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 |
|---|---|---|---|---|---|---|---|---|---|---|
mod_IirrK G H : H <| G -> cancel (@mod_Iirr G H) (@quo_Iirr G H). | Proof.
move=> nsHG i; apply: irr_inj.
by rewrite quo_IirrE ?mod_IirrE ?cfker_mod // cfModK.
Qed. | Lemma | mod_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",
"cfModK",
"cfker_mod",
"irr_inj",
"mod_Iirr",
"mod_IirrE",
"nsHG",
"quo_Iirr",
"quo_IirrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quo_IirrK G H i :
H <| G -> H \subset cfker 'chi[G]_i -> mod_Iirr (quo_Iirr H i) = i. | Proof.
by move=> nsHG kerH; apply: irr_inj; rewrite mod_IirrE ?quo_IirrE ?cfQuoK.
Qed. | Lemma | quo_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",
"cfQuoK",
"cfker",
"chi",
"irr_inj",
"mod_Iirr",
"mod_IirrE",
"nsHG",
"quo_Iirr",
"quo_IirrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quo_IirrKeq G H :
H <| G ->
forall i, (mod_Iirr (quo_Iirr H i) == i) = (H \subset cfker 'chi[G]_i). | Proof.
move=> nsHG i; apply/eqP/idP=> [<- | ]; last exact: quo_IirrK.
by rewrite mod_IirrE ?cfker_mod.
Qed. | Lemma | quo_IirrKeq | 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",
"cfker",
"cfker_mod",
"chi",
"last",
"mod_Iirr",
"mod_IirrE",
"nsHG",
"quo_Iirr",
"quo_IirrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mod_Iirr_bij H G :
H <| G -> {on [pred i | H \subset cfker 'chi_i], bijective (@mod_Iirr G H)}. | Proof.
by exists (quo_Iirr H) => [i _ | i]; [apply: mod_IirrK | apply: quo_IirrK].
Qed. | Lemma | mod_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",
"cfker",
"mod_Iirr",
"mod_IirrK",
"on",
"quo_Iirr",
"quo_IirrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_norm_irr_quo H G x :
x \in G -> H <| G ->
\sum_i `|'chi[G / H]_i (coset H x)| ^+ 2
= \sum_(i | H \subset cfker 'chi_i) `|'chi[G]_i x| ^+ 2. | Proof.
move=> Gx nsHG; rewrite (reindex _ (mod_Iirr_bij nsHG)) /=.
by apply/esym/eq_big=> [i | i _]; rewrite mod_IirrE ?cfker_mod ?cfModE.
Qed. | Lemma | sum_norm_irr_quo | 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",
"cfModE",
"cfker",
"cfker_mod",
"chi",
"coset",
"eq_big",
"mod_IirrE",
"mod_Iirr_bij",
"nsHG",
"reindex"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cap_cfker_normal G H :
H <| G -> \bigcap_(i | H \subset cfker 'chi[G]_i) (cfker 'chi_i) = H. | Proof.
move=> nsHG; have [sHG nHG] := andP nsHG; set lhs := \bigcap_(i | _) _.
have nHlhs: lhs \subset 'N(H) by rewrite (bigcap_min 0) ?cfker_irr0.
apply/esym/eqP; rewrite eqEsubset (introT bigcapsP) //= -quotient_sub1 //.
rewrite -(TI_cfker_irr (G / H)); apply/bigcapsP=> i _.
rewrite sub_quotient_pre // (bigcap_min (m... | Lemma | cap_cfker_normal | 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",
"... | [
"TI_cfker_irr",
"apply",
"bigcap_min",
"bigcapsP",
"cfker",
"cfker_irr0",
"cfker_mod",
"cfker_morph",
"chi",
"eqEsubset",
"mod_Iirr",
"mod_IirrE",
"nHG",
"nsHG",
"quotient_sub1",
"sHG",
"sub_quotient_pre",
"subsetIr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_reg_quo G H : H <| G -> cfker (cfReg (G / H)%g %% H) = H. | Proof.
move=> nsHG; have [sHG nHG] := andP nsHG.
apply/setP=> x; rewrite cfkerEchar ?cfMod_char ?cfReg_char //.
rewrite -[in RHS in _ = RHS](setIidPr sHG) !inE; apply: andb_id2l => Gx.
rewrite !cfModE // !cfRegE // morph1 eqxx.
rewrite (sameP eqP (kerP _ (subsetP nHG x Gx))) ker_coset.
by rewrite -!mulrnA eqr_nat eqn_p... | Lemma | cfker_reg_quo | 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",
"can_eq",
"cardG_gt0",
"cfModE",
"cfMod_char",
"cfReg",
"cfRegE",
"cfReg_char",
"cfker",
"cfkerEchar",
"eqb_id",
"eqn_pmul2l",
"eqr_nat",
"eqxx",
"inE",
"kerP",
"ker_coset",
"morph1",
"mulrnA",
"nHG",
"nsHG",
"oddb",
"sHG",
"setIidPr",
"setP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_irr_der1 G i :
('chi_i \is a linear_char) = (G^`(1)%g \subset cfker 'chi[G]_i). | Proof.
apply/idP/idP=> [|sG'K]; first exact: lin_char_der1.
have nsG'G: G^`(1) <| G := der_normal 1 G.
rewrite qualifE/= irr_char -[i](quo_IirrK nsG'G) // mod_IirrE //=.
by rewrite cfModE // morph1 lin_char1 //; apply/char_abelianP/der_abelian.
Qed. | Lemma | lin_irr_der1 | 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",
"cfModE",
"cfker",
"char_abelianP",
"chi",
"der_abelian",
"der_normal",
"irr_char",
"lin_char1",
"lin_char_der1",
"linear_char",
"mod_IirrE",
"morph1",
"quo_IirrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subGcfker G i : (G \subset cfker 'chi[G]_i) = (i == 0). | Proof.
rewrite -irr_eq1; apply/idP/eqP=> [chiG1 | ->]; last by rewrite cfker_cfun1.
apply/cfun_inP=> x Gx; rewrite cfun1E Gx cfker1 ?(subsetP chiG1) ?lin_char1 //.
by rewrite lin_irr_der1 (subset_trans (der_sub 1 G)).
