statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
submod_mx_faithful : mx_faithful rU -> mx_faithful rG. | Proof. by apply: subset_trans; rewrite rker_submod rstabS ?submx1. Qed. | Lemma | submod_mx_faithful | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"mx_faithful",
"rG",
"rU",
"rker_submod",
"rstabS",
"submx1",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | rker_factmod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"factmod_mx",
"inE",
"mul1mx",
"mulmx1",
"rG",
"rU'",
"rker",
"rkerP",
"subsetP",
"val_factmodK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
factmod_mx_faithful : mx_faithful rU' -> mx_faithful rG. | Proof. exact: subset_trans rker_factmod. Qed. | Lemma | factmod_mx_faithful | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_faithful",
"rG",
"rU'",
"rker_factmod",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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_sub... | Lemma | submod_mx_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqmx0",
"eqmx_eq0",
"eqmx_module",
"genmxE",
"in_submod",
"in_submodK",
"in_submod_module",
"inj_eq",
"linear0",
"lt0n",
"mx_irrP",
"mx_irreducible",
"mxrank1",
"mxrank_eq0",
"mxsimple",
"rG",
"rU",
"simU",
"split",
"sub1mx",
"submx0",
"submx1",
"val_submod",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rGB | := (rconj_repr rG uB). | Notation | rGB | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"rG",
"rconj_repr",
"uB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 rew... | Lemma | rfix_conj | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqmxMfull",
"eqmxP",
"gT",
"invmx",
"mulmxA",
"mulmxK",
"mulmxKV",
"rG",
"rGB",
"rconj_mxJ",
"rfix_mx",
"rfix_mxP",
"rfix_mx_id",
"row_full_unit",
"split",
"submxMr",
"uB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_conj m (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 | rstabs_conj | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inE",
"mulmxA",
"mulmxK",
"rG",
"rGB",
"rconj_mxE",
"row_free_unit",
"rstabs",
"setP",
"submxMfree",
"uB",
"unitmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_conj m (U : 'M_(m, n)) : mxmodule rGB U = mxmodule rG (U *m B). | Proof. by rewrite /mxmodule rstabs_conj. Qed. | Lemma | mxmodule_conj | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxmodule",
"rG",
"rGB",
"rstabs_conj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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: ... | Lemma | conj_mx_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"irrG",
"mulmxKV",
"mx_irrP",
"mx_irreducible",
"mxmodule_conj",
"mxrankMfree",
"mxrank_eq0",
"n_gt0",
"rG",
"rGB",
"row_free",
"row_free_unit",
"row_full",
"split",
"uB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rGH | := (quo_repr krH nHG). | Notation | rGH | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"krH",
"nHG",
"quo_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
E_ r | := (enveloping_algebra_mx r). | Notation | E_ | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"enveloping_algebra_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 rewri... | Lemma | quo_mx_quotient | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_",
"apply",
"coset",
"enum_rankK_in",
"enum_rank_in",
"enum_valP",
"eq_row_sub",
"eqmxP",
"krH",
"mem_quotient",
"morphimP",
"nHG",
"quo_mx_coset",
"quo_repr_coset",
"rG",
"rGH",
"rowK",
"row_subP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | rfix_quo | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqmxP",
"gT",
"group",
"krH",
"mem_morphim",
"morphimP",
"nHG",
"quo_mx_coset",
"quo_repr_coset",
"rG",
"rGH",
"rfix_mx",
"rfix_mxP",
"rfix_mx_id",
"sKG",
"split",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_quo m (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 | rstabs_quo | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inE",
"mem_morphim",
"morphimP",
"quo_repr_coset",
"rG",
"rGH",
"rstabs",
"setIdP",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_quo m (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 | mxmodule_quo | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inE",
"krH",
"mul1mx",
"mulmx1",
"mxmodule",
"quotientSGK",
"rG",
"rGH",
"rstabs_quo",
"subsetP",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | quo_mx_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"irrG",
"mx_irrP",
"mx_irreducible",
"mxmodule_quo",
"n_gt0",
"rG",
"rGH",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
group_splitting_field gT (G : {group gT}) | :=
forall n (rG : mx_representation F G n),
mx_irreducible rG -> mx_absolutely_irreducible rG. | Definition | group_splitting_field | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"gT",
"group",
"mx_absolutely_irreducible",
"mx_irreducible",
"mx_representation",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
group_closure_field gT | :=
forall G : {group gT}, group_splitting_field G. | Definition | group_closure_field | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"gT",
"group",
"group_splitting_field"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_splitting_field gT (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 | quotient_splitting_field | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"gT",
"group",
"group_splitting_field",
"morphim_mx_abs_irr",
"morphim_mx_irr",
"nHG",
"rGH",
"splitG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coset_splitting_field gT (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 | coset_splitting_field | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"coset",
"coset_of",
"gT",
"group_closure_field",
"morphpreK",
"quotient",
"quotient_splitting_field",
"sub_im_coset",
"subsetIl",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_faithful_irr_center_cyclic n (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 := _ - _.
