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 |
|---|---|---|---|---|---|---|---|---|---|---|
irrRepr i : cfRepr 'Chi_i = 'chi_i. | Proof.
rewrite irr.unlock (tnth_nth 0) nth_mkseq // -[<<G>>]/(gval _) genGidG.
by rewrite cfRes_id inord_val.
Qed. | Lemma | irrRepr | 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",
"cfRes_id",
"genGidG",
"inord_val",
"irr",
"nth_mkseq",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr0 : 'chi[G]_0 = 1. | Proof.
apply/cfun_inP=> x Gx; rewrite -irrRepr cfun1E cfunE Gx.
by rewrite socle_Iirr0 irr1_repr // mxtrace1 degree_irr1.
Qed. | Lemma | irr0 | 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",
"cfun1E",
"cfunE",
"cfun_inP",
"chi",
"degree_irr1",
"irr1_repr",
"irrRepr",
"mxtrace1",
"socle_Iirr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun1_irr : 1 \in irr G. | Proof. by rewrite -irr0 mem_tnth. Qed. | Lemma | cfun1_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",
"... | [
"irr",
"irr0",
"mem_tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_irr i : 'chi_i \in irr G. | Proof. exact: mem_tnth. Qed. | Lemma | mem_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",
"... | [
"irr",
"mem_tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irrP xi : reflect (exists i, xi = 'chi_i) (xi \in irr G). | Proof.
apply: (iffP idP) => [/(nthP 0)[i] | [i ->]]; last exact: mem_irr.
rewrite size_tuple => lt_i_G <-.
by exists (Ordinal lt_i_G); rewrite (tnth_nth 0).
Qed. | Lemma | irrP | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"irr",
"last",
"mem_irr",
"nthP",
"size_tuple",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
C'G | := algC'G_pchar G. | Let | C'G | 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'G_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
closG | := @groupC _ G. | Let | closG | 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",
"... | [
"groupC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
W i | := (@socle_of_Iirr _ G i). | Notation | W | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"socle_of_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''n_' i" | := 'n_(W i). | Notation | ''n_' i | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"n_"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''R_' i" | := 'R_(W i). | Notation | ''R_' i | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''e_' i" | := 'e_(W i). | Notation | ''e_' i | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr1_degree i : 'chi_i 1%g = ('n_i)%:R. | Proof. by rewrite -irrRepr cfRepr1. Qed. | Lemma | irr1_degree | 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",
"... | [
"cfRepr1",
"irrRepr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cnat_irr1 i : 'chi_i 1%g \in Num.nat. | Proof. by rewrite irr1_degree rpred_nat. Qed. | Lemma | Cnat_irr1 | 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",
"... | [
"irr1_degree",
"nat",
"rpred_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr1_gt0 i : 0 < 'chi_i 1%g. | Proof. by rewrite irr1_degree ltr0n irr_degree_gt0. Qed. | Lemma | irr1_gt0 | 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",
"... | [
"irr1_degree",
"irr_degree_gt0",
"ltr0n"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr1_neq0 i : 'chi_i 1%g != 0. | Proof. by rewrite eq_le lt_geF ?irr1_gt0. Qed. | Lemma | irr1_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",
"... | [
"eq_le",
"irr1_gt0",
"lt_geF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_neq0 i : 'chi_i != 0. | Proof. by apply: contraNneq (irr1_neq0 i) => ->; rewrite cfunE. Qed. | Lemma | 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",
"... | [
"apply",
"cfunE",
"contraNneq",
"irr1_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIirr_key : unit. | Proof. by []. Qed. | Remark | cfIirr_key | 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",
"... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIirr : forall B, 'CF(B) -> Iirr B | :=
locked_with cfIirr_key (fun B chi => inord (index chi (irr B))). | Definition | cfIirr | 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",
"cfIirr_key",
"chi",
"index",
"inord",
"irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIirrE chi : chi \in irr G -> 'chi_(cfIirr chi) = chi. | Proof.
move=> chi_irr; rewrite (tnth_nth 0) [cfIirr]unlock inordK ?nth_index //.
by rewrite -index_mem size_tuple in chi_irr.
Qed. | Lemma | cfIirrE | 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",
"index_mem",
"inordK",
"irr",
"nth_index",
"size_tuple",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIirrPE J (f : J -> 'CF(G)) (P : pred J) :
(forall j, P j -> f j \in irr G) ->
forall j, P j -> 'chi_(cfIirr (f j)) = f j. | Proof. by move=> irr_f j /irr_f; apply: cfIirrE. Qed. | Lemma | cfIirrPE | 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",
"cfIirr",
"cfIirrE",
"irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_sum_square : \sum_i ('chi[G]_i 1%g) ^+ 2 = #|G|%:R. | Proof.
rewrite -(sum_irr_degree_pchar sG) // natr_sum.
rewrite (reindex _ (socle_of_Iirr_bij _)) /=.
by apply: eq_bigr => i _; rewrite irr1_degree natrX.
