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 |
|---|---|---|---|---|---|---|---|---|---|---|
cfSdprod_char chi :
(cfSdprod defG chi \is a character) = (chi \is a character). | Proof. by rewrite unlock cfMorph_charE // cfIsom_char. Qed. | Lemma | cfSdprod_char | 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",
"... | [
"cfIsom_char",
"cfMorph_charE",
"cfSdprod",
"character",
"chi",
"defG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprod_lin_char chi :
(cfSdprod defG chi \is a linear_char) = (chi \is a linear_char). | Proof. by rewrite qualifE/= cfSdprod_char cfSdprod1. Qed. | Lemma | cfSdprod_lin_char | 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",
"... | [
"cfSdprod",
"cfSdprod1",
"cfSdprod_char",
"chi",
"defG",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfSdprod_irr chi : (cfSdprod defG chi \in irr G) = (chi \in irr H). | Proof. by rewrite !irrEchar cfSdprod_char cfSdprod_iso. Qed. | Lemma | cfSdprod_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",
"... | [
"cfSdprod",
"cfSdprod_char",
"cfSdprod_iso",
"chi",
"defG",
"irr",
"irrEchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sdprod_Iirr j | := cfIirr (cfSdprod defG 'chi_j). | Definition | sdprod_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",
"... | [
"cfIirr",
"cfSdprod",
"defG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sdprod_IirrE j : 'chi_(sdprod_Iirr j) = cfSdprod defG 'chi_j. | Proof. by rewrite cfIirrE ?cfSdprod_irr ?mem_irr. Qed. | Lemma | sdprod_IirrE | 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",
"... | [
"cfIirrE",
"cfSdprod",
"cfSdprod_irr",
"defG",
"mem_irr",
"sdprod_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sdprod_IirrK : cancel sdprod_Iirr (Res_Iirr H). | Proof. by move=> j; rewrite /Res_Iirr sdprod_IirrE cfSdprodK irrK. Qed. | Lemma | sdprod_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",
"... | [
"Res_Iirr",
"cfSdprodK",
"irrK",
"sdprod_Iirr",
"sdprod_IirrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sdprod_Iirr_inj : injective sdprod_Iirr. | Proof. exact: can_inj sdprod_IirrK. Qed. | Lemma | sdprod_Iirr_inj | 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",
"... | [
"sdprod_Iirr",
"sdprod_IirrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sdprod_Iirr_eq0 i : (sdprod_Iirr i == 0) = (i == 0). | Proof. by rewrite -!irr_eq1 sdprod_IirrE cfSdprod_eq1. Qed. | Lemma | sdprod_Iirr_eq0 | 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",
"... | [
"cfSdprod_eq1",
"irr_eq1",
"sdprod_Iirr",
"sdprod_IirrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sdprod_Iirr0 : sdprod_Iirr 0 = 0. | Proof. by apply/eqP; rewrite sdprod_Iirr_eq0. Qed. | Lemma | sdprod_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",
"... | [
"apply",
"sdprod_Iirr",
"sdprod_Iirr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Res_sdprod_irr phi :
K \subset cfker phi -> phi \in irr G -> 'Res phi \in irr H. | Proof.
move=> kerK /irrP[i Dphi]; rewrite irrEchar -(cfSdprod_iso defG).
by rewrite cfRes_sdprodK // Dphi cfnorm_irr cfRes_char ?irr_char /=.
Qed. | Lemma | Res_sdprod_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",
"... | [
"cfRes_char",
"cfRes_sdprodK",
"cfSdprod_iso",
"cfker",
"cfnorm_irr",
"defG",
"irr",
"irrEchar",
"irrP",
"irr_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sdprod_Res_IirrE i :
K \subset cfker 'chi[G]_i -> 'chi_(Res_Iirr H i) = 'Res 'chi_i. | Proof. by move=> kerK; rewrite cfIirrE ?Res_sdprod_irr ?mem_irr. Qed. | Lemma | sdprod_Res_IirrE | 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",
"... | [
"Res_Iirr",
"Res_sdprod_irr",
"cfIirrE",
"cfker",
"chi",
"mem_irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sdprod_Res_IirrK i :
K \subset cfker 'chi_i -> sdprod_Iirr (Res_Iirr H i) = i. | Proof.
by move=> kerK; rewrite /sdprod_Iirr sdprod_Res_IirrE ?cfRes_sdprodK ?irrK.