d... | Lemma | mx_faithful_irr_center_cyclic | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"centP",
"center_abelian",
"center_sub",
"centgmxP",
"cyclic",
"div_ring_mul_group_cyclic",
"inE",
"injG",
"irrG",
"last",
"mul1mx",
"mulmx1",
"mulmxBl",
"mulmxBr",
"mx_Schur",
"mx_faithful",
"mx_irrP",
"mx_irreducible",
"mx_representation",
"rG",
"rZ",
"repr_mx1... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_faithful_irr_abelian_cyclic n (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. | Lemma | mx_faithful_irr_abelian_cyclic | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"abelian",
"cGG",
"cyclic",
"injG",
"irrG",
"mx_faithful",
"mx_faithful_irr_center_cyclic",
"mx_irreducible",
"mx_representation",
"rG",
"setIidPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
splitG : group_splitting_field G. | Hypothesis | splitG | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"group_splitting_field"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
mx_irr_abelian_linear n (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 | mx_irr_abelian_linear | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"abelian",
"abelian_abs_irr",
"apply",
"cGG",
"irrG",
"mx_irreducible",
"mx_representation",
"rG",
"splitG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple_abelian_linear n (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 | mxsimple_abelian_linear | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"abelian",
"cGG",
"mx_irr_abelian_linear",
"mx_representation",
"mxsimple",
"rG",
"rank",
"submod_mx_irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linear_mxsimple n (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 | linear_mxsimple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"linear_mx_abs_irr",
"mx_abs_irrW",
"mx_representation",
"mxmodule",
"mxsimple",
"rG",
"rank",
"submod_mx_irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | center_kquo_cyclic | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"cyclic",
"irrG",
"kquo_mx_faithful",
"mx_faithful_irr_center_cyclic",
"mx_irreducible",
"quo_mx_irr",
"rG",
"rker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | der1_sub_rker | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"abelian_abs_irr",
"apply",
"group_splitting_field",
"irrG",
"kquo_repr",
"last",
"mx_irreducible",
"quo_mx_irr",
"quotient_splitting_field",
"rG",
"rker",
"rker_linear",
"rker_norm",
"splitG",
"sub_der1_abelian"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reprG | := (mx_representation F G). | Notation | reprG | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_representation"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim n1 (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 | mx_rsim | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"reprG",
"row_free"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_rsim n1 n2 (rG1 : reprG n1) (rG2 : reprG n2) :
mx_rsim rG1 rG2 -> n1 = n2. | Proof. by case. Qed. | Lemma | mxrank_rsim | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_rsim",
"reprG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_refl n (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 | mx_rsim_refl | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mul1mx",
"mulmx1",
"mx_rsim",
"rG",
"reprG",
"row_free_unit",
"unitmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_sym n1 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 | mx_rsim_sym | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"invmx",
"mulKmx",
"mulmxA",
"mulmxK",
"mx_rsim",
"reprG",
"row_free_unit",
"unitmx_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_trans n1 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 | mx_rsim_trans | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mulmxA",
"mx_rsim",
"mxrankMfree",
"reprG",
"row_free"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_def n1 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 | mx_rsim_def | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"invmx",
"mulVmx",
"mulmxK",
"mx_rsim",
"reprG",
"row_free_unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_iso n (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 => //.