Qed. | Lemma | subGcfker | 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",
"cfker",
"cfker1",
"cfker_cfun1",
"cfun1E",
"cfun_inP",
"chi",
"der_sub",
"irr_eq1",
"last",
"lin_char1",
"lin_irr_der1",
"subsetP",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_prime_injP G i :
prime #|G| -> reflect {in G &, injective 'chi[G]_i} (i != 0). | Proof.
move=> pr_G; apply: (iffP idP) => [nz_i | inj_chi].
apply: fful_lin_char_inj (irr_prime_lin i pr_G) _.
by rewrite cfaithfulE -(setIidPr (cfker_sub _)) prime_TIg // subGcfker.
have /trivgPn[x Gx ntx]: G :!=: 1%g by rewrite -cardG_gt1 prime_gt1.
apply: contraNneq ntx => i0; apply/eqP/inj_chi=> //.
by rewrite i... | Lemma | irr_prime_injP | 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",
"cardG_gt1",
"cfaithfulE",
"cfker_sub",
"cfun1E",
"chi",
"contraNneq",
"fful_lin_char_inj",
"group1",
"i0",
"irr0",
"irr_prime_lin",
"prime",
"prime_TIg",
"prime_gt1",
"setIidPr",
"subGcfker",
"trivgPn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cap_cfker_lin_irr G :
\bigcap_(i | 'chi[G]_i \is a linear_char) (cfker 'chi_i) = G^`(1)%g. | Proof.
rewrite -(cap_cfker_normal (der_normal 1 G)).
by apply: eq_bigl => i; rewrite lin_irr_der1.
Qed. | Lemma | cap_cfker_lin_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",
"cap_cfker_normal",
"cfker",
"chi",
"der_normal",
"eq_bigl",
"lin_irr_der1",
"linear_char"
] | This is Isaacs (2.23)(a). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
card_lin_irr G :
#|[pred i | 'chi[G]_i \is a linear_char]| = #|G : G^`(1)%g|. | Proof.
have nsG'G := der_normal 1 G; rewrite (eq_card (@lin_irr_der1 G)).
rewrite -(on_card_preimset (mod_Iirr_bij nsG'G)).
rewrite -card_quotient ?normal_norm //.
move: (der_abelian 0 G); rewrite card_classes_abelian; move/eqP<-.
rewrite -NirrE -[RHS]card_ord.
by apply: eq_card => i; rewrite !inE mod_IirrE ?cfker_mod.... | Lemma | card_lin_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",
"... | [
"NirrE",
"apply",
"card_classes_abelian",
"card_ord",
"card_quotient",
"cfker_mod",
"chi",
"der_abelian",
"der_normal",
"eq_card",
"inE",
"lin_irr_der1",
"linear_char",
"mod_IirrE",
"mod_Iirr_bij",
"normal_norm",
"on_card_preimset"
] | This is Isaacs (2.23)(b) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
solvable_has_lin_char G :
G :!=: 1%g -> solvable G ->
exists2 i, 'chi[G]_i \is a linear_char & 'chi_i != 1. | Proof.
move=> ntG solG.
suff /subsetPn[i]: ~~ ([pred i | 'chi[G]_i \is a linear_char] \subset pred1 0).
by rewrite !inE -(inj_eq irr_inj) irr0; exists i.
rewrite (contra (@subset_leq_card _ _ _)) // -ltnNge card1 card_lin_irr.
by rewrite indexg_gt1 proper_subn // (sol_der1_proper solG).
Qed. | Lemma | solvable_has_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",
"... | [
"card1",
"card_lin_irr",
"chi",
"inE",
"indexg_gt1",
"inj_eq",
"irr0",
"irr_inj",
"linear_char",
"ltnNge",
"pred1",
"proper_subn",
"sol_der1_proper",
"solvable",
"subsetPn",
"subset_leq_card"
] | A non-trivial solvable group has a nonprincipal linear character. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
lin_char_group G :
{linG : finGroupType & {cF : linG -> 'CF(G) |
[/\ injective cF, #|linG| = #|G : G^`(1)|,
forall u, cF u \is a linear_char
& forall phi, phi \is a linear_char -> exists u, phi = cF u]
& [/\ cF 1%g = 1%R,
{morph cF : u v / (u * v)%g >-> (u * v)%R},... | Proof.
pose linT := {i : Iirr G | 'chi_i \is a linear_char}.
pose cF (u : linT) := 'chi_(sval u).
have cFlin u: cF u \is a linear_char := svalP u.
have cFinj: injective cF := inj_comp irr_inj val_inj.
have inT xi : xi \is a linear_char -> {u | cF u = xi}.
move=> lin_xi; have /irrP/sig_eqW[i Dxi] := lin_char_irr lin_x... | Lemma | lin_char_group | 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",
"... | [
"Build",
"Iirr",
"Sub",
"apply",
"card_image",
"card_lin_irr",
"cfIirr",
"codomP",
"dvdn_cforder",
"eq_card",
"eqn_dvd",
"exp_cforder",
"expgS",
"expg_order",
"exprS",
"inE",
"inj_eq",
"insubd",
"insubdK",
"inv",
"irrK",
"irrP",
"irr_inj",
"lin_char_irr",
"lin_char_un... | A combinatorial group isommorphic to the linear characters. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfExp_prime_transitive G (i j : Iirr G) :
prime #|G| -> i != 0 -> j != 0 ->
exists2 k, coprime k #['chi_i]%CF & 'chi_j = 'chi_i ^+ k. | Proof.
set p := #|G| => pr_p nz_i nz_j; have cycG := prime_cyclic pr_p.
have [L [h [injh oL Lh h_ontoL]] [h1 hM hX _ o_h]] := lin_char_group G.
rewrite (derG1P (cyclic_abelian cycG)) indexg1 -/p in oL.
have /fin_all_exists[h' h'K] := h_ontoL _ (irr_cyclic_lin _ cycG).
have o_h' k: k != 0 -> #[h' k] = p.
rewrite -cfor... | Lemma | cfExp_prime_transitive | 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",
"cardsT",
"cforder_irr_eq1",
"cforder_lin_char_dvdG",
"coprime",
"coprime_sym",
"cycleP",
"cycle_generator",
"cyclic_abelian",
"derG1P",
"eqEcard",
"eq_sym",
"fin_all_exists",
"generator",
"generator_coprime",
"indexg1",
"irr_cyclic_lin",
"lin_char_group",
"pr_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_subcent1_coset G H x :
x \in G -> H <| G -> (#|'C_(G / H)[coset H x]| <= #|'C_G[x]|)%N. | Proof.
move=> Gx nsHG; rewrite -leC_nat.
move: (second_orthogonality_relation x Gx); rewrite mulrb class_refl => <-.