Qed. | Lemma | sdprod_Res_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",
"... | [
"Res_Iirr",
"cfRes_sdprodK",
"cfker",
"irrK",
"sdprod_Iirr",
"sdprod_Res_IirrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
KxH : K \x H = G. | Hypothesis | KxH | 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 | ||
cfDprodKl_abelian j : abelian H -> cancel ((cfDprod KxH)^~ 'chi_j) 'Res. | Proof. by move=> cHH; apply: cfDprodKl; apply/lin_char1/char_abelianP. Qed. | Lemma | cfDprodKl_abelian | 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",
"... | [
"KxH",
"abelian",
"apply",
"cfDprod",
"cfDprodKl",
"char_abelianP",
"lin_char1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodKr_abelian i : abelian K -> cancel (cfDprod KxH 'chi_i) 'Res. | Proof. by move=> cKK; apply: cfDprodKr; apply/lin_char1/char_abelianP. Qed. | Lemma | cfDprodKr_abelian | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
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"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"KxH",
"abelian",
"apply",
"cfDprod",
"cfDprodKr",
"char_abelianP",
"lin_char1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodl_char phi :
(cfDprodl KxH phi \is a character) = (phi \is a character). | Proof. exact: cfSdprod_char. Qed. | Lemma | cfDprodl_char | 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",
"... | [
"KxH",
"cfDprodl",
"cfSdprod_char",
"character"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodr_char psi :
(cfDprodr KxH psi \is a character) = (psi \is a character). | Proof. exact: cfSdprod_char. Qed. | Lemma | cfDprodr_char | 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",
"... | [
"KxH",
"cfDprodr",
"cfSdprod_char",
"character"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprod_char phi psi :
phi \is a character -> psi \is a character ->
cfDprod KxH phi psi \is a character. | Proof. by move=> Nphi Npsi; rewrite rpredM ?cfDprodl_char ?cfDprodr_char. Qed. | Lemma | cfDprod_char | 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",
"... | [
"KxH",
"cfDprod",
"cfDprodl_char",
"cfDprodr_char",
"character",
"rpredM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprod_eq1 phi psi :
phi \is a character -> psi \is a character ->
(cfDprod KxH phi psi == 1) = (phi == 1) && (psi == 1). | Proof.
move=> /Cnat_char1 Nphi /Cnat_char1 Npsi.
apply/eqP/andP=> [phi_psi_1 | [/eqP-> /eqP->]]; last by rewrite cfDprod_cfun1.
have /andP[/eqP phi1 /eqP psi1]: (phi 1%g == 1) && (psi 1%g == 1).
by rewrite -natr_mul_eq1 // -(cfDprod1 KxH) phi_psi_1 cfun11.
rewrite -[phi](cfDprodKl KxH psi1) -{2}[psi](cfDprodKr KxH ph... | Lemma | cfDprod_eq1 | 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",
"... | [
"Cnat_char1",
"KxH",
"apply",
"cfDprod",
"cfDprod1",
"cfDprodKl",
"cfDprodKr",
"cfDprod_cfun1",
"cfun11",
"character",
"last",
"natr_mul_eq1",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodl_lin_char phi :
(cfDprodl KxH phi \is a linear_char) = (phi \is a linear_char). | Proof. exact: cfSdprod_lin_char. Qed. | Lemma | cfDprodl_lin_char | 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",
"... | [
"KxH",
"cfDprodl",
"cfSdprod_lin_char",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodr_lin_char psi :
(cfDprodr KxH psi \is a linear_char) = (psi \is a linear_char). | Proof. exact: cfSdprod_lin_char. Qed. | Lemma | cfDprodr_lin_char | 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",
"... | [
"KxH",
"cfDprodr",
"cfSdprod_lin_char",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprod_lin_char phi psi :
phi \is a linear_char -> psi \is a linear_char ->
cfDprod KxH phi psi \is a linear_char. | Proof. by move=> Nphi Npsi; rewrite rpredM ?cfSdprod_lin_char. Qed. | Lemma | cfDprod_lin_char | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
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"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"KxH",
"cfDprod",
"cfSdprod_lin_char",
"linear_char",
"rpredM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodl_irr chi : (cfDprodl KxH chi \in irr G) = (chi \in irr K). | Proof. exact: cfSdprod_irr. Qed. | Lemma | cfDprodl_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",
"... | [
"KxH",
"cfDprodl",
"cfSdprod_irr",
"chi",
"irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprodr_irr chi : (cfDprodr KxH chi \in irr G) = (chi \in irr H). | Proof. exact: cfSdprod_irr. Qed. | Lemma | cfDprodr_irr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
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"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"KxH",
"cfDprodr",
"cfSdprod_irr",
"chi",
"irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodl_Iirr i | := cfIirr (cfDprodl KxH 'chi_i). | Definition | dprodl_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",
"... | [
"KxH",
"cfDprodl",
"cfIirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodl_IirrE i : 'chi_(dprodl_Iirr i) = cfDprodl KxH 'chi_i. | Proof. exact: sdprod_IirrE. Qed. | Lemma | dprodl_IirrE | 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",
"... | [
"KxH",
"cfDprodl",
"dprodl_Iirr",
"sdprod_IirrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodl_IirrK : cancel dprodl_Iirr (Res_Iirr K). | Proof. exact: sdprod_IirrK. Qed. | Lemma | dprodl_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",
"... | [
"Res_Iirr",
"dprodl_Iirr",
"sdprod_IirrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodl_Iirr_eq0 i : (dprodl_Iirr i == 0) = (i == 0). | Proof. exact: sdprod_Iirr_eq0. Qed. | Lemma | dprodl_Iirr_eq0 | 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",
"... | [
"dprodl_Iirr",
"sdprod_Iirr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodl_Iirr0 : dprodl_Iirr 0 = 0. | Proof. exact: sdprod_Iirr0. Qed. | Lemma | dprodl_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",
"... | [
"dprodl_Iirr",
"sdprod_Iirr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodr_Iirr j | := cfIirr (cfDprodr KxH 'chi_j). | Definition | dprodr_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",
"... | [
"KxH",
"cfDprodr",
"cfIirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodr_IirrE j : 'chi_(dprodr_Iirr j) = cfDprodr KxH 'chi_j. | Proof. exact: sdprod_IirrE. Qed. | Lemma | dprodr_IirrE | 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",
"... | [
"KxH",
"cfDprodr",
"dprodr_Iirr",
"sdprod_IirrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodr_IirrK : cancel dprodr_Iirr (Res_Iirr H). | Proof. exact: sdprod_IirrK. Qed. | Lemma | dprodr_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",
"... | [
"Res_Iirr",
"dprodr_Iirr",
"sdprod_IirrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodr_Iirr_eq0 j : (dprodr_Iirr j == 0) = (j == 0). | Proof. exact: sdprod_Iirr_eq0. Qed. | Lemma | dprodr_Iirr_eq0 | 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",
"... | [
"dprodr_Iirr",
"sdprod_Iirr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodr_Iirr0 : dprodr_Iirr 0 = 0. | Proof. exact: sdprod_Iirr0. Qed. | Lemma | dprodr_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",
"... | [
"dprodr_Iirr",
"sdprod_Iirr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDprod_irr i j : cfDprod KxH 'chi_i 'chi_j \in irr G. | Proof.
rewrite irrEchar cfDprod_char ?irr_char //=.
by rewrite cfdot_dprod !cfdot_irr !eqxx mul1r.
Qed. | Lemma | cfDprod_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",
"... | [
"KxH",
"cfDprod",
"cfDprod_char",
"cfdot_dprod",
"cfdot_irr",
"eqxx",
"irr",
"irrEchar",
"irr_char",
"mul1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_Iirr ij | := cfIirr (cfDprod KxH 'chi_ij.1 'chi_ij.2). | Definition | dprod_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",
"... | [
"KxH",
"cfDprod",
"cfIirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_IirrE i j : 'chi_(dprod_Iirr (i, j)) = cfDprod KxH 'chi_i 'chi_j. | Proof. by rewrite cfIirrE ?cfDprod_irr. Qed. | Lemma | dprod_IirrE | 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",
"... | [
"KxH",
"cfDprod",
"cfDprod_irr",
"cfIirrE",
"dprod_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_IirrEl i : 'chi_(dprod_Iirr (i, 0)) = cfDprodl KxH 'chi_i. | Proof. by rewrite dprod_IirrE /cfDprod irr0 rmorph1 mulr1. Qed. | Lemma | dprod_IirrEl | 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",
"... | [
"KxH",
"cfDprod",
"cfDprodl",
"dprod_Iirr",
"dprod_IirrE",
"irr0",
"mulr1",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_IirrEr j : 'chi_(dprod_Iirr (0, j)) = cfDprodr KxH 'chi_j. | Proof. by rewrite dprod_IirrE /cfDprod irr0 rmorph1 mul1r. Qed. | Lemma | dprod_IirrEr | 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",
"... | [
"KxH",
"cfDprod",
"cfDprodr",
"dprod_Iirr",
"dprod_IirrE",
"irr0",
"mul1r",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_Iirr_inj : injective dprod_Iirr. | Proof.
move=> [i1 j1] [i2 j2] /eqP; rewrite -[_ == _]oddb -(@natrK algC (_ == _)).
rewrite -cfdot_irr !dprod_IirrE cfdot_dprod !cfdot_irr -natrM mulnb.
by rewrite natrK oddb -xpair_eqE => /eqP.
Qed. | Lemma | dprod_Iirr_inj | 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",
"cfdot_dprod",
"cfdot_irr",
"dprod_Iirr",
"dprod_IirrE",
"mulnb",
"natrK",
"natrM",
"oddb",
"xpair_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_Iirr0 : dprod_Iirr (0, 0) = 0. | Proof. by apply/irr_inj; rewrite dprod_IirrE !irr0 cfDprod_cfun1. Qed. | Lemma | dprod_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",
"... | [
"apply",
"cfDprod_cfun1",
"dprod_Iirr",
"dprod_IirrE",
"irr0",
"irr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_Iirr0l j : dprod_Iirr (0, j) = dprodr_Iirr j. | Proof.
by apply/irr_inj; rewrite dprod_IirrE irr0 dprodr_IirrE cfDprod_cfun1l.
