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... | 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 /... | 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 (s... | 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... | 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 re... | 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 -... | 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 ... | 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 ... | 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) _ ... | 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); appl... | 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... | 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 // mu... | 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 mul... | 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: mi... | 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 ... | 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... | 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 (... | 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 ... | 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_submo... | 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|.
el... | 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... | 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... | 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 submx... | 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=>... | 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]] := comp... | 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 (... | 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 : i... | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq path",
"From mathcomp Require Import div choice fintype tuple finfun bigop prime",
"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.
symmet... | 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 <-] /orb... | 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_b... | 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 ... | 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) (subset... | 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... | 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 va... | 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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