Qed. | Corollary | irr_sum_square | 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",
"chi",
"eq_bigr",
"irr1_degree",
"natrX",
"natr_sum",
"reindex",
"sG",
"socle_of_Iirr_bij",
"sum_irr_degree_pchar"
] | This is Isaacs, Corollary (2.7). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfReg_sum : cfReg G = \sum_i 'chi_i 1%g *: 'chi_i. | Proof.
apply/cfun_inP=> x Gx.
rewrite -cfReprReg cfunE Gx (mxtrace_regular_pchar sG) //=.
rewrite sum_cfunE (reindex _ (socle_of_Iirr_bij _)); apply: eq_bigr => i _.
by rewrite -irrRepr cfRepr1 !cfunE Gx mulr_natl.
Qed. | Lemma | cfReg_sum | 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",
"cfReg",
"cfRepr1",
"cfReprReg",
"cfunE",
"cfun_inP",
"eq_bigr",
"irrRepr",
"mulr_natl",
"mxtrace_regular_pchar",
"reindex",
"sG",
"socle_of_Iirr_bij",
"sum_cfunE"
] | This is Isaacs, Lemma (2.11). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
aG | := regular_repr algC G. | Let | aG | 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",
"regular_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R_G | := group_ring algC G. | Let | R_G | 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",
"group_ring"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcfun_annihilate i j A : i != j -> (A \in 'R_j)%MS -> ('chi_i).[A]%CF = 0. | Proof.
move=> neq_ij RjA; rewrite -irrRepr xcfun_repr.
rewrite (irr_repr'_op0_pchar _ _ RjA) ?raddf0 //.
by rewrite eq_sym (can_eq socle_of_IirrK).
Qed. | Lemma | xcfun_annihilate | 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",
"... | [
"can_eq",
"eq_sym",
"irrRepr",
"irr_repr'_op0_pchar",
"raddf0",
"socle_of_IirrK",
"xcfun_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcfunG phi x : x \in G -> phi.[aG x]%CF = phi x. | Proof.
by move=> Gx; rewrite /xcfun /gring_row rowK -rowE !mxE !(gring_indexK, mul1g).
Qed. | Lemma | xcfunG | 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",
"... | [
"aG",
"gring_indexK",
"gring_row",
"mul1g",
"mxE",
"rowE",
"rowK",
"xcfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcfun_mul_id i A :
(A \in R_G)%MS -> ('chi_i).['e_i *m A]%CF = ('chi_i).[A]%CF. | Proof.
move=> RG_A; rewrite -irrRepr !xcfun_repr gring_opM //.
by rewrite op_Wedderburn_id_pchar ?mul1mx.
Qed. | Lemma | xcfun_mul_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",
"... | [
"R_G",
"gring_opM",
"irrRepr",
"mul1mx",
"op_Wedderburn_id_pchar",
"xcfun_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcfun_id i j : ('chi_i).['e_j]%CF = 'chi_i 1%g *+ (i == j). | Proof.
have [<-{j} | /xcfun_annihilate->//] := eqVneq; last exact: Wedderburn_id_mem.
by rewrite -xcfunG // repr_mx1 -(xcfun_mul_id _ (envelop_mx1 _)) mulmx1.
Qed. | Lemma | xcfun_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",
"... | [
"Wedderburn_id_mem",
"envelop_mx1",
"eqVneq",
"last",
"mulmx1",
"repr_mx1",
"xcfunG",
"xcfun_annihilate",
"xcfun_mul_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_free : free (irr G). | Proof.
apply/freeP=> s s0 i; apply: (mulIf (irr1_neq0 i)).
rewrite mul0r -(raddf0 (xcfun_r 'e_i)) -{}s0 raddf_sum /=.
rewrite (bigD1 i)//= -tnth_nth xcfunZl xcfun_id eqxx big1 ?addr0 // => j ne_ji.
by rewrite -tnth_nth xcfunZl xcfun_id (negbTE ne_ji) mulr0.
Qed. | Lemma | irr_free | 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",
"eqxx",
"free",
"freeP",
"irr",
"irr1_neq0",
"mul0r",
"mulIf",
"mulr0",
"raddf0",
"raddf_sum",
"s0",
"tnth_nth",
"xcfunZl",
"xcfun_id",
"xcfun_r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_inj : injective (tnth (irr G)). | Proof. by apply/injectiveP/free_uniq; rewrite map_tnth_enum irr_free. Qed. | Lemma | irr_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",
"free_uniq",
"injectiveP",
"irr",
"irr_free",
"map_tnth_enum",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irrK : cancel (tnth (irr G)) (@cfIirr G). | Proof. by move=> i; apply: irr_inj; rewrite cfIirrE ?mem_irr. Qed. | Lemma | irrK | 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",
"cfIirr",
"cfIirrE",
"irr",
"irr_inj",
"mem_irr",
"tnth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_eq1 i : ('chi_i == 1) = (i == 0). | Proof. by rewrite -irr0 (inj_eq irr_inj). Qed. | Lemma | irr_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",
"... | [
"inj_eq",
"irr0",
"irr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_irr_eq1 i : (#['chi_i]%CF == 1) = (i == 0). | Proof. by rewrite -dvdn1 dvdn_cforder irr_eq1. Qed. | Lemma | cforder_irr_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",
"... | [
"dvdn1",
"dvdn_cforder",
"irr_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_basis : basis_of 'CF(G)%VS (irr G). | Proof.
rewrite /basis_of irr_free andbT -dimv_leqif_eq ?subvf //.
by rewrite dim_cfun (eqnP irr_free) size_tuple NirrE.