have GHx: coset H x \in (G / H)%g by apply: mem_quotient.
move: (second_orthogonality_relation (coset H x) GHx).
rewrite mulrb class_refl => <-.
rewrite -2!(eq_bigr _ (fun _ _ => normCK _)) sum_norm_irr_... | Lemma | card_subcent1_coset | 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",
"... | [
"addrC",
"addrK",
"apply",
"bigID",
"cfker",
"chi",
"class_refl",
"coset",
"eq_bigr",
"leC_nat",
"mem_quotient",
"mul_conjC_ge0",
"mulrb",
"normCK",
"nsHG",
"second_orthogonality_relation",
"subr_ge0",
"sum_norm_irr_quo",
"sumr_ge0"
] | This is Isaacs (2.24). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
det_repr_mx x : 'M_1 | := (\det (rG x))%:M. | Definition | det_repr_mx | 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",
"... | [
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det_is_repr : mx_repr G det_repr_mx. | Proof.
split=> [|g h Gg Gh]; first by rewrite /det_repr_mx repr_mx1 det1.
by rewrite /det_repr_mx repr_mxM // det_mulmx !mulmxE scalar_mxM.
Qed. | Fact | det_is_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",
"... | [
"Gg",
"det1",
"det_mulmx",
"det_repr_mx",
"mulmxE",
"mx_repr",
"repr_mx1",
"repr_mxM",
"scalar_mxM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
det_repr | := MxRepresentation det_is_repr. | Canonical | det_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",
"... | [
"det_is_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
detRepr | := cfRepr det_repr. | Definition | detRepr | 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",
"det_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
detRepr_lin_char : detRepr \is a linear_char. | Proof.
by rewrite qualifE/= cfRepr_char cfunE group1 repr_mx1 mxtrace1 mulr1n /=.
Qed. | Lemma | detRepr_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",
"... | [
"cfRepr_char",
"cfunE",
"detRepr",
"group1",
"linear_char",
"mulr1n",
"mxtrace1",
"repr_mx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDet_unlockable | := Unlockable cfDet.unlock. | Canonical | cfDet_unlockable | 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",
"... | [
"cfDet"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDet | := (@cfDet gT G). | Notation | cfDet | 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 | |
cfDet_lin_char phi : cfDet phi \is a linear_char. | Proof.
rewrite unlock; apply: rpred_prod => i _; apply: rpredX.
exact: detRepr_lin_char.
Qed. | Lemma | cfDet_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",
"cfDet",
"detRepr_lin_char",
"linear_char",
"rpredX",
"rpred_prod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDetD :
{in character &, {morph cfDet : phi psi / phi + psi >-> phi * psi}}. | Proof.
move=> phi psi Nphi Npsi; rewrite unlock /= -big_split; apply: eq_bigr => i _ /=.
by rewrite -exprD cfdotDl truncnD ?nnegrE ?natr_ge0 // Cnat_cfdot_char_irr.
Qed. | Lemma | cfDetD | 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_cfdot_char_irr",
"apply",
"big_split",
"cfDet",
"cfdotDl",
"character",
"eq_bigr",
"exprD",
"natr_ge0",
"nnegrE",
"truncnD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDet0 : cfDet 0 = 1. | Proof. by rewrite unlock big1 // => i _; rewrite cfdot0l truncn0. Qed. | Lemma | cfDet0 | 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",
"cfDet",
"cfdot0l",
"truncn0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDetMn k :
{in character, {morph cfDet : phi / phi *+ k >-> phi ^+ k}}. | Proof.
move=> phi Nphi; elim: k => [|k IHk]; rewrite ?cfDet0 // mulrS exprS -{}IHk.
by rewrite cfDetD ?rpredMn.
Qed. | Lemma | cfDetMn | 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",
"... | [
"cfDet",
"cfDet0",
"cfDetD",
"character",
"exprS",
"mulrS",
"rpredMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDetRepr n rG : cfDet (cfRepr rG) = @detRepr _ _ n rG. | Proof.
transitivity (\prod_W detRepr (socle_repr W) ^+ standard_irr_coef rG W).
rewrite (reindex _ (socle_of_Iirr_bij _)) unlock /=.
apply: eq_bigr => i _; congr (_ ^+ _).
rewrite (cfRepr_sim (mx_rsim_standard rG)) cfRepr_standard.
rewrite cfdot_suml (bigD1 i) ?big1 //= => [j i'j|].
by rewrite cfdotZl cfdot... | Lemma | cfDetRepr | 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",
"... | [
"addr0",
"apply",
"big1",
"bigD1",
"big_ord0",
"big_ord_recl",
"big_rec2",
"cfDet",
"cfRepr",
"cfRepr_sim",
"cfRepr_standard",
"cfdotZl",
"cfdot_irr",
"cfdot_suml",
"cfnorm_irr",
"cfunE",
"cfun_inP",
"det1",
"detRepr",
"det_mulmx",
"det_ublock",
"eq_bigr",
"eqxx",
"exp_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDet_id xi : xi \is a linear_char -> cfDet xi = xi. | Proof.
move=> lin_xi; have /irrP[i Dxi] := lin_char_irr lin_xi.
apply/cfun_inP=> x Gx; rewrite Dxi -irrRepr cfDetRepr !cfunE trace_mx11 mxE.
move: lin_xi (_ x) => /andP[_]; rewrite Dxi irr1_degree pnatr_eq1 => /eqP-> X.
by rewrite {1}[X]mx11_scalar det_scalar1 trace_mx11.
Qed. | Lemma | cfDet_id | 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",
"cfDet",
"cfDetRepr",
"cfunE",
"cfun_inP",
"det_scalar1",
"irr1_degree",
"irrP",
"irrRepr",
"lin_char_irr",
"lin_xi",
"linear_char",
"mx11_scalar",
"mxE",
"pnatr_eq1",
"trace_mx11"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDet_order phi | := #[cfDet phi]%CF. | Definition | cfDet_order | 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",
"... | [
"cfDet"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDet_order_lin xi :
xi \is a linear_char -> cfDet_order xi = #[xi]%CF. | Proof. by rewrite /cfDet_order => /cfDet_id->. Qed. | Definition | cfDet_order_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",
"... | [
"cfDet_id",
"cfDet_order",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDet_order_dvdG phi : cfDet_order phi %| #|G|. | Proof. by rewrite cforder_lin_char_dvdG ?cfDet_lin_char. Qed. | Definition | cfDet_order_dvdG | 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",
"... | [
"cfDet_lin_char",
"cfDet_order",
"cforder_lin_char_dvdG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''o' ( phi )" | := (cfDet_order phi)
(format "''o' ( phi )") : cfun_scope. | Notation | ''o' ( phi ) | 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",
"... | [
"cfDet_order"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDetRes gT (G H : {group gT}) phi :
phi \is a character -> cfDet ('Res[H, G] phi) = 'Res (cfDet phi). | Proof.