app... | Lemma | mx_rsim_iso | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"complete_unitmx",
"eqmxMr",
"eqmxP",
"hom_mxP",
"in_submod",
"in_submodE",
"in_submodJ",
"in_submodK",
"injf",
"mul1mx",
"mulmx1",
"mulmxA",
"mx_iso",
"mx_rsim",
"mxmodule",
"mxmoduleP",
"mxrank1",
"mxrankMfree",
"mxrank_leqif_eq",
"rG",
"rank",
"reprG",
"row_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_irr n1 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/mxmodule... | Lemma | mx_rsim_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqmx_module",
"genmxE",
"injf",
"last",
"mulmxA",
"mx_irrP",
"mx_irreducible",
"mx_rsim",
"mx_rsim_sym",
"mxmoduleP",
"mxrankMfree",
"mxrank_eq0",
"reprG",
"row_full",
"split",
"submxMr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_abs_irr n1 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_l... | Lemma | mx_rsim_abs_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"enum_valP",
"injf",
"invmx",
"invmxK",
"lin_mx",
"mulKmx",
"mul_rV_lin",
"mul_vec_lin",
"mulmx",
"mulmxA",
"mulmxK",
"mulmxKV",
"mulmxr",
"mx_absolutely_irreducible",
"mx_rV_lin",
"mx_rsim",
"mxrankMfree",
"rank",
"reprG",
"row1",
"rowE",
"rowK",
"row_free",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker_mx_rsim n1 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 | rker_mx_rsim | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"can_eq",
"inE",
"injf",
"mul1mx",
"mulKmx",
"mulmxK",
"mx_rsim",
"reprG",
"rker",
"row_free_unit",
"scalar_mxC",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_faithful n1 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 | mx_rsim_faithful | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_faithful",
"mx_rsim",
"reprG",
"rker_mx_rsim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_factmod n (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_subm... | Lemma | mx_rsim_factmod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"add0r",
"add_sub_fact_mod",
"addnK",
"addrK",
"apply",
"cokermx",
"eq_sym",
"factmod_repr",
"hom_mxP",
"in_submod",
"in_submodJ",
"in_submodK",
"mulmxA",
"mulmxBl",
"mulmxDl",
"mx_rsim",
"mxdirect",
"mxdirectP",
"mxdirect_addsP",
"mxmodule",
"mxrank1",
"mxrank_coker",
"p... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_rsim n1 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 | mxtrace_rsim | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mul1mx",
"mulmxA",
"mx_rsim",
"mx_rsim_def",
"mxtrace_mulC",
"reprG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_scalar n1 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 | mx_rsim_scalar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"mx_rsim",
"mx_rsim_sym",
"reprG",
"row_free_inj",
"scalar_mxC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_irr (W : sG) : mx_irreducible (socle_repr W). | Proof. by apply/submod_mx_irr; apply: socle_simple. Qed. | Lemma | socle_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"mx_irreducible",
"sG",
"socle_repr",
"socle_simple",
"submod_mx_irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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. | Lemma | socle_rsimP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"component_mx_isoP",
"mx_rsim",
"mx_rsim_iso",
"sG",
"socle_repr",
"socle_simple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mG U | := (mxmodule rG U). | Notation | mG | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxmodule",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sr modV | := (submod_repr modV). | Notation | sr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"submod_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_in_submod U 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 r... | Lemma | mx_rsim_in_submod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqmx_module",
"genmxE",
"in_submod",
"in_submodE",
"in_submodJ",
"in_submodK",
"in_submod_module",
"mG",
"mul1mx",
"mulmxA",
"mx_rsim",
"mxmodule",
"mxrank_in_submod",
"rank",
"row_freeP",
"sr",
"submxMr",
"val_submod",
"val_submod1",
"val_submodE",
"val_submodJ",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rsim_submod1 U (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 | rsim_submod1 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mG",
"mul1mx",
"mx_rsim",
"mxrank1",
"rG",
"row_free",
"sr",
"val_submod",
"val_submod1",
"val_submodE",
"val_submodJ"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_submod1 U (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 | mxtrace_submod1 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"defU",
"mG",
"mxtrace_rsim",
"rG",
"rsim_submod1",
"sr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_dadd_mod U 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_submod... | Lemma | mxtrace_dadd_mod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addrC",