Qed. | Lemma | irr_basis | 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",
"basis_of",
"dim_cfun",
"dimv_leqif_eq",
"eqnP",
"irr",
"irr_free",
"size_tuple",
"subvf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_sum_nth_irr a : \sum_i a i *: 'chi[G]_i = \sum_i a i *: (irr G)`_i. | Proof. by apply: eq_bigr => i; rewrite -tnth_nth. Qed. | Lemma | eq_sum_nth_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",
"chi",
"eq_bigr",
"irr",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_irr_sum phi : {a | phi = \sum_i a i *: 'chi[G]_i}. | Proof.
rewrite (coord_basis irr_basis (memvf phi)) -eq_sum_nth_irr.
by exists ((coord (irr G))^~ phi).
Qed. | Theorem | cfun_irr_sum | 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",
"coord",
"coord_basis",
"eq_sum_nth_irr",
"irr",
"irr_basis",
"memvf"
] | This is Isaacs, Theorem (2.8). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfRepr_standard n (rG : mx_representation algC G n) :
cfRepr (standard_grepr rG)
= \sum_i (standard_irr_coef rG (W i))%:R *: 'chi_i. | Proof.
rewrite cfRepr_dsum (reindex _ (socle_of_Iirr_bij _)).
by apply: eq_bigr => i _; rewrite scaler_nat cfRepr_muln irrRepr.
Qed. | Lemma | cfRepr_standard | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"algC",
"apply",
"cfRepr",
"cfRepr_dsum",
"cfRepr_muln",
"eq_bigr",
"irrRepr",
"mx_representation",
"rG",
"reindex",
"scaler_nat",
"socle_of_Iirr_bij",
"standard_grepr",
"standard_irr_coef"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr_inj n1 n2 rG1 rG2 :
@cfRepr _ G n1 rG1 = @cfRepr _ G n2 rG2 -> mx_rsim rG1 rG2. | Proof.
move=> eq_repr12; pose c i : algC := (standard_irr_coef _ (W i))%:R.
have [rsim1 rsim2] := (mx_rsim_standard rG1, mx_rsim_standard rG2).
apply: mx_rsim_trans (rsim1) (mx_rsim_sym _).
suffices ->: standard_grepr rG1 = standard_grepr rG2 by [].
apply: eq_bigr => Wi _; congr (muln_grepr _ _); apply/eqP; rewrite -eq... | Lemma | cfRepr_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",
"apply",
"cfRepr",
"cfRepr_sim",
"cfRepr_standard",
"coord_sum_free",
"eqC_nat",
"eq_bigr",
"eq_sum_nth_irr",
"irr_free",
"irr_of_socleK",
"muln_grepr",
"mx_rsim",
"mx_rsim_standard",
"mx_rsim_sym",
"mx_rsim_trans",
"standard_grepr",
"standard_irr_coef"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr_rsimP n1 n2 rG1 rG2 :
reflect (mx_rsim rG1 rG2) (@cfRepr _ G n1 rG1 == @cfRepr _ G n2 rG2). | Proof. by apply: (iffP eqP) => [/cfRepr_inj | /cfRepr_sim]. Qed. | Lemma | cfRepr_rsimP | 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",
"cfRepr_inj",
"cfRepr_sim",
"mx_rsim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_reprP xi :
reflect (exists2 rG : representation _ G, mx_irreducible rG & xi = cfRepr rG)
(xi \in irr G). | Proof.
apply: (iffP (irrP xi)) => [[i ->] | [[n rG] irr_rG ->]].
by exists (Representation 'Chi_i); [apply: socle_irr | rewrite irrRepr].
exists (irr_of_socle (irr_comp sG rG)); rewrite -irrRepr irr_of_socleK /=.
exact/cfRepr_sim/rsim_irr_comp_pchar.
Qed. | Lemma | irr_reprP | 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",
"cfRepr_sim",
"irr",
"irrP",
"irrRepr",
"irr_comp",
"irr_of_socle",
"irr_of_socleK",
"mx_irreducible",
"rG",
"representation",
"rsim_irr_comp_pchar",
"sG",
"socle_irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_id_expansion i :
'e_i = #|G|%:R^-1 *: (\sum_(x in G) 'chi_i 1%g * 'chi_i x^-1%g *: aG x). | Proof.
have Rei: ('e_i \in 'R_i)%MS by apply: Wedderburn_id_mem.
have /envelop_mxP[a def_e]: ('e_i \in R_G)%MS; last rewrite -/aG in def_e.
by move: Rei; rewrite genmxE mem_sub_gring => /andP[].
apply: canRL (scalerK (neq0CG _)) _; rewrite def_e linear_sum /=.
