mxmodule rG V) :
(U + V :=: 1%:M)%MS -> mxdirect (U + V) ->
mx_rsim (factmod_repr modV) (submod_repr modU).
Proof.
move=> addUV dxUV.
have eqUV: \rank U = \rank (cokermx V).
by rewrite mxrank_coker -{3}(mxrank1 F n) -addUV (mxdirectP dxUV) addnK.
have{} dxUV: (U :&: V = 0)%MS by apply/mxdirect_addsP.
exists (in_submod U (val_factmod 1%:M *m proj_mx U V)) => // [|x Gx].
rewrite /row_free -{6}eqUV -[_ == _]sub1mx -val_submodS val_submod1.
rewrite in_submodK ?proj_mx_sub // -{1}[U](proj_mx_id dxUV) //.
rewrite -{1}(add_sub_fact_mod V U) mulmxDl proj_mx_0 ?val_submodP // add0r.
by rewrite submxMr // val_factmodS submx1.
rewrite -in_submodJ ?proj_mx_sub // -(hom_mxP _) //; last first.
by apply: submx_trans (submx1 _) _; rewrite -addUV proj_mx_hom.
rewrite mulmxA; congr (_ *m _); rewrite mulmxA -val_factmodE; apply/eqP.
rewrite eq_sym -subr_eq0 -mulmxBl proj_mx_0 //.
by rewrite -[_ *m rG x](add_sub_fact_mod V) addrK val_submodP.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_rsim_factmod | |
mxtrace_rsimn1 n2 (rG1 : reprG n1) (rG2 : reprG n2) :
mx_rsim rG1 rG2 -> {in G, forall x, \tr (rG1 x) = \tr (rG2 x)}.
Proof.
case/mx_rsim_def=> B [B' B'B def_rG1] x Gx.
by rewrite def_rG1 // mxtrace_mulC mulmxA B'B mul1mx.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxtrace_rsim | |
mx_rsim_scalarn1 n2 (rG1 : reprG n1) (rG2 : reprG n2) x c :
x \in G -> mx_rsim rG1 rG2 -> rG1 x = c%:M -> rG2 x = c%:M.
Proof.
move=> Gx /mx_rsim_sym[B _ Bfree rG2_B] rG1x.
by apply: (row_free_inj Bfree); rewrite rG2_B // rG1x scalar_mxC.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_rsim_scalar | |
socle_irr(W : sG) : mx_irreducible (socle_repr W).
Proof. by apply/submod_mx_irr; apply: socle_simple. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | socle_irr | |
socle_rsimP(W1 W2 : sG) :
reflect (mx_rsim (socle_repr W1) (socle_repr W2)) (W1 == W2).
Proof.
have [simW1 simW2] := (socle_simple W1, socle_simple W2).
by apply: (iffP (component_mx_isoP simW1 simW2)); move/mx_rsim_iso; apply.
Qed.
Local Notation mG U := (mxmodule rG U).
Local Notation sr modV := (submod_repr modV). | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | socle_rsimP | |
mx_rsim_in_submodU V (modU : mG U) (modV : mG V) :
let U' := <<in_submod V U>>%MS in
(U <= V)%MS ->
exists modU' : mxmodule (sr modV) U', mx_rsim (sr modU) (sr modU').
Proof.
move=> U' sUV; have modU': mxmodule (sr modV) U'.
by rewrite (eqmx_module _ (genmxE _)) in_submod_module.
have rankU': \rank U = \rank U' by rewrite genmxE mxrank_in_submod.
pose v1 := val_submod 1%:M; pose U1 := v1 _ U.
have sU1V: (U1 <= V)%MS by rewrite val_submod1.
have sU1U': (in_submod V U1 <= U')%MS by rewrite genmxE submxMr ?val_submod1.
exists modU', (in_submod U' (in_submod V U1)) => // [|x Gx].
apply/row_freeP; exists (v1 _ _ *m v1 _ _ *m in_submod U 1%:M).
rewrite mulmxA [X in X *m _]mulmxA -in_submodE.
by rewrite -!val_submodE !in_submodK ?val_submodK.
rewrite -!in_submodJ // -(val_submodJ modU) // mul1mx.
by rewrite 2!{1}in_submodE mulmxA (mulmxA _ U1) -val_submodE -!in_submodE.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_rsim_in_submod | |
rsim_submod1U (modU : mG U) : (U :=: 1%:M)%MS -> mx_rsim (sr modU) rG.
