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 |
|---|---|---|---|---|---|---|---|---|---|---|
irr_class i | := enum_val (cast_ord (NirrE G) i). | Definition | irr_class | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"cast_ord",
"enum_val"
] | The character table. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
class_Iirr xG | :=
cast_ord (esym (NirrE G)) (enum_rank_in (classes1 G) xG). | Definition | class_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",
"... | [
"NirrE",
"cast_ord",
"classes1",
"enum_rank_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
c | := irr_class. | Notation | c | group_representation | group_representation/character.v | [
"HB",
"structures",
"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_class"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
g i | := (repr (c i)). | Notation | 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",
"... | [
"repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iC | := class_Iirr. | Notation | iC | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"class_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
character_table | := \matrix_(i, j) 'chi[G]_i (g j). | Definition | character_table | group_representation | group_representation/character.v | [
"HB",
"structures",
"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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
X | := character_table. | Notation | X | group_representation | group_representation/character.v | [
"HB",
"structures",
"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_table"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_classP i : c i \in classes G. | Proof. exact: enum_valP. Qed. | Lemma | irr_classP | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"classes",
"enum_valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
repr_irr_classK i : g i ^: G = c i. | Proof. by case/repr_classesP: (irr_classP i). Qed. | Lemma | repr_irr_classK | group_representation | group_representation/character.v | [
"HB",
"structures",
"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_classP",
"repr_classesP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_classK : cancel c iC. | Proof. by move=> i; rewrite /iC enum_valK_in cast_ordK. Qed. | Lemma | irr_classK | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"cast_ordK",
"enum_valK_in",
"iC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
class_IirrK : {in classes G, cancel iC c}. | Proof. by move=> xG GxG; rewrite /c cast_ordKV enum_rankK_in. Qed. | Lemma | class_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",
"... | [
"cast_ordKV",
"classes",
"enum_rankK_in",
"iC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reindex_irr_class R idx (op : @Monoid.com_law R idx) F :
\big[op/idx]_(xG in classes G) F xG = \big[op/idx]_i F (c i). | Proof.
rewrite (reindex c); last by apply: eq_bigl => i; apply: enum_valP.
by exists iC; [apply: in1W; apply: irr_classK | apply: class_IirrK].
Qed. | Lemma | reindex_irr_class | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"class_IirrK",
"classes",
"com_law",
"enum_valP",
"eq_bigl",
"iC",
"irr_classK",
"last",
"reindex"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
X' | := \matrix_(i, j) (#|'C_G[g i]|%:R^-1 * ('chi[G]_j (g i))^*). | Let | X' | group_representation | group_representation/character.v | [
"HB",
"structures",
"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"
] | orthogonality relation. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
XX'_1: X *m X' = 1%:M. | Proof.
apply/matrixP=> i j; rewrite !mxE -first_orthogonality_relation mulr_sumr.
rewrite sum_by_classes => [u v Gu Gv|]; first by rewrite -conjVg !cfunJ.
rewrite reindex_irr_class /=; apply/esym/eq_bigr=> k _.
rewrite !mxE irr_inv // -/(g k) -divg_index -indexgI /=.
rewrite (pchar0_natf_div Cpchar) ?dvdn_indexg // ind... | Let | XX'_1 | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Cpchar",
"X'",
"apply",
"cfunJ",
"conjVg",
"divg_index",
"dvdn_indexg",
"eq_bigr",
"first_orthogonality_relation",
"index_cent1",
"indexgI",
"invfM",
"invrK",
"irr_inv",
"matrixP",
"mulrA",
"mulrCA",
"mulr_sumr",
"mxE",
"pchar0_natf_div",
"reindex_irr_class",
"repr_irr_cla... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
character_table_unit : X \in unitmx. | Proof. by case/mulmx1_unit: XX'_1. Qed. | Lemma | character_table_unit | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"XX'_1",
"mulmx1_unit",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uX | := character_table_unit. | Let | uX | group_representation | group_representation/character.v | [
"HB",
"structures",
"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_table_unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
second_orthogonality_relation x y :
y \in G ->
\sum_i 'chi[G]_i x * ('chi_i y)^* = #|'C_G[x]|%:R *+ (x \in y ^: G). | Proof.
move=> Gy; pose i_x := iC (x ^: G); pose i_y := iC (y ^: G).
have [Gx | notGx] := boolP (x \in G); last first.
rewrite (contraNF (subsetP _ x) notGx) ?class_subG ?big1 // => i _.
by rewrite cfun0 ?mul0r.
transitivity ((#|'C_G[repr (y ^: G)]|%:R *: (X' *m X)) i_y i_x).
rewrite scalemxAl !mxE; apply: eq_bigr... | Theorem | second_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",
"... | [
"X'",
"apply",
"big1",
"can_in_eq",
"cfun0",
"cfun_repr",
"chi",
"class_IirrK",
"class_eqP",
"class_refl",
"class_subG",
"divg_index",
"eqVneq",
"eq_bigr",
"iC",
"index_cent1",
"indexgI",
"last",
"mem_classes",
"mem_repr",
"mul0r",
"mulVKf",
"mulmx1C",
"mulrA",
"mulrC... | This is Isaacs, Theorem (2.18). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eq_irr_mem_classP x y :
y \in G -> reflect (forall i, 'chi[G]_i x = 'chi_i y) (x \in y ^: G). | Proof.
move=> Gy; apply: (iffP idP) => [/imsetP[z Gz ->] i | xGy]; first exact: cfunJ.
have Gx: x \in G.
congr is_true: Gy; apply/eqP; rewrite -(can_eq oddb) -eqC_nat -!cfun1E.
by rewrite -irr0 xGy.
congr is_true: (class_refl G x); apply/eqP; rewrite -(can_eq oddb).