"adds_eqmx",
"addsmxMr",
"addsmx_sub",
"apply",
"eqmxP",
"eqnP",
"genmxE",
"in_submod",
"in_submodK",
"mG",
"mx_rsim_factmod",
"mx_rsim_in_submod",
"mxdirect",
"mxdirectP",
"mxrank1",
"mxrank_in_submod",
"mxtrace_rsim",
"mxtrace_sub_fact_mod",
"simU",
"sr",
"submx1... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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)... | Lemma | mxtrace_dsum_mod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"P0",
"bigD1",
"big_pred0",
"cardD1x",
"eqmx_refl",
"flatmx0",
"last",
"linear0",
"ltnS",
"mG",
"mxdirect",
"mxdirectE",
"mxdirect_addsE",
"mxdirect_addsP",
"mxrank0",
"mxtrace_dadd_mod",
"pickP",
"sr",
"sumsmx_module",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_component U (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 // -def... | Lemma | mxtrace_component | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"component_mx",
"component_mx_iso",
"component_mx_module",
"component_mx_semisimple",
"eq_bigr",
"lt0n",
"mulnK",
"mx_iso",
"mx_rsim_iso",
"mxdirectP",
"mxrank_eq0",
"mxrank_iso",
"mxsimple",
"mxsimple_module",
"mxtrace_dsum_mod",
"mxtrace_rsim",
"rG",
"rank",
"simU",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | mxtrace_Socle | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Socle_direct",
"Socle_module",
"apply",
"component_mx_module",
"eq_bigr",
"eqmx_refl",
"mxtrace_component",
"mxtrace_dsum_mod",
"rG",
"sG",
"socle_base",
"socle_mult",
"socle_repr",
"socle_simple",
"sr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nHG | := normal_norm nsHG. | Let | nHG | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"normal_norm",
"nsHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rH | := subg_repr rG sHG. | Let | rH | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"rG",
"sHG",
"subg_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Clifford_simple M 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,... | Lemma | Clifford_simple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Hh",
"apply",
"can_eq",
"conjgCV",
"groupJ",
"groupV",
"groupVr",
"mem_conjg",
"mul0mx",
"mulmxA",
"mxmodule",
"mxmoduleP",
"mxsimple",
"nHG",
"normsP",
"rG",
"rH",
"repr_mxK",
"repr_mxKV",
"repr_mxM",
"sHG",
"split",
"submxMr",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Clifford_hom x 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 | Clifford_hom | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Hh",
"apply",
"centP",
"dom_hom_mx",
"hom_mxP",
"mulmxA",
"rG",
"rH",
"rV_subP",
"repr_mxM",
"sHG",
"setIP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Clifford_iso x 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 | Clifford_iso | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_hom",
"mx_iso",
"rG",
"rH",
"repr_mx_unit",
"setIP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Clifford_iso2 x 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 ... | Lemma | Clifford_iso2 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Hh",
"apply",
"conjgCV",
"eqmxMr",
"groupJ",
"groupM",
"groupVr",
"hom_mxP",
"injf",
"invgK",
"last",
"mem_conjg",
"mulmxA",
"mx_iso",
"nHG",
"normsP",
"rG",
"rH",
"repr_mxK",
"repr_mxKV",
"repr_mxM",
"repr_mx_unit",
"sHG",
"subsetP",
"unitmx_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Clifford_componentJ M 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_... | Lemma | Clifford_componentJ | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_iso2",
"Clifford_simple",
"apply",
"component_mx",
"component_mx_def",
"eqmxP",
"groupV",
"mx_iso_component",
"mxsimple",
"rG",
"rH",
"repr_mxK",
"repr_mxKV",
"submxMr",
"sumsmxMr",
"sumsmx_subP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Clifford_basis M : 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... | Lemma | Clifford_basis | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_simple",
"apply",
"eq_bigl",
"eqmxP",
"gT",
"groupM",
"imsetP",
"irrG",
"last",
"mulmx1",
"mulmxA",
"mxdirect",
"mxdirectE",
"mxmoduleP",
"mxsimple",
"on",
"rG",
"rH",
"reindex",
"repr_mx1",
"repr_mxM",
"sgval",
"sgvalK",
"subg",
"subgK",
"submx0",
"subm... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Clifford_act (W : sH) x | :=
let Gx := subgP (subg G x) in
PackSocle (component_socle sH (Clifford_simple (socle_simple W) Gx)). | Definition | Clifford_act | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_simple",
"component_socle",
"socle_simple",
"subg",
"subgP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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. | Let | valWact | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_act",
"Clifford_componentJ",
"PackSocleK",
"apply",
"rG",
"sgval",
"socle_simple",
"subg",
"subgP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 _ _)).
