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 |
|---|---|---|---|---|---|---|---|---|---|---|
third_isom : {f : {morphism (G / H) / (K / H) >-> coset_of K} | 'injm f
& forall A : {set gT}, A \subset G -> f @* (A / H / (K / H)) = A / K}. | Proof.
have [[sKG nKG] [sHG nHG]] := (andP snKG, andP snHG).
have sHker: 'ker (coset H) \subset 'ker (restrm nKG (coset K)).
by rewrite ker_restrm !ker_coset subsetI sHG.
have:= first_isom_loc (factm_morphism sHker nHG) (subxx _) => /=.
rewrite ker_factm_loc ker_restrm ker_coset !(setIidPr sKG) /= -!quotientE.
case=>... | Theorem | third_isom | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"coset",
"coset_of",
"factm_morphism",
"first_isom_loc",
"gT",
"injf",
"ker",
"ker_coset",
"ker_factm_loc",
"ker_restrm",
"morphimS",
"morphim_factm",
"morphim_restrm",
"morphism",
"nHG",
"nKG",
"quotientE",
"restrm",
"sAG",
"sHG",
"sKG",
"setIidPr",
"snHG",
"snKG",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
third_isog : (G / H / (K / H)) \isog (G / K). | Proof.
by case: third_isom => f inj_f im_f; apply/isogP; exists f; rewrite ?im_f.
Qed. | Theorem | third_isog | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"apply",
"inj_f",
"isog",
"isogP",
"third_isom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_from_quotient (gT : finGroupType) (G H K : {group gT}) :
H <| K -> H \char G -> K / H \char G / H -> K \char G. | Proof.
case/andP=> sHK nHK chHG.
have nsHG := char_normal chHG; have [sHG nHG] := andP nsHG.
case/charP; rewrite quotientSGK // => sKG /= chKG.
apply/charP; split=> // f injf Gf; apply/morphim_fixP => //.
rewrite -(quotientSGK _ sHK); first by rewrite -morphimIim Gf subIset ?nHG.
have{chHG} Hf: f @* H = H by case/charP... | Lemma | char_from_quotient | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"Gf",
"Hf",
"apply",
"char",
"charP",
"char_normal",
"gT",
"group",
"injf",
"injm_quotm",
"morphimIim",
"morphim_fixP",
"morphim_quotm",
"nHG",
"nHK",
"nsHG",
"quotientSGK",
"quotm_morphism",
"sHG",
"sHK",
"sKG",
"split",
"subIset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_morphim G : #|f @* G| = #|D :&: G : 'ker f|. | Proof.
rewrite -morphimIdom -indexgI -card_quotient.
by rewrite normsI ?normG ?subIset ?ker_norm.
by apply: esym (card_isog _); rewrite first_isog_loc ?subsetIl.
Qed. | Lemma | card_morphim | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"apply",
"card_isog",
"card_quotient",
"first_isog_loc",
"indexgI",
"ker",
"ker_norm",
"morphimIdom",
"normG",
"normsI",
"subIset",
"subsetIl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_morphim G : #|f @* G| %| #|G|. | Proof.
rewrite card_morphim (dvdn_trans (dvdn_indexg _ _)) //.
by rewrite cardSg ?subsetIr.
Qed. | Lemma | dvdn_morphim | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"cardSg",
"card_morphim",
"dvdn_indexg",
"dvdn_trans",
"subsetIr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
logn_morphim p G : logn p #|f @* G| <= logn p #|G|. | Proof. by rewrite dvdn_leq_log ?dvdn_morphim. Qed. | Lemma | logn_morphim | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"dvdn_leq_log",
"dvdn_morphim",
"logn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprime_morphl G p : coprime #|G| p -> coprime #|f @* G| p. | Proof. exact: coprime_dvdl (dvdn_morphim G). Qed. | Lemma | coprime_morphl | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"coprime",
"coprime_dvdl",
"dvdn_morphim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprime_morphr G p : coprime p #|G| -> coprime p #|f @* G|. | Proof. exact: coprime_dvdr (dvdn_morphim G). Qed. | Lemma | coprime_morphr | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"coprime",
"coprime_dvdr",
"dvdn_morphim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprime_morph G H : coprime #|G| #|H| -> coprime #|f @* G| #|f @* H|. | Proof. by move=> coGH; rewrite coprime_morphl // coprime_morphr. Qed. | Lemma | coprime_morph | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"coprime",
"coprime_morphl",
"coprime_morphr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_morphim_ker G H :
H \subset G -> G \subset D ->
(#|f @* G : f @* H| * #|'ker_G f : H|)%N = #|G : H|. | Proof.
move=> sHG sGD; apply/eqP.
rewrite -(eqn_pmul2l (cardG_gt0 (f @* H))) mulnA Lagrange ?morphimS //.
rewrite !card_morphim (setIidPr sGD) (setIidPr (subset_trans sHG sGD)).
rewrite -(eqn_pmul2l (cardG_gt0 ('ker_H f))) /=.
by rewrite -{1}(setIidPr sHG) setIAC mulnCA mulnC mulnA !LagrangeI Lagrange.