move=> Nphi; have [sGH | not_sHG] := boolP (H \subset G); last first.
have /natrP[n Dphi1] := Cnat_char1 Nphi.
rewrite !cfResEout // Dphi1 lin_char1 ?cfDet_lin_char // scale1r.
by rewrite scaler_nat cfDetMn ?cfDet_id ?rpred1 // expr1n.
have [rG ->] := char_reprP Nphi; rewrite !(=^~ cfRepr_sub, cfDetRepr) /... | Lemma | cfDetRes | 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",
"cfDet",
"cfDetMn",
"cfDetRepr",
"cfDet_id",
"cfDet_lin_char",
"cfRepr_sim",
"cfRepr_sub",
"cfResEout",
"char_reprP",
"character",
"expr1n",
"gT",
"group",
"last",
"lin_char1",
"mul1mx",
"mulmx1",
"natrP",
"rG",
"row_free_unit",
"rpred1",
"sGH",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDetMorph aT rT (D G : {group aT}) (f : {morphism D >-> rT})
(phi : 'CF(f @* G)) :
phi \is a character -> cfDet (cfMorph phi) = cfMorph (cfDet phi). | Proof.
move=> Nphi; have [sGD | not_sGD] := boolP (G \subset D); last first.
have /natrP[n Dphi1] := Cnat_char1 Nphi.
rewrite !cfMorphEout // Dphi1 lin_char1 ?cfDet_lin_char // scale1r.
by rewrite scaler_nat cfDetMn ?cfDet_id ?rpred1 // expr1n.
have [rG ->] := char_reprP Nphi; rewrite !(=^~ cfRepr_morphim, cfDetR... | Lemma | cfDetMorph | 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",
"aT",
"apply",
"cfDet",
"cfDetMn",
"cfDetRepr",
"cfDet_id",
"cfDet_lin_char",
"cfMorph",
"cfMorphEout",
"cfRepr_morphim",
"cfRepr_sim",
"char_reprP",
"character",
"expr1n",
"group",
"last",
"lin_char1",
"morphism",
"mul1mx",
"mulmx1",
"natrP",
"rG",
"row_f... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDetIsom aT rT (G : {group aT}) (R : {group rT})
(f : {morphism G >-> rT}) (isoGR : isom G R f) phi :
cfDet (cfIsom isoGR phi) = cfIsom isoGR (cfDet phi). | Proof.
rewrite unlock rmorph_prod (reindex (isom_Iirr isoGR)).
by exists (isom_Iirr (isom_sym isoGR)) => i; rewrite ?isom_IirrK ?isom_IirrKV.
apply: eq_bigr=> i; rewrite -!cfDetRepr !irrRepr isom_IirrE rmorphXn cfIsom_iso.
by rewrite /= ![in cfIsom _]unlock cfDetMorph ?cfRes_char ?cfDetRes ?irr_char.
Qed. | Lemma | cfDetIsom | 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",
"... | [
"aT",
"apply",
"cfDet",
"cfDetMorph",
"cfDetRepr",
"cfDetRes",
"cfIsom",
"cfIsom_iso",
"cfRes_char",
"eq_bigr",
"group",
"irrRepr",
"irr_char",
"isoGR",
"isom",
"isom_Iirr",
"isom_IirrE",
"isom_IirrK",
"isom_IirrKV",
"isom_sym",
"morphism",
"reindex",
"rmorphXn",
"rmorp... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfDet_mul_lin gT (G : {group gT}) (lambda phi : 'CF(G)) :
lambda \is a linear_char -> phi \is a character ->
cfDet (lambda * phi) = lambda ^+ Num.truncn (phi 1%g) * cfDet phi. | Proof.
case/andP=> /char_reprP[[n1 rG1] ->] /= n1_1 /char_reprP[[n2 rG2] ->] /=.
do [rewrite !cfRepr1 pnatr_eq1 natrK; move/eqP] in n1_1 *.
rewrite {n1}n1_1 in rG1 *; rewrite cfRepr_prod cfDetRepr.
apply/cfun_inP=> x Gx; rewrite !cfunE cfDetRepr cfunE Gx !mulrb !trace_mx11.
rewrite !mxE prod_repr_lin ?mulrb //=; case: ... | Lemma | cfDet_mul_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",
"cfDet",
"cfDetRepr",
"cfRepr1",
"cfRepr_prod",
"cfun1E",
"cfunE",
"cfun_inP",
"char_reprP",
"character",
"detZ",
"expS_cfunE",
"gT",
"group",
"linear_char",
"mulrb",
"mxE",
"natrK",
"pnatr_eq1",
"prod_repr_lin",
"trace_mx11",
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfcenter (gT : finGroupType) (G : {set gT}) (phi : 'CF(G)) | :=
if phi \is a character then [set g in G | `|phi g| == phi 1%g] else cfker phi. | Definition | cfcenter | 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",
"... | [
"cfker",
"character",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Z' ( phi )" | := (cfcenter phi) : cfun_scope. | Notation | ''Z' ( phi ) | 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",
"... | [
"cfcenter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfcenter_repr n (rG : mx_representation algC G n) :
'Z(cfRepr rG)%CF = rcenter rG. | Proof.
rewrite /cfcenter /rcenter cfRepr_char /=.
apply/setP=> x /[!inE]; apply/andb_id2l=> Gx.
apply/eqP/is_scalar_mxP=> [|[c rG_c]].
by case/max_cfRepr_norm_scalar=> // c; exists c.
rewrite -(sqrCK (char1_ge0 (cfRepr_char rG))) normC_def; congr (sqrtC _).
rewrite expr2 -{2}(mulgV x) -char_inv ?cfRepr_char ?cfunE ?g... | Lemma | cfcenter_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",
"... | [
"algC",
"apply",
"cfRepr",
"cfRepr_char",
"cfcenter",
"cfunE",
"char1_ge0",
"char_inv",
"expr2",
"group1",
"groupM",
"groupV",
"inE",
"invmx_scalar",
"is_scalar_mxP",
"max_cfRepr_norm_scalar",
"mulgV",
"mulr_natl",
"mulrb",
"mulrnAl",
"mulrnAr",
"mx_representation",
"mxtr... | This is Isaacs (2.27)(a). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfcenter_group_set phi : group_set ('Z(phi))%CF. | Proof.
have [[rG ->] | /negbTE notNphi] := altP (@char_reprP _ G phi).
by rewrite cfcenter_repr groupP.
by rewrite /cfcenter notNphi groupP.