... | Fact | Clifford_is_action | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_act",
"apply",
"eqmxMr",
"eqmxP",
"eqmx_sym",
"eqmx_trans",
"groupM",
"is_action",
"mulmxA",
"rG",
"repr_mxK",
"repr_mxM",
"socleP",
"split",
"subg",
"subgK",
"subgP",
"valWact"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Clifford_action | := Action Clifford_is_action. | Definition | Clifford_action | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_is_action"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'Cl" | := Clifford_action : action_scope. | Notation | 'Cl | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_action"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_Clifford_act W 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 | val_Clifford_act | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"act",
"apply",
"eqmx_trans",
"rG",
"subgK",
"valWact"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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_simpl... | Lemma | Clifford_atrans | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_basis",
"Clifford_componentJ",
"Clifford_simple",
"PackSocleK",
"apply",
"component_mx_isoP",
"component_socle",
"eqmxP",
"eqmx_sym",
"eqmx_trans",
"genmxP",
"genmx_component",
"hom_mxsemisimple_iso",
"imsetP",
"inE",
"irrG",
"mulmx1",
"mxmodule1",
"mxsimple_exists",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | Clifford_Socle1 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_atrans",
"Clifford_basis",
"Clifford_simple",
"Socle",
"apply",
"imsetP",
"intro_mxsemisimple",
"mxsemisimple_reducible",
"reducible_Socle1",
"sXG",
"socle_simple",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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_Cliffor... | Lemma | Clifford_rank_components | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_Socle1",
"Clifford_atrans",
"Socle_direct",
"apply",
"eq_bigr",
"imsetP",
"inE",
"mxdirectP",
"mxrank1",
"mxrankMfree",
"orbit",
"orbitP",
"rank",
"repr_mx_free",
"sum_nat_const",
"val_Clifford_act"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Clifford_component_basis M : 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.
app... | Theorem | Clifford_component_basis | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_Socle1",
"Clifford_basis",
"Clifford_rank_components",
"Clifford_simple",
"PackSocleK",
"Socle",
"Socle_direct",
"addsmxS",
"apply",
"bigD1",
"bigID",
"big_pred0",
"cardD1",
"cast_ord",
"cast_ord_id",
"component_mx_id",
"component_socle",
"enum_val",
"enum_valP",
"enu... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 /= PackSocle... | Lemma | Clifford_astab | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_iso",
"Clifford_simple",
"Hh",
"PackSocleK",
"apply",
"component_mx_id",
"component_mx_iso",
"component_mx_isoP",
"eq_sym",
"inE",
"join_subG",
"mxmoduleP",
"mxsimple",
"rG",
"rH",
"sHG",
"setIP",
"socle_base",
"socle_simple",
"split",
"subgK",
"submx_trans",
"s... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | Clifford_astab1 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"inE",
"mxrankMfree",
"mxrank_leqif_sup",
"rG",
"repr_mx_free",
"rstabs",
"setP",
"socleP",
"sub1set",
"val_Clifford_act"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 rew... | Lemma | Clifford_rstabs_simple | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Clifford_astab",
"Clifford_astab1",
"Clifford_component_basis",
"Hh",
"act",
"actKin",
"apply",
"astab_act",
"canF_eq",
"component_mx_disjoint",
"eqVneq",
"eqmxMr",
"groupV",
"inE",
"last",
"mem_gen",
"mxmodule",
"mxmoduleP",
"mxsimple",
"mxsimple_exists",
"nz_socle",
"rG"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modG | := ((mxmodule rG) n). | Notation | modG | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxmodule",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | section_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addsmx_module",
"eqmx_module",
"factmod_repr",
"genmxE",
"in_factmod",
"in_factmod_module",
"modG",
"mxmodule"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
section_repr U V (modU : modG U) (modV : modG V) | :=