Qed. | Lemma | index_morphim_ker | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"Lagrange",
"LagrangeI",
"apply",
"cardG_gt0",
"card_morphim",
"eqn_pmul2l",
"morphimS",
"mulnA",
"mulnC",
"mulnCA",
"sGD",
"sHG",
"setIAC",
"setIidPr",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_morphim G H : G :&: H \subset D -> #|f @* G : f @* H| %| #|G : H|. | Proof.
move=> dGH; rewrite -(indexgI G) -(setIidPr dGH) setIA.
apply: dvdn_trans (indexSg (subsetIl _ H) (subsetIr D G)).
rewrite -index_morphim_ker ?subsetIl ?subsetIr ?dvdn_mulr //= morphimIdom.
by rewrite indexgS ?morphimS ?subsetIr.
Qed. | Lemma | index_morphim | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"apply",
"dvdn_mulr",
"dvdn_trans",
"indexSg",
"index_morphim_ker",
"indexgI",
"indexgS",
"morphimIdom",
"morphimS",
"setIA",
"setIidPr",
"subsetIl",
"subsetIr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_injm G H : 'injm f -> G \subset D -> #|f @* G : f @* H| = #|G : H|. | Proof.
move=> injf dG; rewrite -{2}(setIidPr dG) -(indexgI _ H) /=.
rewrite -index_morphim_ker ?subsetIl ?subsetIr //= setIAC morphimIdom setIC.
rewrite injmI ?subsetIr // indexgI /= morphimIdom setIC ker_injm //.
by rewrite -(indexgI (1 :&: _)) /= -setIA !(setIidPl (sub1G _)) indexgg muln1.
Qed. | Lemma | index_injm | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"index_morphim_ker",
"indexgI",
"indexgg",
"injf",
"injmI",
"ker_injm",
"morphimIdom",
"muln1",
"setIA",
"setIAC",
"setIC",
"setIidPl",
"setIidPr",
"sub1G",
"subsetIl",
"subsetIr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_morphpre L : L \subset f @* D -> #|f @*^-1 L| = (#|'ker f| * #|L|)%N. | Proof.
move/morphpreK=> {2} <-; rewrite card_morphim morphpreIdom.
by rewrite Lagrange // morphpreS ?sub1G.
Qed. | Lemma | card_morphpre | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"Lagrange",
"card_morphim",
"ker",
"morphpreIdom",
"morphpreK",
"morphpreS",
"sub1G"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_morphpre L M :
L \subset f @* D -> #|f @*^-1 L : f @*^-1 M| = #|L : M|. | Proof.
move=> dL; rewrite -!divgI -morphpreI /= card_morphpre //.
have: L :&: M \subset f @* D by rewrite subIset ?dL.
by move/card_morphpre->; rewrite divnMl ?cardG_gt0.
Qed. | Lemma | index_morphpre | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"cardG_gt0",
"card_morphpre",
"divgI",
"divnMl",
"morphpreI",
"subIset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_homg (aT rT : finGroupType) (G : {group aT}) (R : {group rT}) :
G \homg R -> #|G| %| #|R|. | Proof. by case/homgP=> f <-; rewrite card_morphim setIid dvdn_indexg. Qed. | Lemma | card_homg | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"aT",
"card_morphim",
"dvdn_indexg",
"group",
"homg",
"homgP",
"setIid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_quotient : #|G / H| %| #|G|. | Proof. exact: dvdn_morphim. Qed. | Lemma | dvdn_quotient | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"dvdn_morphim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_quotient_ker :
K \subset G -> G \subset 'N(H) ->
(#|G / H : K / H| * #|G :&: H : K|)%N = #|G : K|. | Proof. by rewrite -{5}(ker_coset H); apply: index_morphim_ker. Qed. | Lemma | index_quotient_ker | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"apply",
"index_morphim_ker",
"ker_coset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_quotient : G :&: K \subset 'N(H) -> #|G / H : K / H| %| #|G : K|. | Proof. exact: index_morphim. Qed. | Lemma | index_quotient | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"index_morphim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_quotient_eq :
G :&: H \subset K -> K \subset G -> G \subset 'N(H) ->
#|G / H : K / H| = #|G : K|. | Proof.
move=> sGH_K sKG sGN; rewrite -index_quotient_ker {sKG sGN}//.
by rewrite -(indexgI _ K) (setIidPl sGH_K) indexgg muln1.