Proof.
move=> U1; exists (val_submod 1%:M) => [||x Gx]; first by rewrite U1 mxrank1.
by rewrite /row_free val_submod1.
by rewrite -(val_submodJ modU) // mul1mx -val_submodE.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | rsim_submod1 | |
mxtrace_submod1U (modU : mG U) :
(U :=: 1%:M)%MS -> {in G, forall x, \tr (sr modU x) = \tr (rG x)}.
Proof. by move=> defU; apply: mxtrace_rsim (rsim_submod1 modU defU). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxtrace_submod1 | |
mxtrace_dadd_modU V W (modU : mG U) (modV : mG V) (modW : mG W) :
(U + V :=: W)%MS -> mxdirect (U + V) ->
{in G, forall x, \tr (sr modU x) + \tr (sr modV x) = \tr (sr modW x)}.
Proof.
move=> defW dxW x Gx; have [sUW sVW]: (U <= W)%MS /\ (V <= W)%MS.
by apply/andP; rewrite -addsmx_sub defW.
pose U' := <<in_submod W U>>%MS; pose V' := <<in_submod W V>>%MS.
have addUV': (U' + V' :=: 1%:M)%MS.
apply/eqmxP; rewrite submx1 /= (adds_eqmx (genmxE _) (genmxE _)).
by rewrite -addsmxMr -val_submodS val_submod1 in_submodK ?defW.
have dxUV': mxdirect (U' + V').
apply/eqnP; rewrite /= addUV' mxrank1 !genmxE !mxrank_in_submod //.
by rewrite -(mxdirectP dxW) /= defW.
have [modU' simU] := mx_rsim_in_submod modU modW sUW.
have [modV' simV] := mx_rsim_in_submod modV modW sVW.
rewrite (mxtrace_rsim simU) // (mxtrace_rsim simV) //.
rewrite -(mxtrace_sub_fact_mod modV') addrC; congr (_ + _).
by rewrite (mxtrace_rsim (mx_rsim_factmod modU' modV' addUV' dxUV')).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxtrace_dadd_mod | |
mxtrace_dsum_mod(I : finType) (P : pred I) U W
(modU : forall i, mG (U i)) (modW : mG W) :
let S := (\sum_(i | P i) U i)%MS in (S :=: W)%MS -> mxdirect S ->
{in G, forall x, \sum_(i | P i) \tr (sr (modU i) x) = \tr (sr modW x)}.
Proof.
move=> /= sumS dxS x Gx; have [m lePm] := ubnP #|P|.
elim: m => // m IHm in P lePm W modW sumS dxS *.
have [j /= Pj | P0] := pickP P; last first.
case: sumS (_ x); rewrite !big_pred0 // mxrank0 => <- _ rWx.
by rewrite [rWx]flatmx0 linear0.
rewrite ltnS (cardD1x Pj) in lePm.
rewrite mxdirectE /= !(bigD1 j Pj) -mxdirectE mxdirect_addsE /= in dxS sumS *.
have [_ dxW' dxW] := and3P dxS; rewrite (sameP eqP mxdirect_addsP) in dxW.
rewrite (IHm _ _ _ (sumsmx_module _ (fun i _ => modU i)) (eqmx_refl _)) //.
exact: mxtrace_dadd_mod.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxtrace_dsum_mod | |
mxtrace_componentU (simU : mxsimple rG U) :
let V := component_mx rG U in
let modV := component_mx_module rG U in let modU := mxsimple_module simU in
{in G, forall x, \tr (sr modV x) = \tr (sr modU x) *+ (\rank V %/ \rank U)}.