rewrite -(eqn_pmul2l (cardG_gt0 'C_G[x])) -eqC_na... | Lemma | eq_irr_mem_classP | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"cfun1E",
"cfunJ",
"chi",
"class_refl",
"eqC_nat",
"eq_bigr",
"eqn_pmul2l",
"imsetP",
"irr0",
"mulrnA",
"oddb",
"second_orthogonality_relation"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_afix_irr_classes (ito : action A (Iirr G)) (cto : action A _) a :
a \in A -> [acts A, on classes G | cto] ->
(forall i x y, x \in G -> y \in cto (x ^: G) a ->
'chi_i x = 'chi_(ito i a) y) ->
#|'Fix_ito[a]| = #|'Fix_(classes G | cto)[a]|. | Proof.
move=> Aa actsAG stabAchi; apply/eqP; rewrite -eqC_nat; apply/eqP.
have [[cP cK] iCK] := (irr_classP, irr_classK, class_IirrK).
pose icto b i := iC (cto (c i) b).
have Gca i: cto (c i) a \in classes G by rewrite (acts_act actsAG).
have inj_qa: injective (icto a).
by apply: can_inj (icto a^-1%g) _ => i; rewrite... | Lemma | card_afix_irr_classes | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"actKin",
"action",
"actperm",
"acts_act",
"algC",
"apply",
"big_filter",
"big_filter_cond",
"big_mkcond",
"can_eq",
"cardsE",
"class_IirrK",
"classes",
"col_permE",
"eqC_nat",
"eq_bigr",
"iC",
"inE",
"invgK",
"irr_classK",
"irr_classP",
"matrixP",
"me... | This is Isaacs, Theorem (6.32) (due to Brauer). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfdot_irr i j : '['chi_i, 'chi_j]_G = (i == j)%:R. | Proof.
rewrite -first_orthogonality_relation; congr (_ * _).
by apply: eq_bigr => x Gx; rewrite irr_inv.
Qed. | Lemma | cfdot_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",
"eq_bigr",
"first_orthogonality_relation",
"irr_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm_irr i : '['chi[G]_i] = 1. | Proof. by rewrite cfdot_irr eqxx. Qed. | Lemma | cfnorm_irr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"cfdot_irr",
"chi",
"eqxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_orthonormal : orthonormal (irr G). | Proof.
apply/orthonormalP; split; first exact: free_uniq (irr_free G).
move=> _ _ /irrP[i ->] /irrP[j ->].
by rewrite cfdot_irr (inj_eq irr_inj).
Qed. | Lemma | irr_orthonormal | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"cfdot_irr",
"free_uniq",
"inj_eq",
"irr",
"irrP",
"irr_free",
"irr_inj",
"orthonormal",
"orthonormalP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coord_cfdot phi i : coord (irr G) i phi = '[phi, 'chi_i]. | Proof.
rewrite {2}(coord_basis (irr_basis G) (memvf phi)).
rewrite cfdot_suml (bigD1 i) // cfdotZl /= -tnth_nth cfdot_irr eqxx mulr1.
rewrite big1 ?addr0 // => j neq_ji; rewrite cfdotZl /= -tnth_nth cfdot_irr.
by rewrite (negbTE neq_ji) mulr0.
Qed. | Lemma | coord_cfdot | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"big1",
"bigD1",
"cfdotZl",
"cfdot_irr",
"cfdot_suml",
"coord",
"coord_basis",
"eqxx",
"irr",
"irr_basis",
"memvf",
"mulr0",
"mulr1",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_sum_cfdot phi : phi = \sum_i '[phi, 'chi_i]_G *: 'chi_i. | Proof.
rewrite {1}(coord_basis (irr_basis G) (memvf phi)).
by apply: eq_bigr => i _; rewrite coord_cfdot -tnth_nth.
Qed. | Lemma | cfun_sum_cfdot | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"coord_basis",
"coord_cfdot",
"eq_bigr",
"irr_basis",
"memvf",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_sum_irr phi psi :
'[phi, psi]_G = \sum_i '[phi, 'chi_i] * '[psi, 'chi_i]^*. | Proof.
rewrite {1}[phi]cfun_sum_cfdot cfdot_suml; apply: eq_bigr => i _.
by rewrite cfdotZl -cfdotC.
Qed. | Lemma | cfdot_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",
"... | [
"apply",
"cfdotC",
"cfdotZl",
"cfdot_suml",
"cfun_sum_cfdot",
"eq_bigr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cnat_cfdot_char_irr i phi :
phi \is a character -> '[phi, 'chi_i]_G \in Num.nat. | Proof. by move/forallP/(_ i); rewrite coord_cfdot. Qed. | Lemma | Cnat_cfdot_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",
"... | [
"character",
"coord_cfdot",
"forallP",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_char_r phi chi :
chi \is a character -> '[phi, chi]_G = \sum_i '[phi, 'chi_i] * '[chi, 'chi_i]. | Proof.
move=> Nchi; rewrite cfdot_sum_irr; apply: eq_bigr => i _; congr (_ * _).
by rewrite conj_natr ?Cnat_cfdot_char_irr.
Qed. | Lemma | cfdot_char_r | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Cnat_cfdot_char_irr",
"apply",
"cfdot_sum_irr",
"character",
"chi",
"conj_natr",
"eq_bigr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cnat_cfdot_char chi xi :
chi \is a character -> xi \is a character -> '[chi, xi]_G \in Num.nat. | Proof.
move=> Nchi Nxi; rewrite cfdot_char_r ?rpred_sum // => i _.
by rewrite rpredM ?Cnat_cfdot_char_irr.