Qed. | Lemma | index_quotient_eq | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"index_quotient_ker",
"indexgI",
"indexgg",
"muln1",
"sKG",
"setIidPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_cosetpre : #|coset H @*^-1 L| = (#|H| * #|L|)%N. | Proof. by rewrite card_morphpre ?ker_coset ?sub_im_coset. Qed. | Lemma | card_cosetpre | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"card_morphpre",
"coset",
"ker_coset",
"sub_im_coset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_cosetpre : #|coset H @*^-1 L : coset H @*^-1 M| = #|L : M|. | Proof. by rewrite index_morphpre ?sub_im_coset. Qed. | Lemma | index_cosetpre | finite_group | finite_group/quotient.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"prime",
"finset",
"fingroup",
"morphism",
"automorphism"
] | [
"coset",
"index_morphpre",
"sub_im_coset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
groupC : group_closure_field algC gT. | Proof. exact: group_closure_closed_field. Qed. | Lemma | groupC | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"algC",
"gT",
"group_closure_closed_field",
"group_closure_field"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trow (n1 : nat) :
forall (A : 'rV[F]_n1) m2 n2 (B : 'M[F]_(m2,n2)), 'M[F]_(m2,n1 * n2) | :=
if n1 is n'1.+1
then
fun (A : 'M[F]_(1,(1 + n'1))) m2 n2 (B : 'M[F]_(m2,n2)) =>
(row_mx (lsubmx A 0 0 *: B) (trow (rsubmx A) B))
else (fun _ _ _ _ => 0). | Fixpoint | trow | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"lsubmx",
"nat",
"row_mx",
"rsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trow0 n1 m2 n2 B : @trow n1 0 m2 n2 B = 0. | Proof.
elim: n1=> //= n1 IH.
rewrite !mxE scale0r linear0.
rewrite IH //; apply/matrixP=> i j; rewrite !mxE.
by case: split=> *; rewrite mxE.
Qed. | Lemma | trow0 | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"linear0",
"matrixP",
"mxE",
"scale0r",
"split",
"trow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trowb n1 m2 n2 B A | := @trow n1 A m2 n2 B. | Definition | trowb | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"trow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trowbE n1 m2 n2 A B : trowb B A = @trow n1 A m2 n2 B. | Proof. by []. Qed. | Lemma | trowbE | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"trow",
"trowb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trowb_is_linear n1 m2 n2 (B : 'M_(m2,n2)) : linear (@trowb n1 m2 n2 B). | Proof.
elim: n1=> [|n1 IH] //= k A1 A2 /=; first by rewrite scaler0 add0r.
rewrite !linearD /= !linearZ /= IH 2!mxE.
by rewrite scalerDl -scalerA -add_row_mx -scale_row_mx.
Qed. | Lemma | trowb_is_linear | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"add0r",
"add_row_mx",
"linear",
"linearD",
"linearZ",
"mxE",
"scale_row_mx",
"scaler0",
"scalerA",
"scalerDl",
"trowb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trow_is_linear n1 m2 n2 (A : 'rV_n1) : linear (@trow n1 A m2 n2). | Proof.
elim: n1 A => [|n1 IH] //= A k A1 A2 /=; first by rewrite scaler0 add0r.
rewrite linearP /=; apply/matrixP=> i j; rewrite !mxE.
by case: split=> a; rewrite ?IH !mxE.
Qed. | Lemma | trow_is_linear | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"add0r",
"apply",
"linear",
"linearP",
"matrixP",
"mxE",
"scaler0",
"split",
"trow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tprod (m1 : nat) :
forall n1 (A : 'M[F]_(m1,n1)) m2 n2 (B : 'M[F]_(m2,n2)),
'M[F]_(m1 * m2,n1 * n2) | :=
if m1 is m'1.+1
return forall n1 (A : 'M[F]_(m1,n1)) m2 n2 (B : 'M[F]_(m2,n2)),
'M[F]_(m1 * m2,n1 * n2)
then
fun n1 (A : 'M[F]_(1 + m'1,n1)) m2 n2 B =>
(col_mx (trow (usubmx A) B) (tprod (dsubmx A) B))
else (fun _ _ _ _ _ => 0). | Fixpoint | tprod | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"col_mx",
"dsubmx",
"nat",
"trow",
"usubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dsumx_mul m1 m2 n p A B :
dsubmx ((A *m B) : 'M[F]_(m1 + m2, n)) = dsubmx (A : 'M_(m1 + m2, p)) *m B. | Proof.
apply/matrixP=> i j /[!mxE]; apply: eq_bigr=> k _.
by rewrite !mxE.
Qed. | Lemma | dsumx_mul | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"dsubmx",
"eq_bigr",
"matrixP",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
usumx_mul m1 m2 n p A B :
usubmx ((A *m B) : 'M[F]_(m1 + m2, n)) = usubmx (A : 'M_(m1 + m2, p)) *m B. | Proof.
by apply/matrixP=> i j /[!mxE]; apply: eq_bigr=> k _ /[!mxE].
Qed. | Lemma | usumx_mul | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"eq_bigr",
"matrixP",
"mxE",
"usubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trow_mul (m1 m2 n2 p2 : nat)
(A : 'rV_m1) (B1: 'M[F]_(m2,n2)) (B2 :'M[F]_(n2,p2)) :
trow A (B1 *m B2) = B1 *m trow A B2. | Proof.
elim: m1 A => [|m1 IH] A /=; first by rewrite mulmx0.
by rewrite IH mul_mx_row -scalemxAr.