apply: eq_bigr => x Gx; have Gx' := groupVr Gx; rewrite s... | Lemma | Wedderburn_id_expansion | 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",
"... | [
"R_G",
"Wedderburn_id_mem",
"Wedderburn_ideal",
"aG",
"addr0",
"apply",
"big1",
"bigD1",
"cfReg",
"cfRegE",
"cfReg_sum",
"envelop_mxP",
"envelop_mx_id",
"eq_bigr",
"eq_mulgV1",
"eqxx",
"genmxE",
"groupM",
"groupV",
"groupVr",
"last",
"linear_sum",
"mem_mulsmx",
"mem_sub... | This is Isaacs, Theorem (2.12). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
character_pred {G : {set gT}} | :=
fun phi : 'CF(G) => [forall i, coord (irr G) i phi \in Num.nat]. | Definition | character_pred | 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",
"... | [
"coord",
"gT",
"irr",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
character {G : {set gT}} | := [qualify a phi | @character_pred G phi]. | Definition | character | 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",
"... | [
"character_pred",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_char i : 'chi_i \is a character. | Proof. by apply/forallP=> j; rewrite (tnth_nth 0) coord_free ?irr_free. Qed. | Lemma | irr_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",
"character",
"coord_free",
"forallP",
"irr_free",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun1_char : (1 : 'CF(G)) \is a character. | Proof. by rewrite -irr0 irr_char. Qed. | Lemma | cfun1_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",
"... | [
"character",
"irr0",
"irr_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun0_char : (0 : 'CF(G)) \is a character. | Proof. by apply/forallP=> i; rewrite linear0 rpred0. Qed. | Lemma | cfun0_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",
"character",
"forallP",
"linear0",
"rpred0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_nmod_closed : nmod_closed (@character G). | Proof.
split=> [|chi xi /forallP-Nchi /forallP-Nxi]; first exact: cfun0_char.
by apply/forallP=> i; rewrite linearD rpredD /=.
Qed. | Fact | char_nmod_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",
"... | [
"Nxi",
"apply",
"cfun0_char",
"character",
"chi",
"forallP",
"linearD",
"nmod_closed",
"rpredD",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_sum_irrP {phi} :
reflect (exists n, phi = \sum_i (n i)%:R *: 'chi_i) (phi \is a character). | Proof.
apply: (iffP idP)=> [/forallP-Nphi | [n ->]]; last first.
by apply: rpred_sum => i _; rewrite scaler_nat rpredMn // irr_char.
do [have [a ->] := cfun_irr_sum phi] in Nphi *; exists (Num.truncn \o a).
apply: eq_bigr => i _; congr (_ *: _); have:= eqP (Nphi i).
by rewrite eq_sum_nth_irr coord_sum_free ?irr_free.... | Lemma | char_sum_irrP | 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",
"cfun_irr_sum",
"character",
"coord_sum_free",
"eq_bigr",
"eq_sum_nth_irr",
"forallP",
"irr_char",
"irr_free",
"last",
"rpredMn",
"rpred_sum",
"scaler_nat",
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_sum_irr chi :
chi \is a character -> {r | chi = \sum_(i <- r) 'chi_i}. | Proof.
move=> Nchi; apply: sig_eqW; case/char_sum_irrP: Nchi => n {chi}->.
elim/big_rec: _ => [|i _ _ [r ->]]; first by exists nil; rewrite big_nil.
exists (ncons (n i) i r); rewrite scaler_nat.
by elim: {n}(n i) => [|n IHn]; rewrite ?add0r //= big_cons mulrS -addrA IHn.
Qed. | Lemma | char_sum_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",
"... | [
"add0r",
"addrA",
"apply",
"big_cons",
"big_nil",
"big_rec",
"char_sum_irrP",
"character",
"chi",
"mulrS",
"ncons",
"scaler_nat",
"sig_eqW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cnat_char1 chi : chi \is a character -> chi 1%g \in Num.nat. | Proof.
case/char_sum_irr=> r ->{chi}.
by elim/big_rec: _ => [|i chi _ Nchi1]; rewrite cfunE ?rpredD // Cnat_irr1.
Qed. | Lemma | Cnat_char1 | 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_irr1",
"big_rec",
"cfunE",
"char_sum_irr",
"character",
"chi",
"nat",
"rpredD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char1_ge0 chi : chi \is a character -> 0 <= chi 1%g. | Proof. by move/Cnat_char1/natr_ge0. Qed. | Lemma | char1_ge0 | 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",
"character",
"chi",
"natr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char1_eq0 chi : chi \is a character -> (chi 1%g == 0) = (chi == 0). | Proof.
case/char_sum_irr=> r ->; apply/idP/idP=> [|/eqP->]; last by rewrite cfunE.
case: r => [|i r]; rewrite ?big_nil // sum_cfunE big_cons.
rewrite paddr_eq0 ?sumr_ge0 => // [|j _|]; rewrite 1?ltW ?irr1_gt0 //.
by rewrite (negbTE (irr1_neq0 i)).
