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 |
|---|---|---|---|---|---|---|---|---|---|---|
mem_zchar S phi : phi \in S -> phi \in 'Z[S]. | Proof. by move=> Sphi; rewrite mem_zchar_on ?cfun_onT. Qed. | Lemma | mem_zchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfun_onT",
"mem_zchar_on"
] | A special lemma is needed because trivial fails to use the cfun_onT Hint. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
zchar_nth_expansion S A phi :
phi \in 'Z[S, A] ->
{z | forall i, z i \in Num.int & phi = \sum_(i < size S) z i *: S`_i}. | Proof.
case/andP=> _ /sumboolP/sig_eqW[/= z ->]; exists (intr \o z) => //=.
by apply: eq_bigr => i _; rewrite scaler_int.
Qed. | Lemma | zchar_nth_expansion | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"eq_bigr",
"int",
"intr",
"scaler_int",
"sig_eqW",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchar_tuple_expansion n (S : n.-tuple 'CF(G)) A phi :
phi \in 'Z[S, A] ->
{z | forall i, z i \in Num.int & phi = \sum_(i < n) z i *: S`_i}. | Proof. by move/zchar_nth_expansion; rewrite size_tuple. Qed. | Lemma | zchar_tuple_expansion | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"int",
"size_tuple",
"tuple",
"zchar_nth_expansion"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchar_expansion S A phi : uniq S ->
phi \in 'Z[S, A] ->
{z | forall xi, z xi \in Num.int & phi = \sum_(xi <- S) z xi *: xi}. | Proof.
move=> Suniq /zchar_nth_expansion[z Zz ->] /=.
pose zS xi := oapp z 0 (insub (index xi S)).
exists zS => [xi | ]; rewrite {}/zS; first by case: (insub _) => /=.
rewrite (big_nth 0) big_mkord; apply: eq_bigr => i _; congr (_ *: _).
by rewrite index_uniq // valK.
Qed. | Lemma | zchar_expansion | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"big_mkord",
"big_nth",
"eq_bigr",
"index",
"index_uniq",
"insub",
"int",
"uniq",
"valK",
"zchar_nth_expansion"
] | A pure seq version with the extra hypothesis of S's unicity. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
zchar_span S A : {subset 'Z[S, A] <= <<S>>%VS}. | Proof.
move=> _ /zchar_nth_expansion[z Zz ->] /=.
by apply: rpred_sum => i _; rewrite rpredZ // memv_span ?mem_nth.
Qed. | Lemma | zchar_span | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"mem_nth",
"memv_span",
"rpredZ",
"rpred_sum",
"zchar_nth_expansion"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchar_trans S1 S2 A B :
{subset S1 <= 'Z[S2, B]} -> {subset 'Z[S1, A] <= 'Z[S2, A]}. | Proof.
move=> sS12 phi; rewrite !(zchar_split _ A) andbC => /andP[->]; rewrite andbT.
case/zchar_nth_expansion=> z Zz ->; apply: rpred_sum => i _.
by rewrite scale_zchar // (@zcharW _ B) ?sS12 ?mem_nth.
Qed. | Lemma | zchar_trans | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"S1",
"S2",
"apply",
"mem_nth",
"rpred_sum",
"scale_zchar",
"zcharW",
"zchar_nth_expansion",
"zchar_split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchar_trans_on S1 S2 A :
{subset S1 <= 'Z[S2, A]} -> {subset 'Z[S1] <= 'Z[S2, A]}. | Proof.
move=> sS12 _ /zchar_nth_expansion[z Zz ->]; apply: rpred_sum => i _.
by rewrite scale_zchar // sS12 ?mem_nth.
Qed. | Lemma | zchar_trans_on | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"S1",
"S2",
"apply",
"mem_nth",
"rpred_sum",
"scale_zchar",
"zchar_nth_expansion"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchar_sub_irr S A :
{subset S <= 'Z[irr G]} -> {subset 'Z[S, A] <= 'Z[irr G, A]}. | Proof. exact: zchar_trans. Qed. | Lemma | zchar_sub_irr | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"irr",
"zchar_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchar_subset S1 S2 A :
{subset S1 <= S2} -> {subset 'Z[S1, A] <= 'Z[S2, A]}. | Proof.
move=> sS12; apply: zchar_trans setT _ => // f /sS12 S2f.
by rewrite mem_zchar.