Qed. | Let | trow_mul | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"mul_mx_row",
"mulmx0",
"nat",
"scalemxAr",
"trow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tprodE m1 n1 p1 (A1 :'M[F]_(m1,n1)) (A2 :'M[F]_(n1,p1))
m2 n2 p2 (B1 :'M[F]_(m2,n2)) (B2 :'M[F]_(n2,p2)) :
tprod (A1 *m A2) (B1 *m B2) = (tprod A1 B1) *m (tprod A2 B2). | Proof.
elim: m1 n1 p1 A1 A2 m2 n2 p2 B1 B2 => /= [|m1 IH].
by move=> *; rewrite mul0mx.
move=> n1 p1 A1 A2 m2 n2 p2 B1 B2.
rewrite mul_col_mx -IH.
congr col_mx; last by rewrite dsumx_mul.
rewrite usumx_mul.
elim: n1 {A1}(usubmx (A1: 'M_(1 + m1, n1))) p1 A2=> //= [u p1 A2|].
by rewrite [A2](flatmx0) !mulmx0 -trowbE ... | Lemma | tprodE | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"col_mx",
"dsubmx",
"dsumx_mul",
"flatmx0",
"hsubmxK",
"last",
"linear0",
"linearD",
"linearZ",
"lsubmx",
"mul0mx",
"mul_col_mx",
"mul_row_col",
"mul_scalar_mx",
"mulmx0",
"mx11_scalar",
"rsubmx",
"scalemxAl",
"tprod",
"trow_mul",
"trowbE",
"usubmx",
"usumx_mul",
"vsubm... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tprod_tr m1 n1 (A :'M[F]_(m1, 1 + n1)) m2 n2 (B :'M[F]_(m2, n2)) :
tprod A B = row_mx (trow (lsubmx A)^T B^T)^T (tprod (rsubmx A) B). | Proof.
elim: m1 n1 A m2 n2 B=> [|m1 IH] n1 A m2 n2 B //=.
by rewrite trmx0 row_mx0.
rewrite !IH.
pose A1 := A : 'M_(1 + m1, 1 + n1).
have F1: dsubmx (rsubmx A1) = rsubmx (dsubmx A1).
by apply/matrixP=> i j; rewrite !mxE.
have F2: rsubmx (usubmx A1) = usubmx (rsubmx A1).
by apply/matrixP=> i j; rewrite !mxE.
have... | Let | tprod_tr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"F1",
"F2",
"F3",
"apply",
"block_mx",
"block_mxEh",
"block_mxEv",
"dsubmx",
"linearZ",
"lsubmx",
"matrixP",
"mxE",
"row_mx",
"row_mx0",
"rsubmx",
"tprod",
"tr_row_mx",
"trmx0",
"trmxK",
"trmx_dsub",
"trow",
"usubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tprod1 m n : tprod (1%:M : 'M[F]_(m,m)) (1%:M : 'M[F]_(n,n)) = 1%:M. | Proof.
elim: m n => [|m IH] n //=; first by rewrite [1%:M]flatmx0.
rewrite tprod_tr.
set u := rsubmx _; have->: u = 0.
apply/matrixP=> i j; rewrite !mxE.
by case: i; case: j=> /= j Hj; case.
set v := lsubmx (dsubmx _); have->: v = 0.
apply/matrixP=> i j; rewrite !mxE.
by case: i; case: j; case.
set w := rsubmx ... | Lemma | tprod1 | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"block_mxEv",
"dsubmx",
"eqxx",
"flatmx0",
"linear0",
"lsubmx",
"matrixP",
"mxE",
"rsubmx",
"scalar_mx_block",
"scale1r",
"tprod",
"tprod_tr",
"trowbE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_prod m n (A :'M[F]_(m)) (B :'M[F]_(n)) :
\tr (tprod A B) = \tr A * \tr B. | Proof.
elim: m n A B => [|m IH] n A B //=.
by rewrite [A]flatmx0 mxtrace0 mul0r.
rewrite tprod_tr -block_mxEv mxtrace_block IH.
rewrite linearZ/= -mulrDl -trace_mx11; congr (_ * _).
pose A1 := A : 'M_(1 + m).
rewrite -[A in RHS](@submxK _ 1 m 1 m A1).
by rewrite (@mxtrace_block _ _ _ (ulsubmx A1)).
Qed. | Lemma | mxtrace_prod | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"block_mxEv",
"flatmx0",
"linearZ",
"mul0r",
"mulrDl",
"mxtrace0",
"mxtrace_block",
"submxK",
"tprod",
"tprod_tr",
"trace_mx11",
"ulsubmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reprG | := (mx_representation R G). | Notation | reprG | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"mx_representation"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
representation | :=
Representation {rdegree; mx_repr_of_repr :> reprG rdegree}. | Record | representation | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"reprG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_repr0 : mx_repr G (fun _ : gT => 1%:M : 'M[R]_0). | Proof. by split=> // g h Hg Hx; rewrite mulmx1. Qed. | Lemma | mx_repr0 | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"gT",
"mulmx1",
"mx_repr",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
grepr0 | := Representation (MxRepresentation mx_repr0). | Definition | grepr0 | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"mx_repr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_mx_repr (rG1 rG2 : representation) :
mx_repr G (fun g => block_mx (rG1 g) 0 0 (rG2 g)). | Proof.
split=> [|x y Hx Hy]; first by rewrite !repr_mx1 -scalar_mx_block.
by rewrite mulmx_block !(mulmx0, mul0mx, addr0, add0r, repr_mxM).