Proof.
move=> V modV modU x Gx.
have [I W S simW defV dxV] := component_mx_semisimple simU.
rewrite -(mxtrace_dsum_mod (fun i => mxsimple_module (simW i)) modV defV) //.
have rankU_gt0: \rank U > 0 by rewrite lt0n mxrank_eq0; case simU.
have isoW i: mx_iso rG U (W i).
by apply: component_mx_iso; rewrite ?simU // -defV (sumsmx_sup i).
have ->: (\rank V %/ \rank U)%N = #|I|.
symmetry; rewrite -(mulnK #|I| rankU_gt0); congr (_ %/ _)%N.
rewrite -defV (mxdirectP dxV) /= -sum_nat_const.
by apply: eq_bigr => i _; apply: mxrank_iso.
rewrite -sumr_const; apply: eq_bigr => i _; symmetry.
by apply: mxtrace_rsim Gx; apply/mx_rsim_iso; apply: isoW.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxtrace_component | |
mxtrace_Socle: let modS := Socle_module sG in
{in G, forall x,
\tr (sr modS x) = \sum_(W : sG) \tr (socle_repr W x) *+ socle_mult W}.
Proof.
move=> /= x Gx /=; pose modW (W : sG) := component_mx_module rG (socle_base W).
rewrite -(mxtrace_dsum_mod modW _ (eqmx_refl _) (Socle_direct sG)) //.
by apply: eq_bigr => W _; rewrite (mxtrace_component (socle_simple W)).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mxtrace_Socle | |
Clifford_simpleM x : mxsimple rH M -> x \in G -> mxsimple rH (M *m rG x).
Proof.
have modmG m U y: y \in G -> (mxmodule rH) m U -> mxmodule rH (U *m rG y).
move=> Gy modU; apply/mxmoduleP=> h Hh; have Gh := subsetP sHG h Hh.
rewrite -mulmxA -repr_mxM // conjgCV repr_mxM ?groupJ ?groupV // mulmxA.
by rewrite submxMr ?(mxmoduleP modU) // -mem_conjg (normsP nHG).
have nzmG m y (U : 'M_(m, n)): y \in G -> (U *m rG y == 0) = (U == 0).
by move=> Gy; rewrite -{1}(mul0mx m (rG y)) (can_eq (repr_mxK rG Gy)).
case=> [modM nzM simM] Gx; have Gx' := groupVr Gx.
split=> [||U modU sUMx nzU]; rewrite ?modmG ?nzmG //.
rewrite -(repr_mxKV rG Gx U) submxMr //.
by rewrite (simM (U *m _)) ?modmG ?nzmG // -(repr_mxK rG Gx M) submxMr.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_simple | |
Clifford_homx m (U : 'M_(m, n)) :
x \in 'C_G(H) -> (U <= dom_hom_mx rH (rG x))%MS.
Proof.
case/setIP=> Gx cHx; apply/rV_subP=> v _{U}.
apply/hom_mxP=> h Hh; have Gh := subsetP sHG h Hh.
by rewrite -!mulmxA /= -!repr_mxM // (centP cHx).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_hom | |
Clifford_isox U : x \in 'C_G(H) -> mx_iso rH U (U *m rG x).
Proof.
move=> cHx; have [Gx _] := setIP cHx.
by exists (rG x); rewrite ?repr_mx_unit ?Clifford_hom.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_iso | |
Clifford_iso2x U V :
mx_iso rH U V -> x \in G -> mx_iso rH (U *m rG x) (V *m rG x).
Proof.
case=> [f injf homUf defV] Gx; have Gx' := groupVr Gx.
pose fx := rG (x^-1)%g *m f *m rG x; exists fx; last 1 first.
- by rewrite !mulmxA repr_mxK //; apply: eqmxMr.