Qed. | Lemma | zchar_subset | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"S1",
"S2",
"S2f",
"apply",
"mem_zchar",
"setT",
"zchar_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchar_subseq S1 S2 A :
subseq S1 S2 -> {subset 'Z[S1, A] <= 'Z[S2, A]}. | Proof. by move/mem_subseq; apply: zchar_subset. Qed. | Lemma | zchar_subseq | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"S1",
"S2",
"apply",
"mem_subseq",
"subseq",
"zchar_subset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchar_filter S A (p : pred 'CF(G)) :
{subset 'Z[filter p S, A] <= 'Z[S, A]}. | Proof. by apply: zchar_subset=> f; apply/mem_subseq/filter_subseq. Qed. | Lemma | zchar_filter | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"filter",
"filter_subseq",
"mem_subseq",
"zchar_subset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
char_vchar chi : chi \is a character -> chi \in 'Z[irr G]. | Proof.
case/char_sum_irr=> r ->; apply: rpred_sum => i _.
by rewrite mem_zchar ?mem_tnth.
Qed. | Lemma | char_vchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"char_sum_irr",
"character",
"chi",
"irr",
"mem_tnth",
"mem_zchar",
"rpred_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_vchar i : 'chi[G]_i \in 'Z[irr G]. | Proof. exact/char_vchar/irr_char. Qed. | Lemma | irr_vchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"char_vchar",
"chi",
"irr",
"irr_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfun1_vchar : 1 \in 'Z[irr G]. | Proof. by rewrite -irr0 irr_vchar. Qed. | Lemma | cfun1_vchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"irr",
"irr0",
"irr_vchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vcharP phi :
reflect (exists2 chi1, chi1 \is a character
& exists2 chi2, chi2 \is a character & phi = chi1 - chi2)
(phi \in 'Z[irr G]). | Proof.
apply: (iffP idP) => [| [a Na [b Nb ->]]]; last by rewrite rpredB ?char_vchar.
case/zchar_tuple_expansion=> z Zz ->; rewrite (bigID (fun i => 0 <= z i)) /=.
set chi1 := \sum_(i | _) _; set nchi2 := \sum_(i | _) _.
exists chi1; last exists (- nchi2); last by rewrite opprK.
apply: rpred_sum => i zi_ge0; rewrite ... | Lemma | vcharP | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Rreal_int",
"apply",
"bigID",
"char_vchar",
"character",
"irr",
"irr_char",
"last",
"ltW",
"natrEint",
"opprK",
"oppr_ge0",
"real_ltNge",
"rpredB",
"rpredN",
"rpredZ_nat",
"rpred_sum",
"scaleNr",
"sumrN",
"tnth_nth",
"zchar_tuple_expansion"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aint_vchar phi x : phi \in 'Z[irr G] -> phi x \in Aint. | Proof.
case/vcharP=> [chi1 Nchi1 [chi2 Nchi2 ->]].
by rewrite !cfunE rpredB ?Aint_char.
Qed. | Lemma | Aint_vchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Aint",
"Aint_char",
"cfunE",
"irr",
"rpredB",
"vcharP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cint_vchar1 phi : phi \in 'Z[irr G] -> phi 1%g \in Num.int. | Proof.
case/vcharP=> phi1 Nphi1 [phi2 Nphi2 ->].
by rewrite !cfunE rpredB // rpred_nat_num ?Cnat_char1.
Qed. | Lemma | Cint_vchar1 | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cnat_char1",
"cfunE",
"int",
"irr",
"rpredB",
"rpred_nat_num",
"vcharP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cint_cfdot_vchar_irr i phi :
phi \in 'Z[irr G] -> '[phi, 'chi_i] \in Num.int. | Proof.
case/vcharP=> chi1 Nchi1 [chi2 Nchi2 ->].
by rewrite cfdotBl rpredB // rpred_nat_num ?Cnat_cfdot_char_irr.