Qed. | Lemma | dprod_Iirr0l | 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",
"cfDprod_cfun1l",
"dprod_Iirr",
"dprod_IirrE",
"dprodr_Iirr",
"dprodr_IirrE",
"irr0",
"irr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_Iirr0r i : dprod_Iirr (i, 0) = dprodl_Iirr i. | Proof.
by apply/irr_inj; rewrite dprod_IirrE irr0 dprodl_IirrE cfDprod_cfun1r.
Qed. | Lemma | dprod_Iirr0r | 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",
"cfDprod_cfun1r",
"dprod_Iirr",
"dprod_IirrE",
"dprodl_Iirr",
"dprodl_IirrE",
"irr0",
"irr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_Iirr_eq0 i j : (dprod_Iirr (i, j) == 0) = (i == 0) && (j == 0). | Proof. by rewrite -xpair_eqE -(inj_eq dprod_Iirr_inj) dprod_Iirr0. Qed. | Lemma | dprod_Iirr_eq0 | 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",
"... | [
"dprod_Iirr",
"dprod_Iirr0",
"dprod_Iirr_inj",
"inj_eq",
"xpair_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_dprod_irr i1 i2 j1 j2 :
'['chi_(dprod_Iirr (i1, j1)), 'chi_(dprod_Iirr (i2, j2))]
= ((i1 == i2) && (j1 == j2))%:R. | Proof. by rewrite cfdot_irr (inj_eq dprod_Iirr_inj). Qed. | Lemma | cfdot_dprod_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",
"... | [
"cfdot_irr",
"dprod_Iirr",
"dprod_Iirr_inj",
"inj_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_Iirr_onto k : k \in codom dprod_Iirr. | Proof.
set D := codom _; have Df: dprod_Iirr _ \in D := codom_f dprod_Iirr _.
have: 'chi_k 1%g ^+ 2 != 0 by rewrite mulf_neq0 ?irr1_neq0.
apply: contraR => notDk; move/eqP: (irr_sum_square G).
rewrite (bigID [in D]) (reindex _ (bij_on_codom dprod_Iirr_inj (0, 0))) /=.
have ->: #|G|%:R = \sum_i \sum_j 'chi_(dprod_Iirr (... | Lemma | dprod_Iirr_onto | 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",
"... | [
"KxH",
"addrC",
"addrK",
"apply",
"bigID",
"bij_on_codom",
"cfDprodE",
"codom",
"codom_f",
"dprod_Iirr",
"dprod_IirrE",
"dprod_Iirr_inj",
"dprod_card",
"eq_bigl",
"eq_bigr",
"exprMn",
"irr1_degree",
"irr1_neq0",
"irr_sum_square",
"ler0n",
"mulf_neq0",
"mulg1",
"mulr_suml"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv_dprod_Iirr i | := iinv (dprod_Iirr_onto i). | Definition | inv_dprod_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",
"... | [
"dprod_Iirr_onto",
"iinv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_IirrK : cancel dprod_Iirr inv_dprod_Iirr. | Proof. by move=> p; apply: (iinv_f dprod_Iirr_inj). Qed. | Lemma | dprod_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",
"... | [
"apply",
"dprod_Iirr",
"dprod_Iirr_inj",
"iinv_f",
"inv_dprod_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv_dprod_IirrK : cancel inv_dprod_Iirr dprod_Iirr. | Proof. by move=> i; apply: f_iinv. Qed. | Lemma | inv_dprod_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",
"... | [
"apply",
"dprod_Iirr",
"f_iinv",
"inv_dprod_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv_dprod_Iirr0 : inv_dprod_Iirr 0 = (0, 0). | Proof. by apply/(canLR dprod_IirrK); rewrite dprod_Iirr0. Qed. | Lemma | inv_dprod_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",
"... | [
"apply",
"dprod_Iirr0",
"dprod_IirrK",
"inv_dprod_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_IirrC (gT : finGroupType) (G K H : {group gT})
(KxH : K \x H = G) (HxK : H \x K = G) i j :
dprod_Iirr KxH (i, j) = dprod_Iirr HxK (j, i). | Proof. by apply: irr_inj; rewrite !dprod_IirrE; apply: cfDprodC. Qed. | Lemma | dprod_IirrC | 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",
"... | [
"KxH",
"apply",
"cfDprodC",
"dprod_Iirr",
"dprod_IirrE",
"gT",
"group",
"irr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
defG : \big[dprod/1%g]_(i | P i) A i = G. | Hypothesis | defG | 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",
"... | [
"dprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
sAG i : P i -> A i \subset G. | Proof. by move=> Pi; rewrite -(bigdprodWY defG) (bigD1 i) ?joing_subl. Qed. | Let | sAG | 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",
"... | [
"bigD1",
"bigdprodWY",
"defG",
"joing_subl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodi_char i (phi : 'CF(A i)) :
phi \is a character -> cfBigdprodi defG phi \is a character. | Proof. by move=> Nphi; rewrite cfDprodl_char cfRes_char. Qed. | Lemma | cfBigdprodi_char | 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",
"... | [
"cfBigdprodi",
"cfDprodl_char",
"cfRes_char",
"character",
"defG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodi_charE i (phi : 'CF(A i)) :
P i -> (cfBigdprodi defG phi \is a character) = (phi \is a character). | Proof. by move=> Pi; rewrite cfDprodl_char Pi cfRes_id. Qed. | Lemma | cfBigdprodi_charE | 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",
"... | [
"cfBigdprodi",
"cfDprodl_char",
"cfRes_id",
"character",
"defG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprod_char phi :
(forall i, P i -> phi i \is a character) ->
cfBigdprod defG phi \is a character. | Proof.