Qed. | Lemma | add_mx_repr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"add0r",
"addr0",
"block_mx",
"mul0mx",
"mulmx0",
"mulmx_block",
"mx_repr",
"repr_mx1",
"repr_mxM",
"representation",
"scalar_mx_block",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dadd_grepr rG1 rG2 | :=
Representation (MxRepresentation (add_mx_repr rG1 rG2)). | Definition | dadd_grepr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"add_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_dadd (U V W : 'M_n) (rU rV : representation)
(modU : mxmodule rG U) (modV : mxmodule rG V) (modW : mxmodule rG W) :
(U + V :=: W)%MS -> mxdirect (U + V) ->
mx_rsim (submod_repr modU) rU -> mx_rsim (submod_repr modV) rV ->
mx_rsim (submod_repr modW) (dadd_grepr rU rV). | Proof.
case: rU; case: rV=> nV rV nU rU defW dxUV /=.
have tiUV := mxdirect_addsP dxUV.
move=> [fU def_nU]; rewrite -{nU}def_nU in rU fU * => inv_fU hom_fU.
move=> [fV def_nV]; rewrite -{nV}def_nV in rV fV * => inv_fV hom_fV.
pose pU := in_submod U (proj_mx U V) *m fU.
pose pV := in_submod V (proj_mx V U) *m fV.
exists... | Lemma | mx_rsim_dadd | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Gg",
"add0r",
"add_proj_mx",
"addr0",
"addrC",
"addsmxC",
"apply",
"capmxC",
"col_mx",
"dadd_grepr",
"dom_hom_mx",
"hom_mxP",
"in_submod",
"in_submodE",
"in_submodJ",
"in_submodK",
"invmx",
"mul1mx",
"mulKVmx",
"mul_mx_row",
"mul_row_block",
"mul_row_col",
"mulmx0",
"m... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_dsum (I : finType) (P : pred I) U rU (W : 'M_n)
(modU : forall i, mxmodule rG (U i)) (modW : mxmodule rG W) :
let S := (\sum_(i | P i) U i)%MS in (S :=: W)%MS -> mxdirect S ->
(forall i, mx_rsim (submod_repr (modU i)) (rU i : representation)) ->
mx_rsim (submod_repr modW) (\big[dadd_grepr/grepr0]_... | Proof.
move=> /= defW dxW rsimU.
rewrite mxdirectE /= -!(big_filter _ P) in dxW defW *.
elim: {P}(filter P _) => [|i e IHe] in W modW dxW defW *.
rewrite !big_nil /= in defW *.
by exists 0 => [||? _]; rewrite ?mul0mx ?mulmx0 // /row_free -defW !mxrank0.
rewrite !big_cons /= in dxW defW *.
rewrite 2!(big_nth i) !big... | Lemma | mx_rsim_dsum | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"big_cons",
"big_filter",
"big_mkord",
"big_nil",
"big_nth",
"dadd_grepr",
"eqmx_refl",
"eqxx",
"filter",
"grepr0",
"mul0mx",
"mulmx0",
"mx_rsim",
"mx_rsim_dadd",
"mxdirect",
"mxdirectE",
"mxdirect_addsE",
"mxdirect_addsP",
"mxmodule",
"mxrank0",
"rG",
"rU",
"r... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln_grepr rW k | := \big[dadd_grepr/grepr0]_(i < k) rW. | Definition | muln_grepr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"dadd_grepr",
"grepr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_socle (sG : socleType rG) (W : sG) (rW : representation) :
let modW : mxmodule rG W := component_mx_module rG (socle_base W) in
mx_rsim (socle_repr W) rW ->
mx_rsim (submod_repr modW) (muln_grepr rW (socle_mult W)). | Proof.
set M := socle_base W => modW rsimM.
have simM: mxsimple rG M := socle_simple W.
have rankM_gt0: (\rank M > 0)%N by rewrite lt0n mxrank_eq0; case: simM.
have [I /= U_I simU]: mxsemisimple rG W by apply: component_mx_semisimple.
pose U (i : 'I_#|I|) := U_I (enum_val i).
have reindexI := reindex _ (onW_bij I (enum... | Lemma | mx_rsim_socle | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"component_mx_iso",
"component_mx_module",
"component_mx_semisimple",
"enum_val",
"enum_val_bij",
"eq_bigr",
"lt0n",
"mulnK",
"muln_grepr",
"mx_iso",
"mx_rsim",
"mx_rsim_dsum",
"mx_rsim_iso",
"mx_rsim_sym",
"mx_rsim_trans",
"mxdirectE",
"mxdirectP",
"mxmodule",
"mxrank... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_mx_repr : mx_repr G (fun g => tprod (rG1 g) (rG2 g)). | Proof.
split=>[|i j InG JnG]; first by rewrite !repr_mx1 tprod1.
by rewrite !repr_mxM // tprodE.