submod_repr (section_module modU modV). | Definition | section_repr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"modG",
"section_module",
"submod_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_factmod_sub U 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 | mx_factmod_sub | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addIn",
"addsmx_idPr",
"apply",
"factmod_repr",
"genmxE",
"mul1mx",
"mx_rsim",
"mxmodule1",
"mxrank1",
"mxrank_coker",
"mxrank_in_factmod",
"rG",
"rank",
"rank_leq_row",
"row_free",
"section_repr",
"submx1",
"subnK",
"val_submod",
"val_submod1",
"val_submodE",
"val_submodJ... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_submod (U V : 'M_n) | :=
(U < V)%MS /\ (forall W, ~ [/\ modG W, U < W & W < V])%MS. | Definition | max_submod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"modG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_submodP U 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 (sub... | Lemma | max_submodP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"adds0mx",
"addsmxMr",
"addsmxS",
"addsmxSl",
"addsmx_module",
"addsmx_sub",
"apply",
"eqmx0",
"eqmx_module",
"genmxE",
"in_factmod",
"in_factmodE",
"in_factmod_eq0",
"in_factmod_module",
"in_factmodsK",
"in_submod",
"in_submodK",
"in_submod_module",
"inj_eq",
"linear0",
"lt0... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_submod_eqmx U1 U2 V1 V2 :
(U1 :=: U2)%MS -> (V1 :=: V2)%MS -> max_submod U1 V1 -> max_submod U2 V2. | Proof.
move=> eqU12 eqV12 [ltUV1 maxU1].
by split=> [|W]; rewrite -(lt_eqmx eqU12) -(lt_eqmx eqV12).
Qed. | Lemma | max_submod_eqmx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"lt_eqmx",
"max_submod",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_subseries | := all modG. | Definition | mx_subseries | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"all",
"modG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_composition_series V | :=
mx_subseries V /\ (forall i, i < size V -> max_submod (0 :: V)`_i V`_i). | Definition | mx_composition_series | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"max_submod",
"mx_subseries",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_series | := mx_composition_series. | Notation | mx_series | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_composition_series"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_subseries_module V i : mx_subseries V -> mxmodule rG V`_i. | Proof.
move=> modV; have [|leVi] := ltnP i (size V); first exact: all_nthP.
by rewrite nth_default ?mxmodule0.
Qed. | Fact | mx_subseries_module | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"all_nthP",
"ltnP",
"mx_subseries",
"mxmodule",
"mxmodule0",
"nth_default",
"rG",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_subseries_module' V i : mx_subseries V -> mxmodule rG (0 :: V)`_i. | Proof. by move=> modV; rewrite mx_subseries_module //= mxmodule0. Qed. | Fact | mx_subseries_module' | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_subseries",
"mx_subseries_module",
"mxmodule",
"mxmodule0",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseries_repr V i (modV : all modG V) | :=
section_repr (mx_subseries_module' i modV) (mx_subseries_module i modV). | Definition | subseries_repr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"all",
"modG",
"mx_subseries_module",
"mx_subseries_module'",
"section_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
series_repr V i (compV : mx_composition_series V) | :=
subseries_repr i (proj1 compV). | Definition | series_repr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_composition_series",
"subseries_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_series_lt V : mx_composition_series V -> path ltmx 0 V. | Proof. by case=> _ compV; apply/(pathP 0)=> i /compV[]. Qed. | Lemma | mx_series_lt | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"ltmx",
"mx_composition_series",
"path",
"pathP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_size_mx_series (V : seq 'M[F]_n) :
path ltmx 0 V -> size V <= \rank (last 0 V). | Proof.
rewrite -[size V]addn0 -(mxrank0 F n n); elim: V 0 => //= V1 V IHV V0.
rewrite ltmxErank -andbA => /and3P[_ ltV01 ltV].
by apply: leq_trans (IHV _ ltV); rewrite addSnnS leq_add2l.