- by rewrite !unitmx_mul andbC !repr_mx_unit.
apply/hom_mxP=> h Hh; have Gh := subsetP sHG h Hh.
rewrite -(mulmxA U) -repr_mxM // conjgCV repr_mxM ?groupJ // !mulmxA.
rewrite !repr_mxK // (hom_mxP homUf) -?mem_conjg ?(normsP nHG) //=.
by rewrite !repr_mxM ?invgK ?groupM // !mulmxA repr_mxKV.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_iso2 | |
Clifford_componentJM x :
mxsimple rH M -> x \in G ->
(component_mx rH (M *m rG x) :=: component_mx rH M *m rG x)%MS.
Proof.
set simH := mxsimple rH; set cH := component_mx rH.
have actG: {in G, forall y M, simH M -> cH M *m rG y <= cH (M *m rG y)}%MS.
move=> {M} y Gy /= M simM; have [I [U isoU def_cHM]] := component_mx_def simM.
rewrite /cH def_cHM sumsmxMr; apply/sumsmx_subP=> i _.
by apply: mx_iso_component; [apply: Clifford_simple | apply: Clifford_iso2].
move=> simM Gx; apply/eqmxP; rewrite actG // -/cH.
rewrite -{1}[cH _](repr_mxKV rG Gx) submxMr // -{2}[M](repr_mxK rG Gx).
by rewrite actG ?groupV //; apply: Clifford_simple.
Qed.
Hypothesis irrG : mx_irreducible rG. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_componentJ | |
Clifford_basisM : mxsimple rH M ->
{X : {set gT} | X \subset G &
let S := \sum_(x in X) M *m rG x in S :=: 1%:M /\ mxdirect S}%MS.
Proof.
move=> simM. have simMG (g : [subg G]) : mxsimple rH (M *m rG (val g)).
by case: g => x Gx; apply: Clifford_simple.
have [|XG [defX1 dxX1]] := sum_mxsimple_direct_sub simMG (_ : _ :=: 1%:M)%MS.
apply/eqmxP; case irrG => _ _ ->; rewrite ?submx1 //; last first.
rewrite -submx0; apply/sumsmx_subP; move/(_ 1%g (erefl _)); apply: negP.
by rewrite submx0 repr_mx1 mulmx1; case simM.
apply/mxmoduleP=> x Gx; rewrite sumsmxMr; apply/sumsmx_subP=> [[y Gy]] /= _.
by rewrite (sumsmx_sup (subg G (y * x)))// subgK ?groupM// -mulmxA repr_mxM.
exists (val @: XG); first by apply/subsetP=> ?; case/imsetP=> [[x Gx]] _ ->.
have bij_val: {on val @: XG, bijective (@sgval _ G)}.
exists (subg G) => [g _ | x]; first exact: sgvalK.
by case/imsetP=> [[x' Gx]] _ ->; rewrite subgK.
have defXG g: (val g \in val @: XG) = (g \in XG).
by apply/imsetP/idP=> [[h XGh] | XGg]; [move/val_inj-> | exists g].
by rewrite /= mxdirectE /= !(reindex _ bij_val) !(eq_bigl _ _ defXG).
Qed.
Variable sH : socleType rH. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_basis | |
Clifford_act(W : sH) x :=
let Gx := subgP (subg G x) in
PackSocle (component_socle sH (Clifford_simple (socle_simple W) Gx)).
Let valWact W x : (Clifford_act W x :=: W *m rG (sgval (subg G x)))%MS.
Proof.
rewrite PackSocleK; apply: Clifford_componentJ (subgP _).
exact: socle_simple.
Qed.
Fact Clifford_is_action : is_action G Clifford_act.
Proof.
split=> [x W W' eqWW' | W x y Gx Gy].
pose Gx := subgP (subg G x); apply/socleP; apply/eqmxP.
rewrite -(repr_mxK rG Gx W) -(repr_mxK rG Gx W'); apply: eqmxMr.
apply: eqmx_trans (eqmx_sym _) (valWact _ _).
by rewrite -eqWW'; apply: valWact.
apply/socleP; rewrite !{1}valWact 2!{1}(eqmxMr _ (valWact _ _)).
by rewrite !subgK ?groupM ?repr_mxM ?mulmxA ?andbb.