Qed. | Lemma | Cnat_cfdot_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_irr",
"Nxi",
"cfdot_char_r",
"character",
"chi",
"nat",
"rpredM",
"rpred_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdotC_char chi xi :
chi \is a character-> xi \is a character -> '[chi, xi]_G = '[xi, chi]. | Proof. by move=> Nchi Nxi; rewrite cfdotC conj_natr ?Cnat_cfdot_char. Qed. | Lemma | cfdotC_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",
"Nxi",
"cfdotC",
"character",
"chi",
"conj_natr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irrEchar chi : (chi \in irr G) = (chi \is a character) && ('[chi] == 1). | Proof.
apply/irrP/andP=> [[i ->] | [Nchi]]; first by rewrite irr_char cfnorm_irr.
rewrite cfdot_sum_irr => /eqP/natr_sum_eq1[i _| i [_ ci1 cj0]].
by rewrite rpredM // ?conj_natr ?Cnat_cfdot_char_irr.
exists i; rewrite [chi]cfun_sum_cfdot (bigD1 i) //=.
rewrite -(normr_idP (natr_ge0 (Cnat_cfdot_char_irr i Nchi))).
rew... | Lemma | irrEchar | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"addr0",
"apply",
"big1",
"bigD1",
"cfdot_sum_irr",
"cfnorm_irr",
"cfun_sum_cfdot",
"character",
"chi",
"conj_natr",
"irr",
"irrP",
"irr_char",
"natr_ge0",
"natr_sum_eq1",
"normC_def",
"normr_eq0",
"normr_idP",
"rpredM",
"scale0r",
"scale1r",
"sqrtC... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irrWchar chi : chi \in irr G -> chi \is a character. | Proof. by rewrite irrEchar => /andP[]. Qed. | Lemma | irrWchar | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"chi",
"irr",
"irrEchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irrWnorm chi : chi \in irr G -> '[chi] = 1. | Proof. by rewrite irrEchar => /andP[_ /eqP]. Qed. | Lemma | irrWnorm | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"irrEchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_lin_irr xi chi :
xi \is a linear_char -> chi \in irr G -> xi * chi \in irr G. | Proof.
move=> Lxi; rewrite !irrEchar => /andP[Nphi /eqP <-].
rewrite rpredM // ?lin_charW //=; apply/eqP; congr (_ * _).
apply: eq_bigr=> x Gx; rewrite !cfunE rmorphM/= mulrACA -(lin_charV_conj Lxi)//.
by rewrite -lin_charM ?groupV // mulgV lin_char1 ?mul1r.
Qed. | Lemma | mul_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",
"cfunE",
"chi",
"eq_bigr",
"groupV",
"irr",
"irrEchar",
"lin_char1",
"lin_charM",
"lin_charV_conj",
"lin_charW",
"linear_char",
"mul1r",
"mulgV",
"mulrACA",
"rmorphM",
"rpredM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_scaled_irr a b i j :
(a *: 'chi[G]_i == b *: 'chi_j) = (a == b) && ((a == 0) || (i == j)). | Proof.
apply/eqP/andP=> [|[/eqP-> /pred2P[]-> //]]; last by rewrite !scale0r.
move/(congr1 (cfdotr 'chi__)) => /= eq_ai_bj.
move: {eq_ai_bj}(eq_ai_bj i) (esym (eq_ai_bj j)); rewrite !cfdotZl !cfdot_irr.
by rewrite !mulr_natr !mulrb !eqxx eq_sym orbC; case: ifP => _ -> //= ->.
Qed. | Lemma | eq_scaled_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",
"cfdotZl",
"cfdot_irr",
"cfdotr",
"chi",
"eq_sym",
"eqxx",
"last",
"mulr_natr",
"mulrb",
"pred2P",
"scale0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_signed_irr (s t : bool) i j :
((-1) ^+ s *: 'chi[G]_i == (-1) ^+ t *: 'chi_j) = (s == t) && (i == j). | Proof. by rewrite eq_scaled_irr signr_eq0 (inj_eq signr_inj). Qed. | Lemma | eq_signed_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",
"... | [
"chi",
"eq_scaled_irr",
"inj_eq",
"signr_eq0",
"signr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_scale_irr a (i j : Iirr G) :
(a *: 'chi_i == a *: 'chi_j) = (a == 0) || (i == j). | Proof. by rewrite eq_scaled_irr eqxx. Qed. | Lemma | eq_scale_irr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Iirr",
"eq_scaled_irr",
"eqxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_addZ_irr a b (i j r t : Iirr G) :
(a *: 'chi_i + b *: 'chi_j == a *: 'chi_r + b *: 'chi_t)
= [|| [&& (a == 0) || (i == r) & (b == 0) || (j == t)],
[&& i == t, j == r & a == b] | [&& i == j, r == t & a == - b]]. | Proof.
rewrite -!eq_scale_irr; apply/eqP/idP; last first.
case/orP; first by case/andP=> /eqP-> /eqP->.
case/orP=> /and3P[/eqP-> /eqP-> /eqP->]; first by rewrite addrC.
by rewrite !scaleNr !addNr.
have [-> /addrI/eqP-> // | /=] := eqVneq.
rewrite eq_scale_irr => /norP[/negP nz_a /negPf neq_ir].
move/(congr1 (cfdo... | Lemma | eq_addZ_irr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Iirr",
"add0r",
"addNr",
"addr0",
"addrC",
"addrI",
"addr_eq0",
"apply",
"cfdotDl",
"cfdotZl",
"cfdot_irr",
"cfdotr",
"eqVneq",
"eq_scale_irr",
"eq_sym",
"eqxx",
"last",
"mulr_natr",
"mulrb",
"scaleNr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_subZnat_irr (a b : nat) (i j r t : Iirr G) :
(a%:R *: 'chi_i - b%:R *: 'chi_j == a%:R *: 'chi_r - b%:R *: 'chi_t)
= [|| a == 0 | i == r] && [|| b == 0 | j == t]
|| [&& i == j, r == t & a == b]. | Proof.
rewrite -!scaleNr eq_addZ_irr oppr_eq0 opprK -addr_eq0 -natrD eqr_nat.
by rewrite !pnatr_eq0 addn_eq0; case: a b => [|a] [|b]; rewrite ?andbF.
Qed. | Lemma | eq_subZnat_irr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Iirr",
"addn_eq0",
"addr_eq0",
"eq_addZ_irr",
"eqr_nat",
"nat",
"natrD",
"opprK",
"oppr_eq0",
"pnatr_eq0",
"scaleNr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char1_ge_norm (chi : 'CF(G)) x :
chi \is a character -> `|chi x| <= chi 1%g. | Proof.
case/char_reprP=> rG ->; case Gx: (x \in G); last first.
by rewrite cfunE cfRepr1 Gx normr0 ler0n.
by have [e [_ _ []]] := repr_rsim_diag rG Gx.