Qed. | Lemma | prod_mx_repr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"mx_repr",
"repr_mx1",
"repr_mxM",
"split",
"tprod",
"tprod1",
"tprodE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_repr | := MxRepresentation prod_mx_repr. | Definition | prod_repr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"prod_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_repr_lin n2 (rG1 : reprG 1) (rG2 : reprG n2) :
{in G, forall x, let cast_n2 := esym (mul1n n2) in
prod_repr rG1 rG2 x = castmx (cast_n2, cast_n2) (rG1 x 0 0 *: rG2 x)}. | Proof.
move=> x Gx /=; set cast_n2 := esym _; rewrite /prod_repr /= !mxE !lshift0.
apply/matrixP=> i j; rewrite castmxE /=.
do 2![rewrite mxE; case: splitP => [? ? | []//]].
by congr ((_ *: rG2 x) _ _); apply: val_inj.
Qed. | Lemma | prod_repr_lin | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"castmx",
"castmxE",
"lshift0",
"matrixP",
"mul1n",
"mxE",
"prod_repr",
"reprG",
"splitP",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr_subproof n (rG : mx_representation algC G n) :
is_class_fun <<G>> [ffun x => \tr (rG x) *+ (x \in G)]. | Proof.
rewrite genGid; apply: intro_class_fun => [x y Gx Gy | _ /negbTE-> //].
by rewrite groupJr // !repr_mxM ?groupM ?groupV // mxtrace_mulC repr_mxK.
Qed. | Fact | cfRepr_subproof | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"algC",
"apply",
"genGid",
"groupJr",
"groupM",
"groupV",
"intro_class_fun",
"is_class_fun",
"mx_representation",
"mxtrace_mulC",
"rG",
"repr_mxK",
"repr_mxM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr n rG | := Cfun 0 (@cfRepr_subproof n rG). | Definition | cfRepr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Cfun",
"cfRepr_subproof",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr1 n rG : @cfRepr n rG 1%g = n%:R. | Proof. by rewrite cfunE group1 repr_mx1 mxtrace1. Qed. | Lemma | cfRepr1 | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"cfRepr",
"cfunE",
"group1",
"mxtrace1",
"rG",
"repr_mx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr_sim n1 n2 rG1 rG2 :
mx_rsim rG1 rG2 -> @cfRepr n1 rG1 = @cfRepr n2 rG2. | Proof.
case/mx_rsim_def=> f12 [f21] fK def_rG1; apply/cfun_inP=> x Gx.
by rewrite !cfunE def_rG1 // mxtrace_mulC mulmxA fK mul1mx.
Qed. | Lemma | cfRepr_sim | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"cfRepr",
"cfunE",
"cfun_inP",
"fK",
"mul1mx",
"mulmxA",
"mx_rsim",
"mx_rsim_def",
"mxtrace_mulC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr0 : cfRepr grepr0 = 0. | Proof. by apply/cfun_inP=> x Gx; rewrite !cfunE Gx mxtrace1. Qed. | Lemma | cfRepr0 | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"cfRepr",
"cfunE",
"cfun_inP",
"grepr0",
"mxtrace1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr_dadd rG1 rG2 :
cfRepr (dadd_grepr rG1 rG2) = cfRepr rG1 + cfRepr rG2. | Proof. by apply/cfun_inP=> x Gx; rewrite !cfunE Gx mxtrace_block. Qed. | Lemma | cfRepr_dadd | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"cfRepr",
"cfunE",
"cfun_inP",
"dadd_grepr",
"mxtrace_block"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr_dsum I r (P : pred I) rG :
cfRepr (\big[dadd_grepr/grepr0]_(i <- r | P i) rG i)
= \sum_(i <- r | P i) cfRepr (rG i). | Proof. exact: (big_morph _ cfRepr_dadd cfRepr0). Qed. | Lemma | cfRepr_dsum | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"big_morph",
"cfRepr",
"cfRepr0",
"cfRepr_dadd",
"dadd_grepr",
"grepr0",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr_muln rG k : cfRepr (muln_grepr rG k) = cfRepr rG *+ k. | Proof. by rewrite cfRepr_dsum /= sumr_const card_ord. Qed. | Lemma | cfRepr_muln | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"card_ord",
"cfRepr",
"cfRepr_dsum",
"muln_grepr",
"rG",
"sumr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sG | := DecSocleType rG. | Let | sG | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"DecSocleType",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iG : irrType algC G | := DecSocleType _. | Let | iG | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"DecSocleType",
"algC",
"irrType"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
standard_irr (W : sG) | := irr_comp iG (socle_repr W). | Definition | standard_irr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"iG",
"irr_comp",
"sG",
"socle_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
standard_socle i | := pick [pred W | standard_irr W == i]. | Definition | standard_socle | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"pick",
"standard_irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
soc | := standard_socle. | Notation | soc | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"standard_socle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
standard_irr_coef i | := oapp (fun W => socle_mult W) 0 (soc i). | Definition | standard_irr_coef | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"soc",
"socle_mult"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
standard_grepr | :=
\big[dadd_grepr/grepr0]_i
muln_grepr (Representation (socle_repr i)) (standard_irr_coef i). | Definition | standard_grepr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"dadd_grepr",
"grepr0",
"muln_grepr",
"socle_repr",
"standard_irr_coef"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_rsim_standard : mx_rsim rG standard_grepr. | Proof.