Qed. | Lemma | Cint_cfdot_vchar_irr | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cnat_cfdot_char_irr",
"cfdotBl",
"int",
"irr",
"rpredB",
"rpred_nat_num",
"vcharP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_vchar_r phi psi :
psi \in 'Z[irr G] -> '[phi, psi] = \sum_i '[phi, 'chi_i] * '[psi, 'chi_i]. | Proof.
move=> Zpsi; rewrite cfdot_sum_irr; apply: eq_bigr => i _; congr (_ * _).
by rewrite aut_intr ?Cint_cfdot_vchar_irr.
Qed. | Lemma | cfdot_vchar_r | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cint_cfdot_vchar_irr",
"apply",
"aut_intr",
"cfdot_sum_irr",
"eq_bigr",
"irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cint_cfdot_vchar :
{in 'Z[irr G] &, forall phi psi, '[phi, psi] \in Num.int}. | Proof.
move=> phi psi Zphi Zpsi; rewrite /= cfdot_vchar_r // rpred_sum // => k _.
by rewrite rpredM ?Cint_cfdot_vchar_irr.
Qed. | Lemma | Cint_cfdot_vchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cint_cfdot_vchar_irr",
"cfdot_vchar_r",
"int",
"irr",
"rpredM",
"rpred_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cnat_cfnorm_vchar : {in 'Z[irr G], forall phi, '[phi] \in Num.nat}. | Proof. by move=> phi Zphi; rewrite /= natrEint cfnorm_ge0 Cint_cfdot_vchar. Qed. | Lemma | Cnat_cfnorm_vchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cint_cfdot_vchar",
"cfnorm_ge0",
"irr",
"nat",
"natrEint"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vchar_mulr_closed : mulr_closed 'Z[irr G]. | Proof.
split; first exact: cfun1_vchar.
move=> _ _ /vcharP[xi1 Nxi1 [xi2 Nxi2 ->]] /vcharP[xi3 Nxi3 [xi4 Nxi4 ->]].
by rewrite mulrBl !mulrBr !(rpredB, rpredD) // char_vchar ?rpredM.
Qed. | Fact | vchar_mulr_closed | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfun1_vchar",
"char_vchar",
"irr",
"mulrBl",
"mulrBr",
"mulr_closed",
"rpredB",
"rpredD",
"rpredM",
"split",
"vcharP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_vchar A :
{in 'Z[irr G, A] &, forall phi psi, phi * psi \in 'Z[irr G, A]}. | Proof.
move=> phi psi; rewrite zchar_split => /andP[Zphi Aphi] /zcharW Zpsi.
rewrite zchar_split rpredM //; apply/cfun_onP=> x A'x.
by rewrite cfunE (cfun_onP Aphi) ?mul0r.
Qed. | Lemma | mul_vchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfunE",
"cfun_onP",
"irr",
"mul0r",
"rpredM",
"zcharW",
"zchar_split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(Inu : {in 'Z[S] &, isometry nu}) (oSS : pairwise_orthogonal S). | Hypotheses | Inu | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"isometry",
"oSS",
"pairwise_orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
freeS | := orthogonal_free oSS. | Let | freeS | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"oSS",
"orthogonal_free"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniqS : uniq S | := free_uniq freeS. | Let | uniqS | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"freeS",
"free_uniq",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Z_S : {subset S <= 'Z[S]}. | Proof. by move=> phi; apply: mem_zchar. Qed. | Let | Z_S | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"mem_zchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
notS0 : 0 \notin S. | Proof. by case/andP: oSS. Qed. | Let | notS0 | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"oSS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dotSS | := proj2 (pairwise_orthogonalP oSS). | Let | dotSS | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"oSS",
"pairwise_orthogonalP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_pairwise_orthogonal : pairwise_orthogonal (map nu S). | Proof.
have inj_nu: {in S &, injective nu}.
move=> phi psi Sphi Spsi /= eq_nu; apply: contraNeq (memPn notS0 _ Sphi).
by rewrite -cfnorm_eq0 -Inu ?Z_S // {2}eq_nu Inu ?Z_S // => /dotSS->.
have notSnu0: 0 \notin map nu S.