Qed. | Fact | cfcenter_group_set | 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",
"... | [
"cfcenter",
"cfcenter_repr",
"char_reprP",
"groupP",
"group_set",
"rG"
] | This is part of Isaacs (2.27)(b). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfcenter_group f | := Group (cfcenter_group_set f). | Canonical | cfcenter_group | 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",
"... | [
"cfcenter_group_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_cfcenterE chi x :
chi \is a character -> x \in G ->
(x \in ('Z(chi))%CF) = (`|chi x| == chi 1%g). | Proof. by move=> Nchi Gx; rewrite /cfcenter Nchi inE Gx. Qed. | Lemma | char_cfcenterE | 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",
"... | [
"cfcenter",
"character",
"chi",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_cfcenterE i x :
x \in G -> (x \in 'Z('chi[G]_i)%CF) = (`|'chi_i x| == 'chi_i 1%g). | Proof. by move/char_cfcenterE->; rewrite ?irr_char. Qed. | Lemma | irr_cfcenterE | 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",
"... | [
"char_cfcenterE",
"chi",
"irr_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfcenter_sub phi : ('Z(phi))%CF \subset G. | Proof. by rewrite /cfcenter /cfker !setIdE -fun_if subsetIl. Qed. | Lemma | cfcenter_sub | 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",
"... | [
"cfcenter",
"cfker",
"setIdE",
"subsetIl"
] | This is also Isaacs (2.27)(b). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfker_center_normal phi : cfker phi <| 'Z(phi)%CF. | Proof.
apply: normalS (cfcenter_sub phi) (cfker_normal phi).
rewrite /= /cfcenter; case: ifP => // Hphi; rewrite cfkerEchar //.
apply/subsetP=> x /[!inE] /andP[-> /eqP->] /=.
by rewrite ger0_norm ?char1_ge0.
Qed. | Lemma | cfker_center_normal | 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",
"cfcenter",
"cfcenter_sub",
"cfker",
"cfkerEchar",
"cfker_normal",
"char1_ge0",
"ger0_norm",
"inE",
"normalS",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfcenter_normal phi : 'Z(phi)%CF <| G. | Proof.
have [[rG ->] | /negbTE notNphi] := altP (@char_reprP _ _ phi).
by rewrite cfcenter_repr rcenter_normal.
by rewrite /cfcenter notNphi cfker_normal.
Qed. | Lemma | cfcenter_normal | 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",
"... | [
"cfcenter",
"cfcenter_repr",
"cfker_normal",
"char_reprP",
"rG",
"rcenter_normal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfcenter_Res chi :
exists2 chi1, chi1 \is a linear_char & 'Res['Z(chi)%CF] chi = chi 1%g *: chi1. | Proof.
have [[rG ->] | /negbTE notNphi] := altP (@char_reprP _ _ chi); last first.
exists 1; first exact: cfun1_lin_char.
rewrite /cfcenter notNphi; apply/cfun_inP=> x Kx.
by rewrite cfunE cfun1E Kx mulr1 cfResE ?cfker_sub // cfker1.
rewrite cfcenter_repr -(cfRepr_sub _ (normal_sub (rcenter_normal _))).
case: rG ... | Lemma | cfcenter_Res | 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",
"cfRepr1",
"cfRepr_char",
"cfRepr_sub",
"cfResE",
"cfcenter",
"cfcenter_repr",
"cfker1",
"cfker_sub",
"cfun1E",
"cfun1_lin_char",
"cfunE",
"cfun_inP",
"char_reprP",
"chi",
"eqxx",
"flatmx0",
"group",
"is_scalar_mxP",
"last",
"linear_char",
"mulr1",
"m... | This is Isaacs (2.27)(c). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfcenter_cyclic chi : cyclic ('Z(chi)%CF / cfker chi)%g. | Proof.
case Nchi: (chi \is a character); last first.
by rewrite /cfcenter Nchi trivg_quotient cyclic1.
have [-> | nz_chi] := eqVneq chi 0.
rewrite quotientS1 ?cyclic1 //= /cfcenter cfkerEchar ?cfun0_char //.
by apply/subsetP=> x /setIdP[Gx _]; rewrite inE Gx /= !cfunE.
have [xi Lxi def_chi] := cfcenter_Res chi.
s... | Lemma | cfcenter_cyclic | 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",
"apply",
"cfResE",
"cfcenter",
"cfcenter_Res",
"cfcenter_sub",
"cfker",
"cfkerEchar",
"cfker_center_normal",
"cfker_repr",
"cfun0_char",
"cfunE",
"char1_eq0",
"character",
"chi",
"cyclic",
"cyclic1",
"eqVneq",
"group1",
"inE",
"inj_eq",
"irrG",
"irr_reprP",
"... | This is Isaacs (2.27)(d). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfcenter_subset_center chi :
('Z(chi)%CF / cfker chi)%g \subset 'Z(G / cfker chi)%g. | Proof.
case Nchi: (chi \is a character); last first.
by rewrite /cfcenter Nchi trivg_quotient sub1G.
rewrite subsetI quotientS ?cfcenter_sub // quotient_cents2r //=.
case/char_reprP: Nchi => rG ->{chi}; rewrite cfker_repr cfcenter_repr gen_subG.
apply/subsetP=> _ /imset2P[x y /setIdP[Gx /is_scalar_mxP[c rGx]] Gy ->].... | Lemma | cfcenter_subset_center | 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",
"cfcenter",
"cfcenter_repr",
"cfcenter_sub",
"cfker",
"cfker_repr",
"char_reprP",
"character",
"chi",
"gen_subG",
"groupM",
"groupR",
"groupV",
"imset2P",
"inE",
"is_scalar_mxP",
"last",
"mulmxA",
"quotientS",
"quotient_cents2r",
"rG",
"repr_mxKV",
"repr_mxM",
... | This is Isaacs (2.27)(e). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfcenter_eq_center (i : Iirr G) :
('Z('chi_i)%CF / cfker 'chi_i)%g = 'Z(G / cfker 'chi_i)%g. | Proof.