Qed. | Lemma | char1_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",
"... | [
"apply",
"big_cons",
"big_nil",
"cfunE",
"char_sum_irr",
"character",
"chi",
"irr1_gt0",
"irr1_neq0",
"last",
"ltW",
"paddr_eq0",
"sum_cfunE",
"sumr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char1_gt0 chi : chi \is a character -> (0 < chi 1%g) = (chi != 0). | Proof. by move=> Nchi; rewrite -char1_eq0 // natr_gt0 ?Cnat_char1. Qed. | Lemma | char1_gt0 | 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",
"char1_eq0",
"character",
"chi",
"natr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_reprP phi :
reflect (exists rG : representation algC G, phi = cfRepr rG)
(phi \is a character). | Proof.
apply: (iffP char_sum_irrP) => [[n ->] | [[n rG] ->]]; last first.
exists (fun i => standard_irr_coef rG (socle_of_Iirr i)).
by rewrite -cfRepr_standard (cfRepr_sim (mx_rsim_standard rG)).
exists (\big[dadd_grepr/grepr0]_i muln_grepr (Representation 'Chi_i) (n i)).
rewrite cfRepr_dsum; apply: eq_bigr => i _.... | Lemma | char_reprP | 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_dsum",
"cfRepr_muln",
"cfRepr_sim",
"cfRepr_standard",
"char_sum_irrP",
"character",
"dadd_grepr",
"eq_bigr",
"grepr0",
"irrRepr",
"last",
"muln_grepr",
"mx_rsim_standard",
"rG",
"representation",
"scaler_nat",
"socle_of_Iirr",
"standard_irr... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reprG | := (mx_representation algC G). | Notation | reprG | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"algC",
"mx_representation"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr_char n (rG : reprG n) : cfRepr rG \is a character. | Proof. by apply/char_reprP; exists (Representation rG). Qed. | Lemma | cfRepr_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",
"cfRepr",
"char_reprP",
"character",
"rG",
"reprG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfReg_char : cfReg G \is a character. | Proof. by rewrite -cfReprReg cfRepr_char. Qed. | Lemma | cfReg_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",
"... | [
"cfReg",
"cfReprReg",
"cfRepr_char",
"character"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr_prod n1 n2 (rG1 : reprG n1) (rG2 : reprG n2) :
cfRepr rG1 * cfRepr rG2 = cfRepr (prod_repr rG1 rG2). | Proof. by apply/cfun_inP=> x Gx; rewrite !cfunE /= Gx mxtrace_prod. Qed. | Lemma | cfRepr_prod | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"cfRepr",
"cfunE",
"cfun_inP",
"mxtrace_prod",
"prod_repr",
"reprG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_char : mulr_closed (@character G). | Proof.
split=> [|_ _ /char_reprP[rG1 ->] /char_reprP[rG2 ->]]; first exact: cfun1_char.
apply/char_reprP; exists (Representation (prod_repr rG1 rG2)).
by rewrite cfRepr_prod.
Qed. | Lemma | mul_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",
"cfRepr_prod",
"cfun1_char",
"char_reprP",
"character",
"mulr_closed",
"prod_repr",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr_map u n (rG : mx_representation algC G n) :
cfRepr (map_repr u rG) = cfAut u (cfRepr rG). | Proof. by apply/cfun_inP=> x Gx; rewrite !cfunE Gx map_reprE trace_map_mx. Qed. | Lemma | cfRepr_map | 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",
"cfAut",
"cfRepr",
"cfunE",
"cfun_inP",
"map_repr",
"map_reprE",
"mx_representation",
"rG",
"trace_map_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_char u chi : (cfAut u chi \is a character) = (chi \is a character). | Proof.
without loss /char_reprP[rG ->]: u chi / chi \is a character.
by move=> IHu; apply/idP/idP=> ?; first rewrite -(cfAutK u chi); rewrite IHu.
rewrite cfRepr_char; apply/char_reprP.
by exists (Representation (map_repr u rG)); rewrite cfRepr_map.
Qed. | Lemma | cfAut_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",
"cfAutK",
"cfRepr_char",
"cfRepr_map",
"char_reprP",
"character",
"chi",
"map_repr",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjC_char chi : (chi^*%CF \is a character) = (chi \is a character). | Proof. exact: cfAut_char. Qed. | Lemma | cfConjC_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",
"... | [
"cfAut_char",
"character",
"chi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_char1 u (chi : 'CF(G)) :
chi \is a character -> cfAut u chi 1%g = chi 1%g. | Proof. by move/Cnat_char1=> Nchi1; rewrite cfunE /= aut_natr. Qed. | Lemma | cfAut_char1 | 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",
"aut_natr",
"cfAut",
"cfunE",
"character",
"chi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_irr1 u i : (cfAut u 'chi[G]_i) 1%g = 'chi_i 1%g. | Proof. exact: cfAut_char1 (irr_char i). Qed. | Lemma | cfAut_irr1 | 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",
"cfAut_char1",
"chi",
"irr_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjC_char1 (chi : 'CF(G)) :
chi \is a character -> chi^*%CF 1%g = chi 1%g. | Proof. exact: cfAut_char1. Qed. | Lemma | cfConjC_char1 | 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_char1",
"character",
"chi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjC_irr1 u i : ('chi[G]_i)^*%CF 1%g = 'chi_i 1%g. | Proof. exact: cfAut_irr1. Qed. | Lemma | cfConjC_irr1 | 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_irr1",
"chi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linear_char_pred {B : {set gT}} | :=
fun phi : 'CF(B) => (phi \is a character) && (phi 1%g == 1). | Definition | linear_char_pred | 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",
"... | [
"character",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linear_char {B : {set gT}} | :=
[qualify a phi | @linear_char_pred B phi]. | Definition | linear_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",
"... | [
"gT",
"linear_char_pred"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CFxi : xi \is a linear_char. | Hypothesis | CFxi | 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",
"... | [
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
lin_char1: xi 1%g = 1. | Proof. by case/andP: CFxi => _ /eqP. Qed. | Lemma | lin_char1 | 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",
"... | [
"CFxi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_charW : xi \is a character. | Proof. by case/andP: CFxi. Qed. | Lemma | lin_charW | 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",
"... | [
"CFxi",
"character"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun1_lin_char : (1 : 'CF(G)) \is a linear_char. | Proof. by rewrite qualifE/= cfun1_char /= cfun11. Qed. | Lemma | cfun1_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",
"... | [
"cfun11",
"cfun1_char",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_charM : {in G &, {morph xi : x y / (x * y)%g >-> x * y}}. | Proof.