Qed. | Lemma | char1_ge_norm | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"cfunE",
"char_reprP",
"character",
"chi",
"last",
"ler0n",
"normr0",
"rG",
"repr_rsim_diag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_cfRepr_norm_scalar n (rG : mx_representation algC G n) x :
x \in G -> `|cfRepr rG x| = cfRepr rG 1%g ->
exists2 c, `|c| = 1 & rG x = c%:M. | Proof.
move=> Gx; have [e [[B uB def_x] [_ e1] [-> _] _]] := repr_rsim_diag rG Gx.
rewrite cfRepr1 -[n in n%:R]card_ord -sumr_const -(eq_bigr _ (in1W e1)).
case/normC_sum_eq1=> [i _ | c /eqP norm_c_1 def_e]; first by rewrite e1.
have{} def_e: e = const_mx c by apply/rowP=> i; rewrite mxE def_e ?andbT.
by exists c => //... | Lemma | max_cfRepr_norm_scalar | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"card_ord",
"cfRepr",
"cfRepr1",
"const_mx",
"diag_const_mx",
"eq_bigr",
"mulmxKV",
"mxE",
"mx_representation",
"normC_sum_eq1",
"rG",
"repr_rsim_diag",
"rowP",
"scalar_mxC",
"sumr_const",
"uB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_cfRepr_mx1 n (rG : mx_representation algC G n) x :
x \in G -> cfRepr rG x = cfRepr rG 1%g -> rG x = 1%:M. | Proof.
move=> Gx kerGx; have [|c _ def_x] := @max_cfRepr_norm_scalar n rG x Gx.
by rewrite kerGx cfRepr1 normr_nat.
move/eqP: kerGx; rewrite cfRepr1 cfunE Gx {rG}def_x mxtrace_scalar.
case: n => [_|n]; first by rewrite ![_%:M]flatmx0.
rewrite mulrb -subr_eq0 -mulrnBl -mulr_natl mulf_eq0 pnatr_eq0 /=.
by rewrite subr_... | Lemma | max_cfRepr_mx1 | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"cfRepr",
"cfRepr1",
"cfunE",
"flatmx0",
"max_cfRepr_norm_scalar",
"mulf_eq0",
"mulr_natl",
"mulrb",
"mulrnBl",
"mx_representation",
"mxtrace_scalar",
"normr_nat",
"pnatr_eq0",
"rG",
"subr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_constt (B : {set gT}) phi | := [pred i | '[phi, 'chi_i]_B != 0]. | Definition | irr_constt | group_representation | group_representation/character.v | [
"HB",
"structures",
"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 | |
irr_consttE i phi : (i \in irr_constt phi) = ('[phi, 'chi_i]_G != 0). | Proof. by []. Qed. | Lemma | irr_consttE | group_representation | group_representation/character.v | [
"HB",
"structures",
"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_constt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
constt_charP (i : Iirr G) chi :
chi \is a character ->
reflect (exists2 chi', chi' \is a character & chi = 'chi_i + chi')
(i \in irr_constt chi). | Proof.
move=> Nchi; apply: (iffP idP) => [i_in_chi| [chi' Nchi' ->]]; last first.
rewrite inE /= cfdotDl cfdot_irr eqxx -(eqP (Cnat_cfdot_char_irr i Nchi')).
by rewrite -natrD pnatr_eq0.
exists (chi - 'chi_i); last by rewrite addrC subrK.
apply/forallP=> j; rewrite coord_cfdot cfdotBl cfdot_irr.
have [<- | _] := eq... | Lemma | constt_charP | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"Iirr",
"addrC",
"apply",
"cfdotBl",
"cfdotDl",
"cfdot_irr",
"character",
"chi",
"coord_cfdot",
"eqxx",
"forallP",
"inE",
"irr_constt",
"last",
"lt0n",
"natrB",
"natrD",
"natrP",
"pnatr_eq0",
"rpred_nat",
"subr0",
"subrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun_sum_constt (phi : 'CF(G)) :
phi = \sum_(i in irr_constt phi) '[phi, 'chi_i] *: 'chi_i. | Proof.
rewrite {1}[phi]cfun_sum_cfdot (bigID [pred i | '[phi, 'chi_i] == 0]) /=.
by rewrite big1 ?add0r // => i /eqP->; rewrite scale0r.
Qed. | Lemma | cfun_sum_constt | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"big1",
"bigID",
"cfun_sum_cfdot",
"irr_constt",
"scale0r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
neq0_has_constt (phi : 'CF(G)) :
phi != 0 -> exists i, i \in irr_constt phi. | Proof.
move=> nz_phi; apply/existsP; apply: contra nz_phi => /pred0P phi0.
by rewrite [phi]cfun_sum_constt big_pred0.
Qed. | Lemma | neq0_has_constt | group_representation | group_representation/character.v | [
"HB",
"structures",
"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_pred0",
"cfun_sum_constt",
"existsP",
"irr_constt",
"pred0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
constt_irr i : irr_constt 'chi[G]_i =i pred1 i. | Proof.
by move=> j; rewrite !inE cfdot_irr pnatr_eq0 (eq_sym j); case: (i == j).
Qed. | Lemma | constt_irr | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"cfdot_irr",
"chi",
"eq_sym",
"inE",
"irr_constt",
"pnatr_eq0",
"pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char1_ge_constt (i : Iirr G) chi :
chi \is a character -> i \in irr_constt chi -> 'chi_i 1%g <= chi 1%g. | Proof.
move=> {chi} _ /constt_charP[// | chi Nchi ->].
by rewrite cfunE addrC -subr_ge0 addrK char1_ge0.