apply/eqP; rewrite eqEsubset; rewrite cfcenter_subset_center ?irr_char //.
apply/subsetP=> _ /setIP[/morphimP[x /= _ Gx ->] cGx]; rewrite mem_quotient //=.
rewrite -irrRepr cfker_repr cfcenter_repr inE Gx in cGx *.
apply: mx_abs_irr_cent_scalar 'Chi_i _ _ _; first exact/groupC/socle_irr.
have nKG: G \subset 'N(r... | Lemma | cfcenter_eq_center | 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",
"centP",
"centgmxP",
"cfcenter_repr",
"cfcenter_subset_center",
"cfker",
"cfker_repr",
"eqEsubset",
"groupC",
"inE",
"irrRepr",
"irr_char",
"mem_quotient",
"morphimP",
"mx_abs_irr_cent_scalar",
"nKG",
"quo_repr",
"quo_repr_coset",
"rG",
"repr_mxM",
"rker",
... | This is Isaacs (2.27)(f). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cap_cfcenter_irr : \bigcap_i 'Z('chi[G]_i)%CF = 'Z(G). | Proof.
apply/esym/eqP; rewrite eqEsubset (introT bigcapsP) /= => [i _|].
rewrite -(quotientSGK _ (normal_sub (cfker_center_normal _))).
by rewrite subIset // normal_norm // cfker_normal.
by rewrite cfcenter_eq_center morphim_center.
set Z := \bigcap_i _.
have sZG: Z \subset G by rewrite (bigcap_min 0) ?cfcenter... | Lemma | cap_cfcenter_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",
"... | [
"TI_cfker_irr",
"apply",
"bigcap_inf",
"bigcap_min",
"bigcapsP",
"cfcenter_eq_center",
"cfcenter_sub",
"cfker_center_normal",
"cfker_normal",
"chi",
"commG1P",
"eqEsubset",
"morphim_center",
"normal_norm",
"normal_sub",
"quotientS",
"quotientSGK",
"quotient_cents2",
"subIset",
... | This is Isaacs (2.28). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfnorm_Res_leif H phi :
H \subset G ->
'['Res[H] phi] <= #|G : H|%:R * '[phi] ?= iff (phi \in 'CF(G, H)). | Proof.
move=> sHG; rewrite cfun_onE mulrCA natf_indexg // -mulrA mulKf ?neq0CG //.
rewrite (big_setID H) (setIidPr sHG) /= addrC.
rewrite (mono_leif (ler_pM2l _)) ?invr_gt0 ?gt0CG // -leifBLR -sumrB.
rewrite big1 => [x Hx|]; first by rewrite !cfResE ?subrr.
have ->: (support phi \subset H) = (G :\: H \subset [set x | p... | Lemma | cfnorm_Res_leif | 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",
"... | [
"addrC",
"apply",
"big1",
"big_setID",
"cfResE",
"cfun0",
"cfun_onE",
"eq_subset",
"eq_sym",
"forall_inP",
"gt0CG",
"inE",
"invr_gt0",
"leifBLR",
"leif_0_sum",
"ler_pM2l",
"mono_leif",
"mulKf",
"mul_conjC_eq0",
"mul_conjC_ge0",
"mulrA",
"mulrCA",
"natf_indexg",
"neq0CG"... | This is Isaacs (2.29). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
irr1_bound (i : Iirr G) :
('chi_i 1%g) ^+ 2 <= #|G : 'Z('chi_i)%CF|%:R
?= iff ('chi_i \in 'CF(G, 'Z('chi_i)%CF)). | Proof.
congr (_ <= _ ?= iff _): (cfnorm_Res_leif 'chi_i (cfcenter_sub 'chi_i)).
have [xi Lxi ->] := cfcenter_Res 'chi_i.
have /irrP[j ->] := lin_char_irr Lxi; rewrite cfdotZl cfdotZr cfdot_irr eqxx.
by rewrite mulr1 irr1_degree conjC_nat.
by rewrite cfdot_irr eqxx mulr1.
Qed. | Lemma | irr1_bound | 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",
"cfcenter_Res",
"cfcenter_sub",
"cfdotZl",
"cfdotZr",
"cfdot_irr",
"cfnorm_Res_leif",
"conjC_nat",
"eqxx",
"irr1_degree",
"irrP",
"lin_char_irr",
"mulr1"
] | This is Isaacs (2.30). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
irr1_abelian_bound (i : Iirr G) :
abelian (G / 'Z('chi_i)%CF) -> ('chi_i 1%g) ^+ 2 = #|G : 'Z('chi_i)%CF|%:R. | Proof.
move=> AbGc; apply/eqP; rewrite irr1_bound cfun_onE; apply/subsetP=> x nz_chi_x.
have Gx: x \in G by apply: contraR nz_chi_x => /cfun0->.
have nKx := subsetP (normal_norm (cfker_normal 'chi_i)) _ Gx.
rewrite -(quotientGK (cfker_center_normal _)) inE nKx inE /=.
rewrite cfcenter_eq_center inE mem_quotient //=.
ap... | Lemma | irr1_abelian_bound | 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",
"abelian",
"apply",
"centP",
"cfcenter_eq_center",
"cfcenter_normal",
"cfcenter_repr",
"cfker_center_normal",
"cfker_normal",
"cfker_repr",
"cfun0",
"cfunE",
"cfunJ",
"cfun_onE",
"commgP",
"conjg_mulR",
"coset_id",
"der1_min",
"groupM",
"inE",
"irr1_bound",
"irrRepr... | This is Isaacs (2.31). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
irr_faithful_center i : cfaithful 'chi[G]_i -> cyclic 'Z(G). | Proof.
rewrite (isog_cyclic (isog_center (quotient1_isog G))) /=.
by move/trivgP <-; rewrite -cfcenter_eq_center cfcenter_cyclic.
Qed. | Lemma | irr_faithful_center | 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",
"... | [
"cfaithful",
"cfcenter_cyclic",
"cfcenter_eq_center",
"chi",
"cyclic",
"isog_center",
"isog_cyclic",
"quotient1_isog",
"trivgP"
] | This is Isaacs (2.32)(a). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfcenter_fful_irr i : cfaithful 'chi[G]_i -> 'Z('chi_i)%CF = 'Z(G). | Proof.
move/trivgP=> Ki1; have:= cfcenter_eq_center i; rewrite {}Ki1.
have inj1: 'injm (@coset gT 1%g) by rewrite ker_coset.
by rewrite -injm_center; last apply: injm_morphim_inj; rewrite ?norms1.