Qed. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_act | |
Clifford_action:= Action Clifford_is_action.
Local Notation "'Cl" := Clifford_action : action_scope. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_action | |
val_Clifford_actW x : x \in G -> ('Cl%act W x :=: W *m rG x)%MS.
Proof. by move=> Gx; apply: eqmx_trans (valWact _ _) _; rewrite subgK. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | val_Clifford_act | |
Clifford_atrans: [transitive G, on [set: sH] | 'Cl].
Proof.
have [_ nz1 _] := irrG.
apply: mxsimple_exists (mxmodule1 rH) nz1 _ _ => [[M simM _]].
pose W1 := PackSocle (component_socle sH simM).
have [X sXG [def1 _]] := Clifford_basis simM; move/subsetP: sXG => sXG.
apply/imsetP; exists W1; first by rewrite inE.
symmetry; apply/setP=> W /[1!inE]; have simW := socle_simple W.
have:= submx1 (socle_base W); rewrite -def1 -[(\sum_(x in X) _)%MS]mulmx1.
case/(hom_mxsemisimple_iso simW) => [x Xx _ | | x Xx isoMxW].
- by apply: Clifford_simple; rewrite ?sXG.
- exact: scalar_mx_hom.
have Gx := sXG x Xx; apply/imsetP; exists x => //; apply/socleP/eqmxP/eqmx_sym.
apply: eqmx_trans (val_Clifford_act _ Gx) _; rewrite PackSocleK.
apply: eqmx_trans (eqmx_sym (Clifford_componentJ simM Gx)) _.
apply/eqmxP; rewrite (sameP genmxP eqP) !{1}genmx_component.
by apply/component_mx_isoP=> //; apply: Clifford_simple.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_atrans | |
Clifford_Socle1: Socle sH = 1%:M.
Proof.
case/imsetP: Clifford_atrans => W _ _; have simW := socle_simple W.
have [X sXG [def1 _]] := Clifford_basis simW.
rewrite reducible_Socle1 //; apply: mxsemisimple_reducible.
apply: intro_mxsemisimple def1 _ => x /(subsetP sXG) Gx _.
exact: Clifford_simple.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_Socle1 | |
Clifford_rank_components(W : sH) : (#|sH| * \rank W)%N = n.
Proof.
rewrite -{9}(mxrank1 F n) -Clifford_Socle1.
rewrite (mxdirectP (Socle_direct sH)) /= -sum_nat_const.
apply: eq_bigr => W1 _; have [W0 _ W0G] := imsetP Clifford_atrans.
have{} W0G W': W' \in orbit 'Cl G W0 by rewrite -W0G inE.
have [/orbitP[x Gx <-] /orbitP[y Gy <-]] := (W0G W, W0G W1).
by rewrite !{1}val_Clifford_act // !mxrankMfree // !repr_mx_free.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_rank_components | |
Clifford_component_basisM : mxsimple rH M ->
{t : nat & {x_ : sH -> 'I_t -> gT |
forall W, let sW := (\sum_j M *m rG (x_ W j))%MS in
[/\ forall j, x_ W j \in G, (sW :=: W)%MS & mxdirect sW]}}.
Proof.
move=> simM; pose t := (n %/ #|sH| %/ \rank M)%N; exists t.
have [X /subsetP sXG [defX1 dxX1]] := Clifford_basis simM.
pose sMv (W : sH) x := (M *m rG x <= W)%MS; pose Xv := [pred x in X | sMv _ x].
have sXvG W: {subset Xv W <= G} by move=> x /andP[/sXG].
have defW W: (\sum_(x in Xv W) M *m rG x :=: W)%MS.
apply/eqmxP; rewrite -(geq_leqif (mxrank_leqif_eq _)); last first.
by apply/sumsmx_subP=> x /andP[].
rewrite -(leq_add2r (\sum_(W' | W' != W) \rank W')) -((bigD1 W) predT) //=.