by move=> Nphi; apply: rpred_prod => i /Nphi; apply: cfBigdprodi_char.
Qed. | Lemma | cfBigdprod_char | 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",
"cfBigdprod",
"cfBigdprodi_char",
"character",
"defG",
"rpred_prod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodi_lin_char i (phi : 'CF(A i)) :
phi \is a linear_char -> cfBigdprodi defG phi \is a linear_char. | Proof. by move=> Lphi; rewrite cfDprodl_lin_char ?cfRes_lin_char. Qed. | Lemma | cfBigdprodi_lin_char | 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",
"... | [
"cfBigdprodi",
"cfDprodl_lin_char",
"cfRes_lin_char",
"defG",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodi_lin_charE i (phi : 'CF(A i)) :
P i -> (cfBigdprodi defG phi \is a linear_char) = (phi \is a linear_char). | Proof. by move=> Pi; rewrite qualifE/= cfBigdprodi_charE // cfBigdprodi1. Qed. | Lemma | cfBigdprodi_lin_charE | 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",
"... | [
"cfBigdprodi",
"cfBigdprodi1",
"cfBigdprodi_charE",
"defG",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprod_lin_char phi :
(forall i, P i -> phi i \is a linear_char) ->
cfBigdprod defG phi \is a linear_char. | Proof.
by move=> Lphi; apply/rpred_prod=> i /Lphi; apply: cfBigdprodi_lin_char.
Qed. | Lemma | cfBigdprod_lin_char | 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",
"cfBigdprod",
"cfBigdprodi_lin_char",
"defG",
"linear_char",
"rpred_prod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodi_irr i chi :
P i -> (cfBigdprodi defG chi \in irr G) = (chi \in irr (A i)). | Proof. by move=> Pi; rewrite !irrEchar cfBigdprodi_charE ?cfBigdprodi_iso. Qed. | Lemma | cfBigdprodi_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",
"... | [
"cfBigdprodi",
"cfBigdprodi_charE",
"cfBigdprodi_iso",
"chi",
"defG",
"irr",
"irrEchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprod_irr chi :
(forall i, P i -> chi i \in irr (A i)) -> cfBigdprod defG chi \in irr G. | Proof.
move=> Nchi; rewrite irrEchar cfBigdprod_char => [i /Nchi/irrWchar|] //=.
by rewrite cfdot_bigdprod big1 // => i /Nchi/irrWnorm.
Qed. | Lemma | cfBigdprod_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",
"... | [
"big1",
"cfBigdprod",
"cfBigdprod_char",
"cfdot_bigdprod",
"chi",
"defG",
"irr",
"irrEchar",
"irrWchar",
"irrWnorm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprod_eq1 phi :
(forall i, P i -> phi i \is a character) ->
(cfBigdprod defG phi == 1) = [forall (i | P i), phi i == 1]. | Proof.
move=> Nphi; set Phi := cfBigdprod defG phi.
apply/eqP/eqfun_inP=> [Phi1 i Pi | phi1]; last first.
by apply: big1 => i /phi1->; rewrite rmorph1.
have Phi1_1: Phi 1%g = 1 by rewrite Phi1 cfun1E group1.
have nz_Phi1: Phi 1%g != 0 by rewrite Phi1_1 oner_eq0.
have [_ <-] := cfBigdprodK nz_Phi1 Pi.
rewrite Phi1_1 d... | Lemma | cfBigdprod_eq1 | 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",
"... | [
"Cnat_char1",
"apply",
"big1",
"cfBigdprod",
"cfBigdprodK",
"cfBigdprodiK",
"cfBigdprodi_char",
"cfRes1",
"cfun1E",
"character",
"defG",
"divr1",
"eqfun_inP",
"group1",
"last",
"natr_prod_eq1",
"oner_eq0",
"prod_cfunE",
"rmorph1",
"scale1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprod_Res_lin chi :
chi \is a linear_char -> cfBigdprod defG (fun i => 'Res[A i] chi) = chi. | Proof.
move=> Lchi; apply/cfun_inP=> _ /(mem_bigdprod defG)[x [Ax -> _]].
rewrite (lin_char_prod Lchi) ?cfBigdprodE // => [i Pi|].
by rewrite (subsetP (sAG Pi)) ?Ax.
by apply/eq_bigr=> i Pi; rewrite cfResE ?sAG ?Ax.