pose W i := oapp val 0 (soc i); pose S := (\sum_i W i)%MS.
have C'G: [pchar algC]^'.-group G := algC'G_pchar G.
have [defS dxS]: (S :=: 1%:M)%MS /\ mxdirect S.
rewrite /S mxdirectE /= !(bigID soc xpredT) /=.
rewrite addsmxC big1 => [i|]; first by rewrite /W; case (soc i).
rewrite adds0mx_id addnC (@big1 na... | Lemma | mx_rsim_standard | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"C'G",
"Socle",
"Socle_direct",
"add0n",
"addnC",
"adds0mx_id",
"addsmxC",
"algC",
"algC'G_pchar",
"apply",
"big1",
"bigID",
"big_ord0",
"big_pred0",
"component_mx_module",
"eq_bigl",
"eq_bigr",
"group",
"iG",
"inE",
"irrK",
"irr_comp",
"last",
"mul0mx",
"mulmx0",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfReg (B : {set gT}) : 'CF(B) | := #|B|%:R *: '1_[1]. | Definition | cfReg | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRegE x : @cfReg G x = #|G|%:R *+ (x == 1%g). | Proof. by rewrite cfunE cfuniE ?normal1 // inE mulr_natr. Qed. | Lemma | cfRegE | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"cfReg",
"cfunE",
"cfuniE",
"inE",
"mulr_natr",
"normal1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfReprReg : cfRepr (regular_repr algC G) = cfReg G. | Proof.
apply/cfun_inP=> x Gx; rewrite cfRegE.
have [-> | ntx] := eqVneq x 1%g; first by rewrite cfRepr1.
rewrite cfunE Gx [\tr _]big1 // => i _; rewrite 2!mxE /=.
rewrite -(inj_eq enum_val_inj) gring_indexK ?groupM ?enum_valP //.
by rewrite eq_mulVg1 mulKg (negbTE ntx).
Qed. | Lemma | cfReprReg | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"algC",
"apply",
"big1",
"cfReg",
"cfRegE",
"cfRepr",
"cfRepr1",
"cfunE",
"cfun_inP",
"enum_valP",
"enum_val_inj",
"eqVneq",
"eq_mulVg1",
"gring_indexK",
"groupM",
"inj_eq",
"mulKg",
"mxE",
"regular_repr"
] | This is Isaacs, Lemma (2.10). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
xcfun (chi : 'CF(G)) A | :=
(gring_row A *m (\col_(i < #|G|) chi (enum_val i))) 0 0. | Definition | xcfun | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"chi",
"enum_val",
"gring_row"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcfun_is_zmod_morphism phi : zmod_morphism (xcfun phi). | Proof. by move=> A B; rewrite /xcfun [gring_row _]linearB mulmxBl !mxE. Qed. | Lemma | xcfun_is_zmod_morphism | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"gring_row",
"linearB",
"mulmxBl",
"mxE",
"xcfun",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcfun_is_additive | := xcfun_is_zmod_morphism. | Definition | xcfun_is_additive | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"xcfun_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcfunZr a phi A : xcfun phi (a *: A) = a * xcfun phi A. | Proof. by rewrite /xcfun linearZ -scalemxAl mxE. Qed. | Lemma | xcfunZr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"linearZ",
"mxE",
"scalemxAl",
"xcfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcfun_r A phi | := xcfun phi A. | Definition | xcfun_r | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"xcfun"
] | In order to add a second canonical structure on xcfun | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
xcfun_rE A chi : xcfun_r A chi = xcfun chi A. | Proof. by []. Qed. | Lemma | xcfun_rE | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"chi",
"xcfun",
"xcfun_r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcfun_r_is_zmod_morphism A : zmod_morphism (xcfun_r A). | Proof.
move=> phi psi; rewrite /= /xcfun !mxE -sumrB; apply: eq_bigr => i _.
by rewrite !mxE !cfunE mulrBr.
Qed. | Fact | xcfun_r_is_zmod_morphism | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"cfunE",
"eq_bigr",
"mulrBr",
"mxE",
"sumrB",
"xcfun",
"xcfun_r",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcfun_r_is_additive | := xcfun_r_is_zmod_morphism. | Definition | xcfun_r_is_additive | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"xcfun_r_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcfunZl a phi A : xcfun (a *: phi) A = a * xcfun phi A. | Proof.
rewrite /xcfun !mxE big_distrr; apply: eq_bigr => i _ /=.
by rewrite !mxE cfunE mulrCA.
Qed. | Lemma | xcfunZl | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"big_distrr",
"cfunE",
"eq_bigr",
"mulrCA",
"mxE",
"xcfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcfun_repr n rG A : xcfun (@cfRepr n rG) A = \tr (gring_op rG A). | Proof.
rewrite gring_opE [gring_row A]row_sum_delta !linear_sum /xcfun !mxE.
apply: eq_bigr => i _; rewrite !mxE /= !linearZ cfunE enum_valP /=.
by congr (_ * \tr _); rewrite {A}/gring_mx /= -rowE rowK mxvecK.