rewrite -(mxdirectP (Socle_direct sH)) /= -/(Socle _) Clifford_Socle1 -defX1.
apply: leq_trans (mxrankS _) (mxrank_sum_leqif _).1 => /=.
rewrite (bigID (sMv W))%MS addsmxS //=.
apply/sumsmx_subP=> x /andP[Xx notW_Mx]; have Gx := sXG x Xx.
have simMx := Clifford_simple simM Gx.
pose Wx := PackSocle (component_socle sH simMx).
have sMxWx: (M *m rG x <= Wx)%MS by rewrite PackSocleK component_mx_id.
by rewrite (sumsmx_sup Wx) //; apply: contra notW_Mx => /eqP <-.
have dxXv W: mxdirect (\sum_(x in Xv W) M *m rG x).
move: dxX1; rewrite !mxdirectE /= !(bigID (sMv W) [in X]) /=.
by rewrite -mxdirectE mxdirect_addsE /= => /andP[].
have def_t W: #|Xv W| = t.
rewrite /t -{1}(Clifford_rank_components W) mulKn 1?(cardD1 W) //.
rewrite -defW (mxdirectP (dxXv W)) /= (eq_bigr (fun _ => \rank M)) => [
... | Theorem | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_component_basis | |
Clifford_astab: H <*> 'C_G(H) \subset 'C([set: sH] | 'Cl).
Proof.
rewrite join_subG !subsetI sHG subsetIl /=; apply/andP; split.
apply/subsetP=> h Hh /[1!inE]; have Gh := subsetP sHG h Hh.
apply/subsetP=> W _; have simW := socle_simple W; have [modW _ _] := simW.
have simWh: mxsimple rH (socle_base W *m rG h) by apply: Clifford_simple.
rewrite inE -val_eqE /= PackSocleK eq_sym.
apply/component_mx_isoP; rewrite ?subgK //; apply: component_mx_iso => //.
by apply: submx_trans (component_mx_id simW); move/mxmoduleP: modW => ->.
apply/subsetP=> z cHz /[1!inE]; have [Gz _] := setIP cHz.
apply/subsetP=> W _; have simW := socle_simple W; have [modW _ _] := simW.
have simWz: mxsimple rH (socle_base W *m rG z) by apply: Clifford_simple.
rewrite inE -val_eqE /= PackSocleK eq_sym.
by apply/component_mx_isoP; rewrite ?subgK //; apply: Clifford_iso.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_astab | |
Clifford_astab1(W : sH) : 'C[W | 'Cl] = rstabs rG W.
Proof.
apply/setP=> x /[!inE]; apply: andb_id2l => Gx.
rewrite sub1set inE (sameP eqP socleP) !val_Clifford_act //.
rewrite andb_idr // => sWxW; rewrite -mxrank_leqif_sup //.
by rewrite mxrankMfree ?repr_mx_free.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_astab1 | |
Clifford_rstabs_simple(W : sH) :
mxsimple (subg_repr rG (rstabs_sub rG W)) W.
Proof.
split => [||U modU sUW nzU]; last 2 [exact: nz_socle].
by rewrite /mxmodule rstabs_subg setIid.
have modUH: mxmodule rH U.
apply/mxmoduleP=> h Hh; rewrite (mxmoduleP modU) //.
rewrite /= -Clifford_astab1 !(inE, sub1set) (subsetP sHG) //.
rewrite (astab_act (subsetP Clifford_astab h _)) ?inE //=.
by rewrite mem_gen // inE Hh.
apply: (mxsimple_exists modUH nzU) => [[M simM sMU]].
have [t [x_ /(_ W)[Gx_ defW _]]] := Clifford_component_basis simM.
rewrite -defW; apply/sumsmx_subP=> j _; set x := x_ W j.
have{Gx_} Gx: x \in G by rewrite Gx_.
apply: submx_trans (submxMr _ sMU) _; apply: (mxmoduleP modU).
rewrite inE -val_Clifford_act Gx //; set Wx := 'Cl%act W x.