Qed. | Lemma | cfcenter_fful_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",
"cfaithful",
"cfcenter_eq_center",
"chi",
"coset",
"gT",
"injm_center",
"injm_morphim_inj",
"ker_coset",
"last",
"norms1",
"trivgP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pgroup_cyclic_faithful (p : nat) :
p.-group G -> cyclic 'Z(G) -> exists i, cfaithful 'chi[G]_i. | Proof.
pose Z := 'Ohm_1('Z(G)) => pG cycZG; have nilG := pgroup_nil pG.
have [-> | ntG] := eqsVneq G [1]; first by exists 0; apply: cfker_sub.
have{pG} [[p_pr _ _] pZ] := (pgroup_pdiv pG ntG, pgroupS (center_sub G) pG).
have ntZ: 'Z(G) != [1] by rewrite center_nil_eq1.
have{pZ} oZ: #|Z| = p by apply: Ohm1_cyclic_pgroup... | Lemma | pgroup_cyclic_faithful | 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",
"... | [
"Ohm1_cyclic_pgroup_prime",
"Ohm1_eq1",
"TI_cfker_irr",
"apply",
"bigcapsP",
"center_nil_eq1",
"center_sub",
"cfaithful",
"cfker_normal",
"cfker_sub",
"chi",
"cyclic",
"eqsVneq",
"existsP",
"existsPn",
"group",
"meet_Ohm1",
"meet_center_nil",
"nat",
"oZ",
"pG",
"pZ",
"p_p... | This is Isaacs (2.32)(b). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfInd_char chi : chi \is a character -> 'Ind[G] chi \is a character. | Proof.
move=> Nchi; apply/forallP=> i; rewrite coord_cfdot -Frobenius_reciprocity //.
by rewrite Cnat_cfdot_char ?cfRes_char ?irr_char.
Qed. | Lemma | cfInd_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_cfdot_char",
"Frobenius_reciprocity",
"apply",
"cfRes_char",
"character",
"chi",
"coord_cfdot",
"forallP",
"irr_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfInd_eq0 chi :
H \subset G -> chi \is a character -> ('Ind[G] chi == 0) = (chi == 0). | Proof.
move=> sHG Nchi; rewrite -!(char1_eq0) ?cfInd_char // cfInd1 //.
by rewrite (mulrI_eq0 _ (mulfI _)) ?neq0CiG.
Qed. | Lemma | cfInd_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",
"... | [
"cfInd1",
"cfInd_char",
"char1_eq0",
"character",
"chi",
"mulfI",
"mulrI_eq0",
"neq0CiG",
"sHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Ind_irr_neq0 i : H \subset G -> 'Ind[G, H] 'chi_i != 0. | Proof. by move/cfInd_eq0->; rewrite ?irr_neq0 ?irr_char. Qed. | Lemma | Ind_irr_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",
"... | [
"cfInd_eq0",
"irr_char",
"irr_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Ind_Iirr (A B : {set gT}) i | := cfIirr ('Ind[B, A] 'chi_i). | Definition | Ind_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",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
constt_cfRes_irr i : {j | j \in irr_constt ('Res[H, G] 'chi_i)}. | Proof. apply/sigW/neq0_has_constt/Res_irr_neq0. Qed. | Lemma | constt_cfRes_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",
"... | [
"Res_irr_neq0",
"apply",
"irr_constt",
"neq0_has_constt",
"sigW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
constt_cfInd_irr i :
H \subset G -> {j | j \in irr_constt ('Ind[G, H] 'chi_i)}. | Proof. by move=> sHG; apply/sigW/neq0_has_constt/Ind_irr_neq0. Qed. | Lemma | constt_cfInd_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",
"... | [
"Ind_irr_neq0",
"apply",
"irr_constt",
"neq0_has_constt",
"sHG",
"sigW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_Res phi :
H \subset G -> phi \is a character -> cfker ('Res[H] phi) = H :&: cfker phi. | Proof.
move=> sHG Nphi; apply/setP=> x; rewrite !cfkerEchar ?cfRes_char // !inE.
by apply/andb_id2l=> Hx; rewrite (subsetP sHG) ?cfResE.
Qed. | Lemma | cfker_Res | 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",
"cfResE",
"cfRes_char",
"cfker",
"cfkerEchar",
"character",
"inE",
"sHG",
"setP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_Ind chi :
H \subset G -> chi \is a character -> chi != 0 ->
cfker ('Ind[G, H] chi) = gcore (cfker chi) G. | Proof.
move=> sHG Nchi nzchi; rewrite !cfker_nzcharE ?cfInd_char ?cfInd_eq0 //.
apply/setP=> x; rewrite inE cfIndE // (can2_eq (mulVKf _) (mulKf _)) ?neq0CG //.
rewrite cfInd1 // mulrA -natrM Lagrange // mulr_natl -sumr_const.
apply/eqP/bigcapP=> [/normC_sum_upper ker_chiG_x y Gy | ker_chiG_x].
by rewrite mem_conjg i... | Lemma | cfker_Ind | 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",
"... | [
"Lagrange",
"apply",
"bigcapP",
"can2_eq",
"cfInd1",
"cfIndE",
"cfInd_char",
"cfInd_eq0",
"cfker",
"cfker_nzcharE",
"char1_ge_norm",
"character",
"chi",
"eq_bigr",
"gcore",
"groupV",
"groupVr",
"inE",
"mem_conjg",
"mem_conjgV",
"mulKf",
"mulVKf",
"mulrA",
"mulr_natl",
... | This is Isaacs Lemma (5.11). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfker_Ind_irr i :
H \subset G -> cfker ('Ind[G, H] 'chi_i) = gcore (cfker 'chi_i) G. | Proof. by move/cfker_Ind->; rewrite ?irr_neq0 ?irr_char. Qed. | Lemma | cfker_Ind_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",
"... | [
"cfker",
"cfker_Ind",
"gcore",
"irr_char",
"irr_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
neq0CG G : (#|G|)%:R != 0 :> algC. | Proof. exact: natrG_neq0. Qed. | Lemma | neq0CG | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"algC",
"natrG_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
neq0CiG G B : (#|G : B|)%:R != 0 :> algC. | Proof. exact: natr_indexg_neq0. Qed. | Lemma | neq0CiG | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"algC",
"natr_indexg_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gt0CG G : 0 < #|G|%:R :> algC. | Proof. exact: natrG_gt0. Qed. | Lemma | gt0CG | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"algC",
"natrG_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gt0CiG G B : 0 < #|G : B|%:R :> algC. | Proof. exact: natr_indexg_gt0. Qed. | Lemma | gt0CiG | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"algC",
"natr_indexg_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC'G_pchar G : [pchar algC]^'.-group G. | Proof. by apply/pgroupP=> p _; rewrite inE /= pchar_num. Qed. | Lemma | algC'G_pchar | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"algC",
"apply",
"group",
"inE",
"pchar",
"pchar_num",
"pgroupP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
algC'G | := (algC'G_pchar) (only parsing). | Notation | algC'G | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"algC'G_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_class_fun (B : {set gT}) (f : {ffun gT -> algC}) | :=
[forall x, forall y in B, f (x ^ y) == f x] && (support f \subset B). | Definition | is_class_fun | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"algC",
"gT",
"support"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intro_class_fun (G : {group gT}) f :
{in G &, forall x y, f (x ^ y) = f x} ->
(forall x, x \notin G -> f x = 0) ->
is_class_fun G (finfun f). | Proof.