Qed. | Lemma | xcfun_repr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"cfRepr",
"cfunE",
"enum_valP",
"eq_bigr",
"gring_mx",
"gring_op",
"gring_opE",
"gring_row",
"linearZ",
"linear_sum",
"mxE",
"mxvecK",
"rG",
"rowE",
"rowK",
"row_sum_delta",
"xcfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"phi .[ A ]" | := (xcfun phi A) : cfun_scope. | Notation | phi .[ A ] | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"xcfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pred_Nirr gT B | := #|@classes gT B|.-1. | Definition | pred_Nirr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"classes",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Nirr G | := (pred_Nirr G).+1. | Notation | Nirr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"pred_Nirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Iirr G | := 'I_(Nirr G). | Notation | Iirr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Nirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sG | := DecSocleType (regular_repr algC G). | Let | sG | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"DecSocleType",
"algC",
"regular_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
NirrE : Nirr G = #|classes G|. | Proof. by rewrite /pred_Nirr (cardD1 [1]) classes1. Qed. | Lemma | NirrE | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Nirr",
"cardD1",
"classes",
"classes1",
"pred_Nirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Iirr_cast : Nirr G = #|sG|. | Proof. by rewrite NirrE ?card_irr_pchar ?algC'G_pchar //; apply: groupC. Qed. | Fact | Iirr_cast | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Nirr",
"NirrE",
"algC'G_pchar",
"apply",
"card_irr_pchar",
"groupC",
"sG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
offset | := cast_ord (esym Iirr_cast) (enum_rank [1 sG]%irr). | Let | offset | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Iirr_cast",
"cast_ord",
"enum_rank",
"irr",
"sG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_of_Iirr (i : Iirr G) : sG | :=
enum_val (cast_ord Iirr_cast (i + offset)). | Definition | socle_of_Iirr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Iirr",
"Iirr_cast",
"cast_ord",
"enum_val",
"offset",
"sG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_of_socle (Wi : sG) : Iirr G | :=
cast_ord (esym Iirr_cast) (enum_rank Wi) - offset. | Definition | irr_of_socle | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Iirr",
"Iirr_cast",
"cast_ord",
"enum_rank",
"offset",
"sG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
W | := socle_of_Iirr. | Notation | W | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"socle_of_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_Iirr0 : W 0 = [1 sG]%irr. | Proof. by rewrite /W add0r cast_ordKV enum_rankK. Qed. | Lemma | socle_Iirr0 | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"add0r",
"cast_ordKV",
"enum_rankK",
"irr",
"sG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_of_IirrK : cancel W irr_of_socle. | Proof. by move=> i; rewrite /irr_of_socle enum_valK cast_ordK addrK. Qed. | Lemma | socle_of_IirrK | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"addrK",
"cast_ordK",
"enum_valK",
"irr_of_socle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_of_socleK : cancel irr_of_socle W. | Proof. by move=> Wi; rewrite /W subrK cast_ordKV enum_rankK. Qed. | Lemma | irr_of_socleK | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"cast_ordKV",
"enum_rankK",
"irr_of_socle",
"subrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_of_socle_bij (A : {pred (Iirr G)}) : {on A, bijective irr_of_socle}. | Proof. by apply: onW_bij; exists W. Qed. | Lemma | irr_of_socle_bij | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Iirr",
"apply",
"irr_of_socle",
"on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
socle_of_Iirr_bij (A : {pred sG}) : {on A, bijective W}. | Proof. by apply: onW_bij; exists irr_of_socle. Qed. | Lemma | socle_of_Iirr_bij | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"irr_of_socle",
"on",
"sG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Chi_' i" | := (irr_repr (socle_of_Iirr i))
(at level 8, i at level 2, format "''Chi_' i"). | Notation | ''Chi_' i | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"irr_repr",
"socle_of_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''chi_' i" | := (tnth (irr _) i%R)
(at level 8, i at level 2, format "''chi_' i") : ring_scope. | Notation | ''chi_' i | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"irr",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''chi[' G ]_ i" | := (tnth (irr G) i%R)
(at level 8, i at level 2, only parsing) : ring_scope. | Notation | ''chi[' G ]_ i | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"irr",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
congr_irr i1 i2 : i1 = i2 -> 'chi_i1 = 'chi_i2. | Proof. by move->. Qed. | Lemma | congr_irr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Iirr1_neq0 : G :!=: 1%g -> inord 1 != 0 :> Iirr G. | Proof. by rewrite -classes_gt1 -NirrE -val_eqE /= => /inordK->. Qed. | Lemma | Iirr1_neq0 | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Iirr",
"NirrE",
"classes_gt1",
"inord",
"inordK",
"val_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_nonprincipal_irr : G :!=: 1%g -> {i : Iirr G | i != 0}. | Proof. by move/Iirr1_neq0; exists (inord 1). Qed. | Lemma | has_nonprincipal_irr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Iirr",
"Iirr1_neq0",
"inord"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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