Qed. | Lemma | char1_ge_constt | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"addrC",
"addrK",
"cfunE",
"char1_ge0",
"character",
"chi",
"constt_charP",
"irr_constt",
"subr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
constt_ortho_char (phi psi : 'CF(G)) i j :
phi \is a character -> psi \is a character ->
i \in irr_constt phi -> j \in irr_constt psi ->
'[phi, psi] = 0 -> '['chi_i, 'chi_j] = 0. | Proof.
move=> _ _ /constt_charP[//|phi1 Nphi1 ->] /constt_charP[//|psi1 Npsi1 ->].
rewrite cfdot_irr; case: eqP => // -> /eqP/idPn[].
rewrite cfdotDl !cfdotDr cfnorm_irr -addrA gt_eqF ?ltr_wpDr ?ltr01 //.
by rewrite natr_ge0 ?rpredD ?Cnat_cfdot_char ?irr_char.
Qed. | Lemma | constt_ortho_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",
"addrA",
"cfdotDl",
"cfdotDr",
"cfdot_irr",
"cfnorm_irr",
"character",
"constt_charP",
"gt_eqF",
"irr_char",
"irr_constt",
"ltr01",
"ltr_wpDr",
"natr_ge0",
"rpredD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_repr n (rG : mx_representation algC G n) :
cfker (cfRepr rG) = rker rG. | Proof.
apply/esym/setP=> x; rewrite inE mul1mx /=.
case Gx: (x \in G); last by rewrite inE Gx.
apply/eqP/idP=> Kx; last by rewrite max_cfRepr_mx1 // cfker1.
rewrite inE Gx; apply/forallP=> y; rewrite !cfunE !mulrb groupMl //.
by case: ifP => // Gy; rewrite repr_mxM // Kx mul1mx.
Qed. | Lemma | cfker_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",
"cfker",
"cfker1",
"cfunE",
"forallP",
"groupMl",
"inE",
"last",
"max_cfRepr_mx1",
"mul1mx",
"mulrb",
"mx_representation",
"rG",
"repr_mxM",
"rker",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfkerEchar chi :
chi \is a character -> cfker chi = [set x in G | chi x == chi 1%g]. | Proof.
move=> Nchi; apply/setP=> x; apply/idP/setIdP=> [Kx | [Gx /eqP chi_x]].
by rewrite (subsetP (cfker_sub chi)) // cfker1.
case/char_reprP: Nchi => rG -> in chi_x *; rewrite inE Gx; apply/forallP=> y.
rewrite !cfunE groupMl // !mulrb; case: ifP => // Gy.
by rewrite repr_mxM // max_cfRepr_mx1 ?mul1mx.
Qed. | Lemma | cfkerEchar | group_representation | group_representation/character.v | [
"HB",
"structures",
"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_sub",
"cfunE",
"char_reprP",
"character",
"chi",
"forallP",
"groupMl",
"inE",
"max_cfRepr_mx1",
"mul1mx",
"mulrb",
"rG",
"repr_mxM",
"setIdP",
"setP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_nzcharE chi :
chi \is a character -> chi != 0 -> cfker chi = [set x | chi x == chi 1%g]. | Proof.
move=> Nchi nzchi; apply/setP=> x; rewrite cfkerEchar // !inE andb_idl //.
by apply: contraLR => /cfun0-> //; rewrite eq_sym char1_eq0.
Qed. | Lemma | cfker_nzcharE | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"cfkerEchar",
"cfun0",
"char1_eq0",
"character",
"chi",
"eq_sym",
"inE",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfkerEirr i : cfker 'chi[G]_i = [set x | 'chi_i x == 'chi_i 1%g]. | Proof. by rewrite cfker_nzcharE ?irr_char ?irr_neq0. Qed. | Lemma | cfkerEirr | group_representation | group_representation/character.v | [
"HB",
"structures",
"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_nzcharE",
"chi",
"irr_char",
"irr_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_irr0 : cfker 'chi[G]_0 = G. | Proof. by rewrite irr0 cfker_cfun1. Qed. | Lemma | cfker_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",
"... | [
"cfker",
"cfker_cfun1",
"chi",
"irr0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfaithful_reg : cfaithful (cfReg G). | Proof.
apply/subsetP=> x; rewrite cfkerEchar ?cfReg_char // !inE !cfRegE eqxx.
by case/andP=> _; apply: contraLR => /negbTE->; rewrite eq_sym neq0CG.
Qed. | Lemma | cfaithful_reg | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"cfRegE",
"cfReg_char",
"cfaithful",
"cfkerEchar",
"eq_sym",
"eqxx",
"inE",
"neq0CG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfkerE chi :
chi \is a character ->
cfker chi = G :&: \bigcap_(i in irr_constt chi) cfker 'chi_i. | Proof.
move=> Nchi; rewrite cfkerEchar //; apply/setP=> x; rewrite !inE.
apply: andb_id2l => Gx; rewrite {1 2}[chi]cfun_sum_constt !sum_cfunE.
apply/eqP/bigcapP=> [Kx i Ci | Kx]; last first.
by apply: eq_bigr => i /Kx Kx_i; rewrite !cfunE cfker1.
rewrite cfkerEirr inE /= -(inj_eq (mulfI Ci)).
have:= (normC_sum_upper ... | Lemma | cfkerE | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"bigcapP",
"cfker",
"cfker1",
"cfkerEchar",
"cfkerEirr",
"cfunE",
"cfun_sum_constt",
"char1_ge_norm",
"character",
"chi",
"eq_bigr",
"inE",
"inj_eq",
"irr_char",
"irr_constt",
"last",
"ler_wpM2l",
"mulfI",
"natr_ge0",
"normC_sum_upper",
"no... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
TI_cfker_irr : \bigcap_i cfker 'chi[G]_i = [1]. | Proof.
apply/trivgP; apply: subset_trans cfaithful_reg; rewrite cfkerE ?cfReg_char //.
rewrite subsetI (bigcap_min 0) //=; first by rewrite cfker_irr0.
by apply/bigcapsP=> i _; rewrite bigcap_inf.