Qed. | Lemma | cfBigdprod_Res_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",
"cfBigdprod",
"cfBigdprodE",
"cfResE",
"cfun_inP",
"chi",
"defG",
"eq_bigr",
"lin_char_prod",
"linear_char",
"mem_bigdprod",
"sAG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodKlin phi :
(forall i, P i -> phi i \is a linear_char) ->
forall i, P i -> 'Res (cfBigdprod defG phi) = phi i. | Proof.
move=> Lphi i Pi; have Lpsi := cfBigdprod_lin_char Lphi.
have [_ <-] := cfBigdprodK (lin_char_neq0 Lpsi (group1 G)) Pi.
by rewrite !lin_char1 ?Lphi // divr1 scale1r.
Qed. | Lemma | cfBigdprodKlin | 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",
"... | [
"cfBigdprod",
"cfBigdprodK",
"cfBigdprod_lin_char",
"defG",
"divr1",
"group1",
"lin_char1",
"lin_char_neq0",
"linear_char",
"scale1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfBigdprodKabelian Iphi (phi := fun i => 'chi_(Iphi i)) :
abelian G -> forall i, P i -> 'Res (cfBigdprod defG phi) = 'chi_(Iphi i). | Proof.
move=> /(abelianS _) cGG.
by apply: cfBigdprodKlin => i /sAG/cGG/char_abelianP->.
Qed. | Lemma | cfBigdprodKabelian | 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",
"... | [
"abelian",
"abelianS",
"apply",
"cGG",
"cfBigdprod",
"cfBigdprodKlin",
"char_abelianP",
"defG",
"sAG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC_charAut u (chi : 'CF(G)) x :
chi \is a character -> (u (chi x))^* = u (chi x)^*. | Proof.
have [Gx | /cfun0->] := boolP (x \in G); last by rewrite !rmorph0.
case/char_reprP=> rG ->; have [e [_ [en1 _] [-> _] _]] := repr_rsim_diag rG Gx.
by rewrite !rmorph_sum; apply: eq_bigr => i _; apply: aut_unity_rootC (en1 i).
Qed. | Lemma | conjC_charAut | 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",
"aut_unity_rootC",
"cfun0",
"char_reprP",
"character",
"chi",
"eq_bigr",
"last",
"rG",
"repr_rsim_diag",
"rmorph0",
"rmorph_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC_irrAut u i x : (u ('chi[G]_i x))^* = u ('chi_i x)^*. | Proof. exact: conjC_charAut (irr_char i). Qed. | Lemma | conjC_irrAut | 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",
"conjC_charAut",
"irr_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_aut_char u (phi chi : 'CF(G)) :
chi \is a character -> '[cfAut u phi, cfAut u chi] = u '[phi, chi]. | Proof. by move/conjC_charAut=> Nchi; apply: cfdot_cfAut => _ /mapP[x _ ->]. Qed. | Lemma | cfdot_aut_char | 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",
"cfAut",
"cfdot_cfAut",
"character",
"chi",
"conjC_charAut",
"mapP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_aut_irr u phi i :
'[cfAut u phi, cfAut u 'chi[G]_i] = u '[phi, 'chi_i]. | Proof. exact: cfdot_aut_char (irr_char i). Qed. | Lemma | cfdot_aut_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",
"... | [
"cfAut",
"cfdot_aut_char",
"chi",
"irr_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_irr u chi : (cfAut u chi \in irr G) = (chi \in irr G). | Proof.
rewrite !irrEchar cfAut_char; apply/andb_id2l=> /cfdot_aut_char->.
exact: fmorph_eq1.