move=> x y Gx Gy; case/andP: CFxi => /char_reprP[[n rG] -> /=].
rewrite cfRepr1 pnatr_eq1 => /eqP n1; rewrite {n}n1 in rG *.
rewrite !cfunE Gx Gy groupM //= !mulr1n repr_mxM //.
by rewrite [rG x]mx11_scalar [rG y]mx11_scalar -scalar_mxM !mxtrace_scalar.
Qed. | Lemma | lin_charM | 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",
"... | [
"CFxi",
"cfRepr1",
"cfunE",
"char_reprP",
"groupM",
"mulr1n",
"mx11_scalar",
"mxtrace_scalar",
"pnatr_eq1",
"rG",
"repr_mxM",
"scalar_mxM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_char_prod I r (P : pred I) (x : I -> gT) :
(forall i, P i -> x i \in G) ->
xi (\prod_(i <- r | P i) x i)%g = \prod_(i <- r | P i) xi (x i). | Proof.
move=> Gx; elim/(big_load (fun y => y \in G)): _.
elim/big_rec2: _ => [|i a y Pi [Gy <-]]; first by rewrite lin_char1.
by rewrite groupM ?lin_charM ?Gx.
Qed. | Lemma | lin_char_prod | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"big_load",
"big_rec2",
"gT",
"groupM",
"lin_char1",
"lin_charM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xiMV x : x \in G -> xi x * xi (x^-1)%g = 1. | Proof. by move=> Gx; rewrite -lin_charM ?groupV // mulgV lin_char1. Qed. | Let | xiMV | 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",
"... | [
"groupV",
"lin_char1",
"lin_charM",
"mulgV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_char_neq0 x : x \in G -> xi x != 0. | Proof.
by move/xiMV/(congr1 (predC1 0)); rewrite /= oner_eq0 mulf_eq0 => /norP[].
Qed. | Lemma | lin_char_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",
"... | [
"mulf_eq0",
"oner_eq0",
"predC1",
"xiMV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_charV x : x \in G -> xi x^-1%g = (xi x)^-1. | Proof. by move=> Gx; rewrite -[_^-1]mulr1 -(xiMV Gx) mulKf ?lin_char_neq0. Qed. | Lemma | lin_charV | 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",
"... | [
"lin_char_neq0",
"mulKf",
"mulr1",
"xiMV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_charX x n : x \in G -> xi (x ^+ n)%g = xi x ^+ n. | Proof.
move=> Gx; elim: n => [|n IHn]; first exact: lin_char1.
by rewrite expgS exprS lin_charM ?groupX ?IHn.
Qed. | Lemma | lin_charX | 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",
"... | [
"expgS",
"exprS",
"groupX",
"lin_char1",
"lin_charM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_char_unity_root x : x \in G -> xi x ^+ #[x] = 1. | Proof. by move=> Gx; rewrite -lin_charX // expg_order lin_char1. Qed. | Lemma | lin_char_unity_root | 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",
"... | [
"expg_order",
"lin_char1",
"lin_charX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normC_lin_char x : x \in G -> `|xi x| = 1. | Proof.
move=> Gx; apply/eqP; rewrite -(@pexpr_eq1 _ _ #[x]) //.
by rewrite -normrX // lin_char_unity_root ?normr1.
Qed. | Lemma | normC_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",
"lin_char_unity_root",
"normr1",
"normrX",
"pexpr_eq1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_charV_conj x : x \in G -> xi x^-1%g = (xi x)^*. | Proof.
move=> Gx; rewrite lin_charV // invC_norm mulrC normC_lin_char //.
by rewrite expr1n divr1.
Qed. | Lemma | lin_charV_conj | 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",
"... | [
"divr1",
"expr1n",
"invC_norm",
"lin_charV",
"mulrC",
"normC_lin_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_char_irr : xi \in irr G. | Proof.
case/andP: CFxi => /char_reprP[rG ->]; rewrite cfRepr1 pnatr_eq1 => /eqP n1.
by apply/irr_reprP; exists rG => //; apply/mx_abs_irrW/linear_mx_abs_irr.