Qed. | Lemma | max_size_mx_series | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addSnnS",
"addn0",
"apply",
"last",
"leq_add2l",
"leq_trans",
"ltmx",
"ltmxErank",
"mxrank0",
"path",
"rank",
"seq",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_series_repr_irr V i (compV : mx_composition_series V) :
i < size V -> mx_irreducible (series_repr i compV). | Proof.
case: compV => modV compV /compV maxVi; apply/max_submodP => //.
by apply: ltmxW; case: maxVi.
Qed. | Lemma | mx_series_repr_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"ltmxW",
"max_submodP",
"mx_composition_series",
"mx_irreducible",
"series_repr",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_series_rcons U V :
mx_series (rcons U V) <-> [/\ mx_series U, modG V & max_submod (last 0 U) V]. | Proof.
rewrite /mx_series /mx_subseries all_rcons size_rcons -rcons_cons.
split=> [ [/andP[modU modV] maxU] | [[modU maxU] modV maxV]].
split=> //; last first.
by have:= maxU _ (leqnn _); rewrite !nth_rcons leqnn ltnn eqxx -last_nth.
by split=> // i ltiU; have:= maxU i (ltnW ltiU); rewrite !nth_rcons leqW ltiU.... | Lemma | mx_series_rcons | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"all_rcons",
"apply",
"eqxx",
"last",
"last_nth",
"leqW",
"leq_eqVlt",
"leqnn",
"ltnS",
"ltnW",
"ltnn",
"max_submod",
"modG",
"mx_series",
"mx_subseries",
"nth_rcons",
"rcons",
"rcons_cons",
"size_rcons",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_Schreier U :
mx_subseries U -> path ltmx 0 U ->
classically (exists V, [/\ mx_series V, last 0 V :=: 1%:M & subseq U V])%MS. | Proof.
move: U => U0; set U := {1 2}U0; have: subseq U0 U := subseq_refl U.
pose n' := n.+1; have: n < size U + n' by rewrite leq_addl.
elim: n' U => [|n' IH_U] U ltUn' sU0U modU incU [] // noV.
rewrite addn0 ltnNge in ltUn'; case/negP: ltUn'.
by rewrite (leq_trans (max_size_mx_series incU)) ?rank_leq_row.
apply: (... | Theorem | mx_Schreier | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addSnnS",
"addn0",
"addnS",
"all_cat",
"all_rcons",
"apply",
"cat_cons",
"cat_path",
"cat_subseq",
"cat_take_drop",
"defU",
"drop",
"drop_nth",
"eqmxP",
"last",
"last_nth",
"leq_addl",
"leq_trans",
"leqnn",
"ltmx",
"ltmxEneq",
"ltnNge",
"max_size_mx_series",
"mx_series... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_second_rsim U V (modU : modG U) (modV : modG V) :
let modI := capmx_module modU modV in let modA := addsmx_module modU modV in
mx_rsim (section_repr modI modU) (section_repr modV modA). | Proof.
move=> modI modA; set nI := {1}(\rank _).
have sIU := capmxSl U V; have sVA := addsmxSr U V.
pose valI := val_factmod (val_submod (1%:M : 'M_nI)).
have UvalI: (valI <= U)%MS.
rewrite -(addsmx_idPr sIU) (submx_trans _ (proj_factmodS _ _)) //.
by rewrite submxMr // val_submod1 genmxE.
exists (valI *m in_factmo... | Lemma | mx_second_rsim | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addIn",
"addnA",
"addnC",
"addnCA",
"addsmxC",
"addsmxSr",
"addsmx_idPr",
"addsmx_module",
"apply",
"capmxSl",
"capmxSr",
"capmx_module",
"eq_sym",
"genmxE",
"in_factmod",
"in_factmodE",
"in_factmodJ",
"in_factmod_addsK",
"in_factmod_eq0",
"in_submod",
"in_submodE",
"in_su... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
section_eqmx_add U1 U2 V1 V2 modU1 modU2 modV1 modV2 :
(U1 :=: U2)%MS -> (U1 + V1 :=: U2 + V2)%MS ->
mx_rsim (@section_repr U1 V1 modU1 modV1) (@section_repr U2 V2 modU2 modV2). | Proof.
move=> eqU12 eqV12; set n1 := {1}(\rank _).
pose v1 := val_factmod (val_submod (1%:M : 'M_n1)).
have sv12: (v1 <= U2 + V2)%MS.
rewrite -eqV12 (submx_trans _ (proj_factmodS _ _)) //.
by rewrite submxMr // val_submod1 genmxE.
exists (v1 *m in_factmod _ 1%:M *m in_submod _ 1%:M) => [||x Gx].