move=> fJ Gf; apply/andP; split; last first.
by apply/supportP=> x notAf; rewrite ffunE Gf.
apply/'forall_eqfun_inP=> x y Gy; rewrite !ffunE.
by have [/fJ-> // | notGx] := boolP (x \in G); rewrite !Gf ?groupJr.
Qed. | Lemma | intro_class_fun | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Gf",
"apply",
"ffunE",
"gT",
"group",
"groupJr",
"is_class_fun",
"last",
"split",
"supportP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
G | := <<B>>. | Notation | G | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
classfun : predArgType | :=
Classfun {cfun_val; _ : is_class_fun G cfun_val}. | Record | classfun | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"is_class_fun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
classfun_key : unit. | Proof. by []. Qed. | Fact | classfun_key | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"unit"
] | The default expansion lemma cfunE requires key = 0. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Cfun | := locked_with classfun_key (fun flag : nat => Classfun). | Definition | Cfun | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"classfun_key",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_eqType : eqType | := classfun. | Definition | cfun_eqType | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"classfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fun_of_cfun phi | := cfun_val phi : gT -> algC. | Definition | fun_of_cfun | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"algC",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fun_of_cfun : classfun >-> Funclass. | Coercion | fun_of_cfun | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"classfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
cfunElock k f fP : @Cfun k (finfun f) fP =1 f. | Proof. by rewrite locked_withE; apply: ffunE. Qed. | Lemma | cfunElock | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cfun",
"apply",
"fP",
"ffunE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfunE f fP : @Cfun 0 (finfun f) fP =1 f. | Proof. exact: cfunElock. Qed. | Lemma | cfunE | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cfun",
"cfunElock",
"fP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfunP phi psi : phi =1 psi <-> phi = psi. | Proof. by split=> [/ffunP/val_inj | ->]. Qed. | Lemma | cfunP | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"ffunP",
"split",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun0gen phi x : x \notin G -> phi x = 0. | Proof. by case: phi => f fP; case: (andP fP) => _ /supportP; apply. Qed. | Lemma | cfun0gen | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"fP",
"supportP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_in_genP phi psi : {in G, phi =1 psi} -> phi = psi. | Proof.
move=> eq_phi; apply/cfunP=> x.
by have [/eq_phi-> // | notAx] := boolP (x \in G); rewrite !cfun0gen.
Qed. | Lemma | cfun_in_genP | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfun0gen",
"cfunP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfunJgen phi x y : y \in G -> phi (x ^ y) = phi x. | Proof.
case: phi => f fP Gy; apply/eqP.
by case: (andP fP) => /'forall_forall_inP->.
Qed. | Lemma | cfunJgen | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"fP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_zero_subproof : is_class_fun G (0 : {ffun _}). | Proof. exact: intro_class_fun. Qed. | Fact | cfun_zero_subproof | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"intro_class_fun",
"is_class_fun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_zero | := Cfun 0 cfun_zero_subproof. | Definition | cfun_zero | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cfun",
"cfun_zero_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_comp_subproof f phi :
f 0 = 0 -> is_class_fun G [ffun x => f (phi x)]. | Proof.
by move=> f0; apply: intro_class_fun => [x y _ /cfunJgen | x /cfun0gen] ->.
Qed. | Fact | cfun_comp_subproof | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfun0gen",
"cfunJgen",
"intro_class_fun",
"is_class_fun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_comp f f0 phi | := Cfun 0 (@cfun_comp_subproof f phi f0). | Definition | cfun_comp | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cfun",
"cfun_comp_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_opp | := cfun_comp (oppr0 _). | Definition | cfun_opp | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfun_comp",
"oppr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_add_subproof phi psi : is_class_fun G [ffun x => phi x + psi x]. | Proof.
apply: intro_class_fun => [x y Gx Gy | x notGx]; rewrite ?cfunJgen //.
by rewrite !cfun0gen ?add0r.
Qed. | Fact | cfun_add_subproof | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"add0r",
"apply",
"cfun0gen",
"cfunJgen",
"intro_class_fun",
"is_class_fun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_add phi psi | := Cfun 0 (cfun_add_subproof phi psi). | Definition | cfun_add | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cfun",
"cfun_add_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_indicator_subproof (A : {set gT}) :
is_class_fun G [ffun x => ((x \in G) && (x ^: G \subset A))%:R]. | Proof.
apply: intro_class_fun => [x y Gx Gy | x /negbTE/= -> //].
by rewrite groupJr ?classGidl.
Qed. | Fact | cfun_indicator_subproof | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"classGidl",
"gT",
"groupJr",
"intro_class_fun",
"is_class_fun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_indicator A | := Cfun 1 (cfun_indicator_subproof A). | Definition | cfun_indicator | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cfun",
"cfun_indicator_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''1_' A" | := (cfun_indicator A) : ring_scope. | Notation | ''1_' A | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfun_indicator"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun1Egen x : '1_G x = (x \in G)%:R. | Proof. by rewrite cfunElock andb_idr // => /class_subG->. Qed. | Lemma | cfun1Egen | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfunElock",
"class_subG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_mul_subproof phi psi : is_class_fun G [ffun x => phi x * psi x]. | Proof.
apply: intro_class_fun => [x y Gx Gy | x notGx]; rewrite ?cfunJgen //.
by rewrite cfun0gen ?mul0r.
Qed. | Fact | cfun_mul_subproof | group_representation | group_representation/classfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfun0gen",
"cfunJgen",
"intro_class_fun",
"is_class_fun",
"mul0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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