Qed. | Lemma | lin_char_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",
"... | [
"CFxi",
"apply",
"cfRepr1",
"char_reprP",
"irr",
"irr_reprP",
"linear_mx_abs_irr",
"mx_abs_irrW",
"pnatr_eq1",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_conjC_lin_char : xi * xi^*%CF = 1. | Proof.
apply/cfun_inP=> x Gx.
by rewrite !cfunE cfun1E Gx -normCK normC_lin_char ?expr1n.
Qed. | Lemma | mul_conjC_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",
"cfun1E",
"cfunE",
"cfun_inP",
"expr1n",
"normCK",
"normC_lin_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_char_unitr : xi \in GRing.unit. | Proof. by apply/unitrPr; exists xi^*%CF; apply: mul_conjC_lin_char. Qed. | Lemma | lin_char_unitr | 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",
"mul_conjC_lin_char",
"unit",
"unitrPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invr_lin_char : xi^-1 = xi^*%CF. | Proof. by rewrite -[_^-1]mulr1 -mul_conjC_lin_char mulKr ?lin_char_unitr. Qed. | Lemma | invr_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",
"... | [
"lin_char_unitr",
"mulKr",
"mul_conjC_lin_char",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fful_lin_char_inj : cfaithful xi -> {in G &, injective xi}. | Proof.
move=> fful_phi x y Gx Gy xi_xy; apply/eqP; rewrite eq_mulgV1 -in_set1.
rewrite (subsetP fful_phi) // inE groupM ?groupV //=; apply/forallP=> z.
have [Gz | G'z] := boolP (z \in G); last by rewrite !cfun0 ?groupMl ?groupV.
by rewrite -mulgA lin_charM ?xi_xy -?lin_charM ?groupM ?groupV // mulKVg.
Qed. | Lemma | fful_lin_char_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",
"cfaithful",
"cfun0",
"eq_mulgV1",
"forallP",
"groupM",
"groupMl",
"groupV",
"inE",
"in_set1",
"last",
"lin_charM",
"mulKVg",
"mulgA",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_lin_char u (xi : 'CF(G)) :
(cfAut u xi \is a linear_char) = (xi \is a linear_char). | Proof. by rewrite qualifE/= cfAut_char; apply/andb_id2l=> /cfAut_char1->. Qed. | Lemma | cfAut_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",
"cfAut",
"cfAut_char",
"cfAut_char1",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjC_lin_char (xi : 'CF(G)) :
(xi^*%CF \is a linear_char) = (xi \is a linear_char). | Proof. exact: cfAut_lin_char. Qed. | Lemma | cfConjC_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",
"... | [
"cfAut_lin_char",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_Iirr_abelian : abelian G -> #|Iirr G| = #|G|. | Proof. by rewrite card_ord NirrE card_classes_abelian => /eqP. Qed. | Lemma | card_Iirr_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",
"... | [
"Iirr",
"NirrE",
"abelian",
"card_classes_abelian",
"card_ord"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_Iirr_cyclic : cyclic G -> #|Iirr G| = #|G|. | Proof. by move/cyclic_abelian/card_Iirr_abelian. Qed. | Lemma | card_Iirr_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",
"... | [
"Iirr",
"card_Iirr_abelian",
"cyclic",
"cyclic_abelian"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_abelianP :
reflect (forall i : Iirr G, 'chi_i \is a linear_char) (abelian G). | Proof.
apply: (iffP idP) => [cGG i | CF_G].
rewrite qualifE/= irr_char /= irr1_degree.
by rewrite irr_degree_abelian //; last apply: groupC.
rewrite card_classes_abelian -NirrE -eqC_nat -irr_sum_square //.
rewrite -{1}[Nirr G]card_ord -sumr_const; apply/eqP/eq_bigr=> i _.
by rewrite lin_char1 ?expr1n ?CF_G.
Qed. | Lemma | char_abelianP | 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",
"Nirr",
"NirrE",
"abelian",
"apply",
"cGG",
"card_classes_abelian",
"card_ord",
"eqC_nat",
"eq_bigr",
"expr1n",
"groupC",
"irr1_degree",
"irr_char",
"irr_degree_abelian",
"irr_sum_square",
"last",
"lin_char1",
"linear_char",
"sumr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_repr_lin_char (i : Iirr G) x :
x \in G -> 'chi_i \is a linear_char ->
irr_repr (socle_of_Iirr i) x = ('chi_i x)%:M. | Proof.
move=> Gx CFi; rewrite -irrRepr cfunE Gx.
move: (_ x); rewrite -[irr_degree _](@natrK algC) -irr1_degree lin_char1 //.
by rewrite (natrK 1) => A; rewrite trace_mx11 -mx11_scalar.
Qed. | Lemma | irr_repr_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",
"... | [
"Iirr",
"algC",
"cfunE",
"irr1_degree",
"irrRepr",
"irr_degree",
"irr_repr",
"lin_char1",
"linear_char",
"mx11_scalar",
"natrK",
"socle_of_Iirr",
"trace_mx11"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linear_char_divr : divr_closed (@linear_char G). | Proof.
split=> [|chi xi Lchi Lxi]; first exact: cfun1_lin_char.
rewrite invr_lin_char // qualifE/= cfunE.
by rewrite rpredM ?lin_char1 ?mulr1 ?lin_charW //= cfConjC_lin_char.