- apply: (@addIn (\... | Lemma | section_eqmx_add | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addIn",
"apply",
"eq_sym",
"genmxE",
"in_factmod",
"in_factmodE",
"in_factmodJ",
"in_factmod_addsK",
"in_factmod_eq0",
"in_submod",
"in_submodE",
"in_submodJ",
"in_submodK",
"inj_eq",
"kermx_eq0",
"linear0",
"linearB",
"mul1mx",
"mulmxA",
"mulmx_sub",
"mx_rsim",
"mxrank_in... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
section_eqmx U1 U2 V1 V2 modU1 modU2 modV1 modV2
(eqU : (U1 :=: U2)%MS) (eqV : (V1 :=: V2)%MS) :
mx_rsim (@section_repr U1 V1 modU1 modV1) (@section_repr U2 V2 modU2 modV2). | Proof. by apply: section_eqmx_add => //; apply: adds_eqmx. Qed. | Lemma | section_eqmx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"adds_eqmx",
"apply",
"mx_rsim",
"section_eqmx_add",
"section_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_butterfly U V W modU modV modW :
~~ (U == V)%MS -> max_submod U W -> max_submod V W ->
let modUV := capmx_module modU modV in
max_submod (U :&: V)%MS U
/\ mx_rsim (@section_repr V W modV modW) (@section_repr _ U modUV modU). | Proof.
move=> neUV maxU maxV modUV; have{neUV maxU} defW: (U + V :=: W)%MS.
wlog{neUV modUV} ltUV: U V modU modV maxU maxV / ~~ (V <= U)%MS.
by case/nandP: neUV => ?; first rewrite addsmxC; apply.
apply/eqmxP/idPn=> neUVW; case: maxU => ltUW; case/(_ (U + V)%MS).
rewrite addsmx_module // ltmxE ltmxEneq neUVW ... | Lemma | mx_butterfly | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addsmxC",
"addsmxSl",
"addsmxSr",
"addsmx_module",
"addsmx_sub",
"apply",
"capmxSl",
"capmx_module",
"eqmxP",
"ltmxE",
"ltmxEneq",
"ltmxW",
"max_submod",
"max_submodP",
"mx_rsim",
"mx_rsim_irr",
"mx_rsim_sym",
"mx_rsim_trans",
"mx_second_rsim",
"section_eqmx",
"section_repr"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_JordanHolder_exists U V :
mx_composition_series U -> modG V -> max_submod V (last 0 U) ->
{W : seq 'M_n | mx_composition_series W & last 0 W = V}. | Proof.
elim/last_ind: U V => [|U Um IHU] V compU modV; first by case; rewrite ltmx0.
rewrite last_rcons => maxV; case/mx_series_rcons: compU => compU modUm maxUm.
case eqUV: (last 0 U == V)%MS.
case/lastP: U eqUV compU {maxUm IHU} => [|U' Um'].
by rewrite andbC; move/eqmx0P->; exists [::].
rewrite last_rcons; m... | Lemma | mx_JordanHolder_exists | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"capmxC",
"capmx_module",
"compU",
"eqmx0P",
"eqmxP",
"last",
"lastP",
"last_ind",
"last_nth",
"last_rcons",
"ltmx0",
"max_submod",
"max_submod_eqmx",
"modG",
"mx_butterfly",
"mx_composition_series",
"mx_series_rcons",
"mx_subseries_module'",
"rcons",
"seq",
"split... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rsim_rcons U V compU compUV i : i < size U ->
mx_rsim (@series_repr U i compU) (@series_repr (rcons U V) i compUV). | Proof.
by move=> ltiU; apply: section_eqmx; rewrite -?rcons_cons nth_rcons ?leqW ?ltiU.
Qed. | Let | rsim_rcons | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"compU",
"leqW",
"mx_rsim",
"nth_rcons",
"rcons",
"rcons_cons",
"section_eqmx",
"series_repr",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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