Qed. | Lemma | TI_cfker_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",
"bigcap_inf",
"bigcap_min",
"bigcapsP",
"cfReg_char",
"cfaithful_reg",
"cfker",
"cfkerE",
"cfker_irr0",
"chi",
"subsetI",
"subset_trans",
"trivgP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfker_constt i chi :
chi \is a character -> i \in irr_constt chi ->
cfker chi \subset cfker 'chi[G]_i. | Proof. by move=> Nchi Ci; rewrite cfkerE ?subIset ?(bigcap_min i) ?orbT. Qed. | Lemma | cfker_constt | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"bigcap_min",
"cfker",
"cfkerE",
"character",
"chi",
"irr_constt",
"subIset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_xi : xi \is a linear_char. | Hypothesis | lin_xi | group_representation | group_representation/character.v | [
"HB",
"structures",
"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 | ||
Nxi: xi \is a character. | Proof. by have [] := andP lin_xi. Qed. | Let | Nxi | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"lin_xi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_char_der1 : G^`(1)%g \subset cfker xi. | Proof.
rewrite gen_subG /=; apply/subsetP=> _ /imset2P[x y Gx Gy ->].
rewrite cfkerEchar // inE groupR //= !lin_charM ?lin_charV ?in_group //.
by rewrite mulrCA mulKf ?mulVf ?lin_char_neq0 // lin_char1.
Qed. | Lemma | lin_char_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",
"cfker",
"cfkerEchar",
"gen_subG",
"groupR",
"imset2P",
"inE",
"in_group",
"lin_char1",
"lin_charM",
"lin_charV",
"lin_char_neq0",
"mulKf",
"mulVf",
"mulrCA",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_lin_char : #[xi]%CF = exponent (G / cfker xi)%g. | Proof.
apply/eqP; rewrite eqn_dvd; apply/andP; split.
apply/dvdn_cforderP=> x Gx; rewrite -lin_charX // -cfQuoEker ?groupX //.
rewrite morphX ?(subsetP (cfker_norm xi)) //= expg_exponent ?mem_quotient //.
by rewrite cfQuo1 ?cfker_normal ?lin_char1.
have abGbar: abelian (G / cfker xi) := sub_der1_abelian lin_char_... | Lemma | cforder_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",
"... | [
"abelian",
"abelian_nil",
"apply",
"cfQuo1",
"cfQuoEker",
"cfker",
"cfkerEchar",
"cfker_norm",
"cfker_normal",
"coset_id",
"dvdn_cforderP",
"eqn_dvd",
"expg_exponent",
"exponent",
"exponent_witness",
"groupX",
"inE",
"lin_char1",
"lin_charX",
"lin_char_der1",
"mem_quotient",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_lin_char_dvdG : #[xi]%CF %| #|G|. | Proof.
by rewrite cforder_lin_char (dvdn_trans (exponent_dvdn _)) ?dvdn_morphim.
Qed. | Lemma | cforder_lin_char_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",
"... | [
"cforder_lin_char",
"dvdn_morphim",
"dvdn_trans",
"exponent_dvdn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cforder_lin_char_gt0 : (0 < #[xi]%CF)%N. | Proof. by rewrite cforder_lin_char exponent_gt0. Qed. | Lemma | cforder_lin_char_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",
"... | [
"cforder_lin_char",
"exponent_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr_sub n (rG : mx_representation algC G n) (sHG : H \subset G) :
cfRepr (subg_repr rG sHG) = 'Res[H] (cfRepr rG). | Proof.
by apply/cfun_inP => x Hx; rewrite cfResE // !cfunE Hx (subsetP sHG).
Qed. | Lemma | cfRepr_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",
"... | [
"algC",
"apply",
"cfRepr",
"cfResE",
"cfunE",
"cfun_inP",
"mx_representation",
"rG",
"sHG",
"subg_repr",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_char chi : chi \is a character -> 'Res[H, G] chi \is a character. | Proof.
have [sHG | not_sHG] := boolP (H \subset G).
by case/char_reprP=> rG ->; rewrite -(cfRepr_sub rG sHG) cfRepr_char.
by move/Cnat_char1=> Nchi1; rewrite cfResEout // rpredZ_nat ?rpred1.
Qed. | Lemma | cfRes_char | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Cnat_char1",
"cfRepr_char",
"cfRepr_sub",
"cfResEout",
"char_reprP",
"character",
"chi",
"rG",
"rpred1",
"rpredZ_nat",
"sHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_eq0 phi : phi \is a character -> ('Res[H, G] phi == 0) = (phi == 0). | Proof. by move=> Nchi; rewrite -!char1_eq0 ?cfRes_char // cfRes1. Qed. | Lemma | cfRes_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",
"... | [
"cfRes1",
"cfRes_char",
"char1_eq0",
"character"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_lin_char chi :
chi \is a linear_char -> 'Res[H, G] chi \is a linear_char. | Proof. by case/andP=> Nchi; rewrite qualifE/= cfRes_char ?cfRes1. Qed. | Lemma | cfRes_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",
"... | [
"cfRes1",
"cfRes_char",
"chi",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Res_irr_neq0 i : 'Res[H, G] 'chi_i != 0. | Proof. by rewrite cfRes_eq0 ?irr_neq0 ?irr_char. Qed. | Lemma | Res_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",
"... | [
"cfRes_eq0",
"irr_char",
"irr_neq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_lin_lin (chi : 'CF(G)) :
chi \is a character -> 'Res[H] chi \is a linear_char -> chi \is a linear_char. | Proof. by rewrite !qualifE/= !qualifE/= cfRes1 => -> /andP[]. Qed. | Lemma | cfRes_lin_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",
"... | [
"cfRes1",
"character",
"chi",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_irr_irr chi :
chi \is a character -> 'Res[H] chi \in irr H -> chi \in irr G. | Proof.
have [sHG /char_reprP[rG ->] | not_sHG Nchi] := boolP (H \subset G).
rewrite -(cfRepr_sub _ sHG) => /irr_reprP[rH irrH def_rH]; apply/irr_reprP.
suffices /subg_mx_irr: mx_irreducible (subg_repr rG sHG) by exists rG.