case: (eqVneq Wx W) (simM) => [-> //=|] neWxW [_ /negP[]]; rewrite -submx0.
rewrite (canF_eq (actKin 'Cl Gx)) in neWxW.
rewrite -(component_mx_disjoint _ _ neWxW); try exact: socle_simple.
rewrite sub_capmx {1}(submx_trans sMU sUW) val_Clifford_act ?groupV //.
by rewrite -(eqmxMr _ defW) sumsmxMr (sumsmx_sup j) ?repr_mxK.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | Clifford_rstabs_simple | |
section_module(U V : 'M_n) (modU : modG U) (modV : modG V) :
mxmodule (factmod_repr modU) <<in_factmod U V>>%MS.
Proof.
by rewrite (eqmx_module _ (genmxE _)) in_factmod_module addsmx_module.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | section_module | |
section_reprU V (modU : modG U) (modV : modG V) :=
submod_repr (section_module modU modV). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | section_repr | |
mx_factmod_subU modU :
mx_rsim (@section_repr U _ modU (mxmodule1 rG)) (factmod_repr modU).
Proof.
exists (val_submod 1%:M) => [||x Gx].
- apply: (@addIn (\rank U)); rewrite genmxE mxrank_in_factmod mxrank_coker.
by rewrite (addsmx_idPr (submx1 U)) mxrank1 subnK ?rank_leq_row.
- by rewrite /row_free val_submod1.
by rewrite -[_ x]mul1mx -val_submodE val_submodJ.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | mx_factmod_sub | |
max_submod(U V : 'M_n) :=
(U < V)%MS /\ (forall W, ~ [/\ modG W, U < W & W < V])%MS. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | max_submod | |
max_submodPU V (modU : modG U) (modV : modG V) :
(U <= V)%MS -> (max_submod U V <-> mx_irreducible (section_repr modU modV)).
Proof.
move=> sUV; split=> [[ltUV maxU] | ].
apply/mx_irrP; split=> [|WU modWU nzWU].
by rewrite genmxE lt0n mxrank_eq0 in_factmod_eq0; case/andP: ltUV.
rewrite -sub1mx -val_submodS val_submod1 genmxE.
pose W := (U + val_factmod (val_submod WU))%MS.
suffices sVW: (V <= W)%MS.
rewrite {2}in_factmodE (submx_trans (submxMr _ sVW)) //.
rewrite addsmxMr -!in_factmodE val_factmodK.
by rewrite ((in_factmod U U =P 0) _) ?adds0mx ?in_factmod_eq0.
move/and3P: {maxU}(maxU W); apply: contraR; rewrite /ltmx addsmxSl => -> /=.
move: modWU; rewrite /mxmodule rstabs_submod rstabs_factmod => -> /=.
rewrite addsmx_sub submx_refl -in_factmod_eq0 val_factmodK.
move: nzWU; rewrite -[_ == 0](inj_eq val_submod_inj) linear0 => ->.
rewrite -(in_factmodsK sUV) addsmxS // val_factmodS.
by rewrite -(genmxE (in_factmod U V)) val_submodP.
case/mx_irrP; rewrite lt0n {1}genmxE mxrank_eq0 in_factmod_eq0 => ltUV maxV.
split=> // [|W [modW /andP[sUW ltUW] /andP[sWV /negP[]]]]; first exact/andP.
rewrite -(in_factmodsK sUV) -(in_factmodsK sUW) addsmxS // val_factmodS.
rewrite -genmxE -val_submod1; set VU := <<_>>%MS.
have sW_VU: (in_factmod U W <= VU)%MS.
by rewrite genmxE -val_factmodS !submxMr.
rewrite -(in_submodK sW_VU) val_submodS -(genmxE (in_submod _ _)).
rewrite sub1mx maxV //.
rewrite (eqmx_module _ (genmxE _)) in_submod_module ?genmxE ?submxMr //.
... | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"From mathcomp Require Import ssralg poly polydiv finset fingroup morphism",
"From mathcomp Require Import perm aut... | character/mxrepresentation.v | max_submodP |
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