Qed. | Fact | linear_char_divr | 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",
"... | [
"cfConjC_lin_char",
"cfun1_lin_char",
"cfunE",
"chi",
"divr_closed",
"invr_lin_char",
"lin_char1",
"lin_charW",
"linear_char",
"mulr1",
"rpredM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_cyclic_lin i : cyclic G -> 'chi[G]_i \is a linear_char. | Proof. by move/cyclic_abelian/char_abelianP. Qed. | Lemma | irr_cyclic_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",
"... | [
"char_abelianP",
"chi",
"cyclic",
"cyclic_abelian",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_prime_lin i : prime #|G| -> 'chi[G]_i \is a linear_char. | Proof. by move/prime_cyclic/irr_cyclic_lin. Qed. | Lemma | irr_prime_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",
"... | [
"chi",
"irr_cyclic_lin",
"linear_char",
"prime",
"prime_cyclic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
repr_rsim_diag (G : {group gT}) f (rG : mx_representation algC G f) x :
x \in G -> let chi := cfRepr rG in
exists e,
[/\ (*a*) exists2 B, B \in unitmx & rG x = invmx B *m diag_mx e *m B,
(*b*) (forall i, e 0 i ^+ #[x] = 1) /\ (forall i, `|e 0 i| = 1),
(*c*) chi x = \sum_i e 0 i /\ `|chi x| <= chi 1%g
... | Proof.
move=> Gx; without loss cGG: G rG Gx / abelian G.
have sXG: <[x]> \subset G by rewrite cycle_subG.
move/(_ _ (subg_repr rG sXG) (cycle_id x) (cycle_abelian x)).
by rewrite /= !cfunE !groupV Gx (cycle_id x) !group1.
have [I U W simU W1 dxW]: mxsemisimple rG 1%:M.
rewrite -(reducible_Socle1 (DecSocleType r... | Lemma | repr_rsim_diag | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"DecSocleType",
"Socle_semisimple",
"abelian",
"algC",
"algC'G_pchar",
"apply",
"cGG",
"card_ord",
"cast_ord",
"cast_ordKV",
"cfRepr",
"cfRepr1",
"cfunE",
"chi",
"cycle_abelian",
"cycle_id",
"cycle_subG",
"diag_const_mx",
"diag_mx",
"enum_rank",
"enum_rankK",
"enum_val",
... | This is Isaacs, Lemma (2.15) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
char_inv (chi : 'CF(G)) x : chi \is a character -> chi x^-1%g = (chi x)^*. | Proof.
case Gx: (x \in G); last by rewrite !cfun0 ?rmorph0 ?groupV ?Gx.
by case/char_reprP=> rG ->; have [e [_ _ _]] := repr_rsim_diag rG Gx.
Qed. | Lemma | char_inv | 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",
"... | [
"cfun0",
"char_reprP",
"character",
"chi",
"groupV",
"last",
"rG",
"repr_rsim_diag",
"rmorph0"
] | This is Isaacs, Lemma (2.15) (d). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
irr_inv i x : 'chi[G]_i x^-1%g = ('chi_i x)^*. | Proof. exact/char_inv/irr_char. Qed. | Lemma | irr_inv | 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_inv",
"chi",
"irr_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
generalized_orthogonality_relation y (i j : Iirr G) :
#|G|%:R^-1 * (\sum_(x in G) 'chi_i (x * y)%g * 'chi_j x^-1%g)
= (i == j)%:R * ('chi_i y / 'chi_i 1%g). | Proof.
pose W := @socle_of_Iirr _ G; pose e k := Wedderburn_id (W k).
pose aG := regular_repr algC G.
have [Gy | notGy] := boolP (y \in G); last first.
rewrite cfun0 // mul0r big1 ?mulr0 // => x Gx.
by rewrite cfun0 ?groupMl ?mul0r.
transitivity (('chi_i).[e j *m aG y]%CF / 'chi_j 1%g).
rewrite [e j]Wedderburn_id... | Theorem | generalized_orthogonality_relation | 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",
"Wedderburn_id",
"Wedderburn_id_expansion",
"Wedderburn_id_mem",
"Wedderburn_ideal",
"aG",
"algC",
"apply",
"big1",
"big_distrl",
"cfun0",
"envelop_mx_id",
"eqVneq",
"eq_bigr",
"groupM",
"groupMl",
"irr1_neq0",
"last",
"mem_mulsmx",
"mul0r",
"mulKf",
"mulmx_suml",
... | This is Isaacs, Theorem (2.13). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
first_orthogonality_relation (i j : Iirr G) :
#|G|%:R^-1 * (\sum_(x in G) 'chi_i x * 'chi_j x^-1%g) = (i == j)%:R. | Proof.
have:= generalized_orthogonality_relation 1 i j.
rewrite mulrA mulfK ?irr1_neq0 // => <-; congr (_ * _).
by apply: eq_bigr => x; rewrite mulg1.
Qed. | Corollary | first_orthogonality_relation | 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",
"eq_bigr",
"generalized_orthogonality_relation",
"irr1_neq0",
"mulfK",
"mulg1",
"mulrA"
] | This is Isaacs, Corollary (2.14). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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