by apply: mx_rsim_irr irrH; apply/cfRepr_rsimP/eqP.
rewrite cfResEout // => /irrP[j Dchi_j]... | Lemma | cfRes_irr_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",
"cfRepr_rsimP",
"cfRepr_sub",
"cfResEout",
"cfRes_lin_lin",
"cfunE",
"char_reprP",
"character",
"chi",
"contraNeq",
"eqxx",
"irr",
"irr0",
"irr1_neq0",
"irrP",
"irr_reprP",
"lin_char_irr",
"mx_irreducible",
"mx_rsim_irr",
"rG",
"rH",
"rpred1",
"sHG",
"subg_mx_i... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Res_Iirr (A B : {set gT}) i | := cfIirr ('Res[B, A] 'chi_i). | Definition | Res_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 | |
Res_Iirr0 : Res_Iirr H (0 : Iirr G) = 0. | Proof. by rewrite /Res_Iirr irr0 rmorph1 -irr0 irrK. Qed. | Lemma | Res_Iirr0 | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Iirr",
"Res_Iirr",
"irr0",
"irrK",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lin_Res_IirrE i : 'chi[G]_i 1%g = 1 -> 'chi_(Res_Iirr H i) = 'Res 'chi_i. | Proof.
move=> chi1; rewrite cfIirrE ?lin_char_irr ?cfRes_lin_char //.
by rewrite qualifE/= irr_char /= chi1.
Qed. | Lemma | lin_Res_IirrE | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Res_Iirr",
"cfIirrE",
"cfRes_lin_char",
"chi",
"irr_char",
"lin_char_irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
constt_Ind_Res i j :
(i \in irr_constt ('Ind[G] 'chi_j)) = (j \in irr_constt ('Res[H] 'chi_i)). | Proof. by rewrite !irr_consttE cfdotC conjC_eq0 -cfdot_Res_l. Qed. | Lemma | constt_Ind_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",
"... | [
"cfdotC",
"cfdot_Res_l",
"conjC_eq0",
"irr_constt",
"irr_consttE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_Res_ge_constt i j psi :
psi \is a character -> j \in irr_constt psi ->
'['Res[H, G] 'chi_j, 'chi_i] <= '['Res[H] psi, 'chi_i]. | Proof.
move=> {psi} _ /constt_charP[// | psi Npsi ->].
rewrite linearD cfdotDl addrC -subr_ge0 addrK natr_ge0 //=.
by rewrite Cnat_cfdot_char_irr // cfRes_char.
Qed. | Lemma | cfdot_Res_ge_constt | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"addrC",
"addrK",
"cfRes_char",
"cfdotDl",
"character",
"constt_charP",
"irr_constt",
"linearD",
"natr_ge0",
"subr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
constt_Res_trans j psi :
psi \is a character -> j \in irr_constt psi ->
{subset irr_constt ('Res[H, G] 'chi_j) <= irr_constt ('Res[H] psi)}. | Proof.
move=> Npsi Cj i; apply: contraNneq; rewrite eq_le => {1}<-.
rewrite cfdot_Res_ge_constt ?natr_ge0 ?Cnat_cfdot_char_irr //.
by rewrite cfRes_char ?irr_char.
Qed. | Lemma | constt_Res_trans | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"cfRes_char",
"cfdot_Res_ge_constt",
"character",
"contraNneq",
"eq_le",
"irr_char",
"irr_constt",
"natr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr_morphim n (rfG : mx_representation algC (f @* G) n) sGD :
cfRepr (morphim_repr rfG sGD) = cfMorph (cfRepr rfG). | Proof.
apply/cfun_inP=> x Gx; have Dx: x \in D := subsetP sGD x Gx.
by rewrite cfMorphE // !cfunE ?mem_morphim ?Gx.
Qed. | Lemma | cfRepr_morphim | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Dx",
"algC",
"apply",
"cfMorph",
"cfMorphE",
"cfRepr",
"cfunE",
"cfun_inP",
"mem_morphim",
"morphim_repr",
"mx_representation",
"sGD",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorph_char chi : chi \is a character -> cfMorph chi \is a character. | Proof.
have [sGD /char_reprP[rfG ->] | outGD Nchi] := boolP (G \subset D); last first.
by rewrite cfMorphEout // rpredZ_nat ?rpred1 ?Cnat_char1.
apply/char_reprP; exists (Representation (morphim_repr rfG sGD)).
by rewrite cfRepr_morphim.
Qed. | Lemma | cfMorph_char | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Cnat_char1",
"apply",
"cfMorph",
"cfMorphEout",
"cfRepr_morphim",
"char_reprP",
"character",
"chi",
"last",
"morphim_repr",
"rpred1",
"rpredZ_nat",
"sGD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorph_lin_char chi :
chi \is a linear_char -> cfMorph chi \is a linear_char. | Proof. by case/andP=> Nchi; rewrite qualifE/= cfMorph1 cfMorph_char. Qed. | Lemma | cfMorph_lin_char | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"cfMorph",
"cfMorph1",
"cfMorph_char",
"chi",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorph_charE chi :
G \subset D -> (cfMorph chi \is a character) = (chi \is a character). | Proof.
move=> sGD; apply/idP/idP=> [/char_reprP[[n rG] /=Dfchi] | /cfMorph_char//].
pose H := 'ker_G f; have kerH: H \subset rker rG.
by rewrite -cfker_repr -Dfchi cfker_morph // setIS // ker_sub_pre.
have nHG: G \subset 'N(H) by rewrite normsI // (subset_trans sGD) ?ker_norm.