Qed. | Lemma | cfAut_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",
"... | [
"apply",
"cfAut",
"cfAut_char",
"cfdot_aut_char",
"chi",
"fmorph_eq1",
"irr",
"irrEchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjC_irr i : (('chi_i)^*)%CF \in irr G. | Proof. by rewrite cfAut_irr mem_irr. Qed. | Lemma | cfConjC_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",
"... | [
"cfAut_irr",
"irr",
"mem_irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_aut_closed u : cfAut_closed u (irr G). | Proof. by move=> chi; rewrite /= cfAut_irr. Qed. | Lemma | irr_aut_closed | 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",
"... | [
"cfAut_closed",
"cfAut_irr",
"chi",
"irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_Iirr u i | := cfIirr (cfAut u 'chi[G]_i). | Definition | aut_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",
"... | [
"cfAut",
"cfIirr",
"chi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_IirrE u i : 'chi_(aut_Iirr u i) = cfAut u 'chi_i. | Proof. by rewrite cfIirrE ?cfAut_irr ?mem_irr. Qed. | Lemma | aut_IirrE | 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",
"... | [
"aut_Iirr",
"cfAut",
"cfAut_irr",
"cfIirrE",
"mem_irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC_Iirr | := aut_Iirr Num.conj. | Definition | conjC_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",
"... | [
"aut_Iirr",
"conj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC_IirrE i : 'chi[G]_(conjC_Iirr i) = ('chi_i)^*%CF. | Proof. exact: aut_IirrE. Qed. | Lemma | conjC_IirrE | 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",
"... | [
"aut_IirrE",
"chi",
"conjC_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC_IirrK : involutive conjC_Iirr. | Proof. by move=> i; apply: irr_inj; rewrite !conjC_IirrE cfConjCK. Qed. | Lemma | conjC_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",
"... | [
"apply",
"cfConjCK",
"conjC_Iirr",
"conjC_IirrE",
"irr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_Iirr0 u : aut_Iirr u 0 = 0 :> Iirr G. | Proof. by apply/irr_inj; rewrite aut_IirrE irr0 cfAut_cfun1. Qed. | Lemma | aut_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",
"... | [
"Iirr",
"apply",
"aut_Iirr",
"aut_IirrE",
"cfAut_cfun1",
"irr0",
"irr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC_Iirr0 : conjC_Iirr 0 = 0 :> Iirr G. | Proof. exact: aut_Iirr0. Qed. | Lemma | conjC_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",
"... | [
"Iirr",
"aut_Iirr0",
"conjC_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_Iirr_eq0 u i : (aut_Iirr u i == 0) = (i == 0). | Proof. by rewrite -!irr_eq1 aut_IirrE cfAut_eq1. Qed. | Lemma | aut_Iirr_eq0 | 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",
"... | [
"aut_Iirr",
"aut_IirrE",
"cfAut_eq1",
"irr_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC_Iirr_eq0 i : (conjC_Iirr i == 0 :> Iirr G) = (i == 0). | Proof. exact: aut_Iirr_eq0. Qed. | Lemma | conjC_Iirr_eq0 | 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",
"aut_Iirr_eq0",
"conjC_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_Iirr_inj u : injective (aut_Iirr u). | Proof.
by move=> i j eq_ij; apply/irr_inj/(cfAut_inj u); rewrite -!aut_IirrE eq_ij.
Qed. | Lemma | aut_Iirr_inj | 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",
"aut_Iirr",
"aut_IirrE",
"cfAut_inj",
"irr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfQuo_char G H (chi : 'CF(G)) :
chi \is a character -> (chi / H)%CF \is a character. | Proof.
move=> Nchi; without loss kerH: / H \subset cfker chi.
move/contraNF=> IHchi; apply/wlog_neg=> N'chiH.
suffices ->: (chi / H)%CF = (chi 1%g)%:A.
by rewrite rpredZ_nat ?Cnat_char1 ?rpred1.
by apply/cfunP=> x; rewrite cfunE cfun1E mulr_natr cfunElock IHchi.
without loss nsHG: G chi Nchi kerH / H <| G.
... | Lemma | cfQuo_char | 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",
"... | [
"Cnat_char1",
"apply",
"cfQuoE",
"cfQuoInorm",
"cfRes_char",
"cfker",
"cfker_repr",
"cfker_sub",
"cfun1E",
"cfunE",
"cfunElock",
"cfunP",
"cfun_inP",
"char_reprP",
"character",
"chi",
"mem_quotient",
"morphimP",
"mulr_natr",
"normalSG",
"normal_norm",
"normal_sub",
"nsHG"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfQuo_lin_char G H (chi : 'CF(G)) :
chi \is a linear_char -> (chi / H)%CF \is a linear_char. | Proof. by case/andP=> Nchi; rewrite qualifE/= cfQuo_char ?cfQuo1. Qed. | Lemma | cfQuo_lin_char | 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",
"... | [
"cfQuo1",
"cfQuo_char",
"chi",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMod_char G H (chi : 'CF(G / H)) :
chi \is a character -> (chi %% H)%CF \is a character. | Proof. exact: cfMorph_char. Qed. | Lemma | cfMod_char | 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",
"... | [
"cfMorph_char",
"character",