have [h injh im_h] := first_isom_loc f s... | Lemma | cfMorph_charE | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"Dx",
"apply",
"cfMorph",
"cfMorphE",
"cfMorph_char",
"cfker_morph",
"cfker_repr",
"cfunE",
"cfun_inP",
"char_reprP",
"character",
"chi",
"coset",
"eqg_repr",
"first_isom_loc",
"invm",
"invmE",
"ker_norm",
"ker_sub_pre",
"mem_morphim",
"mem_quotient",
"morphimP",
"morphim... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorph_lin_charE chi :
G \subset D -> (cfMorph chi \is a linear_char) = (chi \is a linear_char). | Proof. by rewrite qualifE/= cfMorph1 => /cfMorph_charE->. Qed. | Lemma | cfMorph_lin_charE | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"cfMorph",
"cfMorph1",
"cfMorph_charE",
"chi",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfMorph_irr chi :
G \subset D -> (cfMorph chi \in irr G) = (chi \in irr (f @* G)). | Proof. by move=> sGD; rewrite !irrEchar cfMorph_charE // cfMorph_iso. Qed. | Lemma | cfMorph_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",
"... | [
"cfMorph",
"cfMorph_charE",
"cfMorph_iso",
"chi",
"irr",
"irrEchar",
"sGD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morph_Iirr i | := cfIirr (cfMorph 'chi[f @* G]_i). | Definition | morph_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",
"cfMorph",
"chi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morph_Iirr0 : morph_Iirr 0 = 0. | Proof. by rewrite /morph_Iirr irr0 rmorph1 -irr0 irrK. Qed. | Lemma | morph_Iirr0 | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"irr0",
"irrK",
"morph_Iirr",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morph_IirrE i : 'chi_(morph_Iirr i) = cfMorph 'chi_i. | Proof. by rewrite cfIirrE ?cfMorph_irr ?mem_irr. Qed. | Lemma | morph_IirrE | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"cfIirrE",
"cfMorph",
"cfMorph_irr",
"mem_irr",
"morph_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morph_Iirr_inj : injective morph_Iirr. | Proof.
by move=> i j eq_ij; apply/irr_inj/cfMorph_inj; rewrite // -!morph_IirrE eq_ij.
Qed. | Lemma | morph_Iirr_inj | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"cfMorph_inj",
"irr_inj",
"morph_Iirr",
"morph_IirrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morph_Iirr_eq0 i : (morph_Iirr i == 0) = (i == 0). | Proof. by rewrite -!irr_eq1 morph_IirrE cfMorph_eq1. Qed. | Lemma | morph_Iirr_eq0 | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"cfMorph_eq1",
"irr_eq1",
"morph_Iirr",
"morph_IirrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom_char chi :
(cfIsom isoGR chi \is a character) = (chi \is a character). | Proof.
rewrite [cfIsom _]locked_withE cfMorph_charE //.
by rewrite (isom_im (isom_sym _)) cfRes_id.
Qed. | Lemma | cfIsom_char | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"cfIsom",
"cfMorph_charE",
"cfRes_id",
"character",
"chi",
"isoGR",
"isom_im",
"isom_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom_lin_char chi :
(cfIsom isoGR chi \is a linear_char) = (chi \is a linear_char). | Proof. by rewrite qualifE/= cfIsom_char cfIsom1. Qed. | Lemma | cfIsom_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",
"... | [
"cfIsom",
"cfIsom1",
"cfIsom_char",
"chi",
"isoGR",
"linear_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfIsom_irr chi : (cfIsom isoGR chi \in irr R) = (chi \in irr G). | Proof. by rewrite !irrEchar cfIsom_char cfIsom_iso. Qed. | Lemma | cfIsom_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",
"... | [
"cfIsom",
"cfIsom_char",
"cfIsom_iso",
"chi",
"irr",
"irrEchar",
"isoGR"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isom_Iirr i | := cfIirr (cfIsom isoGR 'chi_i). | Definition | isom_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",
"cfIsom",
"isoGR"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isom_IirrE i : 'chi_(isom_Iirr i) = cfIsom isoGR 'chi_i. | Proof. by rewrite cfIirrE ?cfIsom_irr ?mem_irr. Qed. | Lemma | isom_IirrE | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"cfIirrE",
"cfIsom",
"cfIsom_irr",
"isoGR",
"isom_Iirr",
"mem_irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isom_Iirr_inj : injective isom_Iirr. | Proof.
by move=> i j eqij; apply/irr_inj/(cfIsom_inj isoGR); rewrite -!isom_IirrE eqij.
Qed. | Lemma | isom_Iirr_inj | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"cfIsom_inj",
"irr_inj",
"isoGR",
"isom_Iirr",
"isom_IirrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isom_Iirr_eq0 i : (isom_Iirr i == 0) = (i == 0). | Proof. by rewrite -!irr_eq1 isom_IirrE cfIsom_eq1. Qed. | Lemma | isom_Iirr_eq0 | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"cfIsom_eq1",
"irr_eq1",
"isom_Iirr",
"isom_IirrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isom_Iirr0 : isom_Iirr 0 = 0. | Proof. by apply/eqP; rewrite isom_Iirr_eq0. Qed. | Lemma | isom_Iirr0 | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"apply",
"isom_Iirr",
"isom_Iirr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isom_IirrK : cancel (isom_Iirr isoGR) (isom_Iirr (isom_sym isoGR)). | Proof. by move=> i; apply: irr_inj; rewrite !isom_IirrE cfIsomK. Qed. | Lemma | isom_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",
"cfIsomK",
"irr_inj",
"isoGR",
"isom_Iirr",
"isom_IirrE",
"isom_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isom_IirrKV : cancel (isom_Iirr (isom_sym isoGR)) (isom_Iirr isoGR). | Proof. by move=> i; apply: irr_inj; rewrite !isom_IirrE cfIsomKV. Qed. | Lemma | isom_IirrKV | group_representation | group_representation/character.v | [
"HB",
"structures",
"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",
"cfIsomKV",
"irr_inj",
"isoGR",
"isom_Iirr",
"isom_IirrE",
"isom_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
defG : K ><| H = G. | Hypothesis | defG | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
nKG: G \subset 'N(K). | Proof. by have [/andP[]] := sdprod_context defG. Qed. | Let | nKG | group_representation | group_representation/character.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"choice",
"ssrnat",
"seq",
"path",
"div",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"gproduct",
"fingroup",
"morphism",
"perm",
"automorphism",
"... | [
"defG",
"sdprod_context"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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