"chi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMod_lin_char G H (chi : 'CF(G / H)) :
chi \is a linear_char -> (chi %% H)%CF \is a linear_char. | Proof. exact: cfMorph_lin_char. Qed. | Lemma | cfMod_lin_char | 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",
"... | [
"cfMorph_lin_char",
"chi",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMod_charE G H (chi : 'CF(G / H)) :
H <| G -> (chi %% H \is a character)%CF = (chi \is a character). | Proof. by case/andP=> _; apply: cfMorph_charE. Qed. | Lemma | cfMod_charE | 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",
"cfMorph_charE",
"character",
"chi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMod_lin_charE G H (chi : 'CF(G / H)) :
H <| G -> (chi %% H \is a linear_char)%CF = (chi \is a linear_char). | Proof. by case/andP=> _; apply: cfMorph_lin_charE. Qed. | Lemma | cfMod_lin_charE | 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",
"cfMorph_lin_charE",
"chi",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfQuo_charE G H (chi : 'CF(G)) :
H <| G -> H \subset cfker chi ->
(chi / H \is a character)%CF = (chi \is a character). | Proof. by move=> nsHG kerH; rewrite -cfMod_charE ?cfQuoK. Qed. | Lemma | cfQuo_charE | 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",
"... | [
"cfMod_charE",
"cfQuoK",
"cfker",
"character",
"chi",
"nsHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfQuo_lin_charE G H (chi : 'CF(G)) :
H <| G -> H \subset cfker chi ->
(chi / H \is a linear_char)%CF = (chi \is a linear_char). | Proof. by move=> nsHG kerH; rewrite -cfMod_lin_charE ?cfQuoK. Qed. | Lemma | cfQuo_lin_charE | 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",
"... | [
"cfMod_lin_charE",
"cfQuoK",
"cfker",
"chi",
"linear_char",
"nsHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMod_irr G H chi :
H <| G -> (chi %% H \in irr G)%CF = (chi \in irr (G / H)). | Proof. by case/andP=> _; apply: cfMorph_irr. Qed. | Lemma | cfMod_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",
"... | [
"apply",
"cfMorph_irr",
"chi",
"irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mod_Iirr G H i | := cfIirr ('chi[G / H]_i %% H)%CF. | Definition | mod_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",
"... | [
"cfIirr",
"chi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mod_Iirr0 G H : mod_Iirr (0 : Iirr (G / H)) = 0. | Proof. exact: morph_Iirr0. Qed. | Lemma | mod_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",
"... | [
"Iirr",
"mod_Iirr",
"morph_Iirr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mod_IirrE G H i : H <| G -> 'chi_(mod_Iirr i) = ('chi[G / H]_i %% H)%CF. | Proof. by move=> nsHG; rewrite cfIirrE ?cfMod_irr ?mem_irr. Qed. | Lemma | mod_IirrE | 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",
"... | [
"cfIirrE",
"cfMod_irr",
"chi",
"mem_irr",
"mod_Iirr",
"nsHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mod_Iirr_eq0 G H i :
H <| G -> (mod_Iirr i == 0) = (i == 0 :> Iirr (G / H)). | Proof. by case/andP=> _ /morph_Iirr_eq0->. Qed. | Lemma | mod_Iirr_eq0 | 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",
"mod_Iirr",
"morph_Iirr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfQuo_irr G H chi :
H <| G -> H \subset cfker chi ->
((chi / H)%CF \in irr (G / H)) = (chi \in irr G). | Proof. by move=> nsHG kerH; rewrite -cfMod_irr ?cfQuoK. Qed. | Lemma | cfQuo_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",
"... | [
"cfMod_irr",
"cfQuoK",
"cfker",
"chi",
"irr",
"nsHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quo_Iirr G H i | := cfIirr ('chi[G]_i / H)%CF. | Definition | quo_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",
"... | [
"cfIirr",
"chi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quo_Iirr0 G H : quo_Iirr H (0 : Iirr G) = 0. | Proof. by rewrite /quo_Iirr irr0 cfQuo_cfun1 -irr0 irrK. Qed. | Lemma | quo_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",
"... | [
"Iirr",
"cfQuo_cfun1",
"irr0",
"irrK",
"quo_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quo_IirrE G H i :
H <| G -> H \subset cfker 'chi[G]_i -> 'chi_(quo_Iirr H i) = ('chi_i / H)%CF. | Proof. by move=> nsHG kerH; rewrite cfIirrE ?cfQuo_irr ?mem_irr. Qed. | Lemma | quo_IirrE | 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",
"... | [
"cfIirrE",
"cfQuo_irr",
"cfker",
"chi",
"mem_irr",
"nsHG",
"quo_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quo_Iirr_eq0 G H i :
H <| G -> H \subset cfker 'chi[G]_i -> (quo_Iirr H i == 0) = (i == 0). | Proof. by move=> nsHG kerH; rewrite -!irr_eq1 quo_IirrE ?cfQuo_eq1. Qed. | Lemma | quo_Iirr_eq0 | 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",
"... | [
"cfQuo_eq1",
"cfker",
"chi",
"irr_eq1",
"nsHG",
"quo_Iirr",
"quo_IirrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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