apply: contra notS0 => /mapP[phi Sphi /esym/eqP].
by rewrite -cfnorm_eq0 Inu ?Z_S // cfnorm... | Lemma | map_pairwise_orthogonal | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Inu",
"Z_S",
"apply",
"cfnorm_eq0",
"contraNeq",
"dotSS",
"inj_in_eq",
"map",
"mapP",
"map_inj_in_uniq",
"memPn",
"notS0",
"pairwise_orthogonal",
"pairwise_orthogonalP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfproj_sum_orthogonal P z phi :
phi \in S ->
'[\sum_(xi <- S | P xi) z xi *: nu xi, nu phi]
= if P phi then z phi * '[phi] else 0. | Proof.
move=> Sphi; have defS := perm_to_rem Sphi.
rewrite cfdot_suml (perm_big _ defS) big_cons /= cfdotZl Inu ?Z_S //.
rewrite big1_seq ?addr0 // => xi; rewrite mem_rem_uniq ?inE //.
by case/and3P=> _ neq_xi Sxi; rewrite cfdotZl Inu ?Z_S // dotSS ?mulr0.
Qed. | Lemma | cfproj_sum_orthogonal | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Inu",
"Z_S",
"addr0",
"big1_seq",
"big_cons",
"cfdotZl",
"cfdot_suml",
"dotSS",
"inE",
"mem_rem_uniq",
"mulr0",
"perm_big",
"perm_to_rem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_sum_orthogonal z1 z2 :
'[\sum_(xi <- S) z1 xi *: nu xi, \sum_(xi <- S) z2 xi *: nu xi]
= \sum_(xi <- S) z1 xi * (z2 xi)^* * '[xi]. | Proof.
rewrite cfdot_sumr; apply: eq_big_seq => phi Sphi.
by rewrite cfdotZr cfproj_sum_orthogonal // mulrCA mulrA.
Qed. | Lemma | cfdot_sum_orthogonal | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfdotZr",
"cfdot_sumr",
"cfproj_sum_orthogonal",
"eq_big_seq",
"mulrA",
"mulrCA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm_sum_orthogonal z :
'[\sum_(xi <- S) z xi *: nu xi] = \sum_(xi <- S) `|z xi| ^+ 2 * '[xi]. | Proof.
by rewrite cfdot_sum_orthogonal; apply: eq_bigr => xi _; rewrite normCK.
Qed. | Lemma | cfnorm_sum_orthogonal | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfdot_sum_orthogonal",
"eq_bigr",
"normCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm_orthogonal : '[\sum_(xi <- S) nu xi] = \sum_(xi <- S) '[xi]. | Proof.
rewrite -(eq_bigr _ (fun _ _ => scale1r _)) cfnorm_sum_orthogonal.
by apply: eq_bigr => xi; rewrite normCK conjC1 !mul1r.
Qed. | Lemma | cfnorm_orthogonal | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfnorm_sum_orthogonal",
"conjC1",
"eq_bigr",
"mul1r",
"normCK",
"scale1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthogonal_span S phi :
pairwise_orthogonal S -> phi \in <<S>>%VS ->
{z | z = fun xi => '[phi, xi] / '[xi] & phi = \sum_(xi <- S) z xi *: xi}. | Proof.
move=> oSS /free_span[|c -> _]; first exact: orthogonal_free.
set z := fun _ => _ : algC; exists z => //; apply: eq_big_seq => u Su.
rewrite /z cfproj_sum_orthogonal // mulfK // cfnorm_eq0.
by rewrite (memPn _ u Su); case/andP: oSS.
Qed. | Lemma | orthogonal_span | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"algC",
"apply",
"cfnorm_eq0",
"cfproj_sum_orthogonal",
"eq_big_seq",
"free_span",
"memPn",
"mulfK",
"oSS",
"orthogonal_free",
"pairwise_orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(Inu : {in 'Z[S] &, isometry nu}) (onS : orthonormal S). | Hypotheses | Inu | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"isometry",
"orthonormal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
oSS | := orthonormal_orthogonal onS. | Let | oSS | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"orthonormal_orthogonal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nS1 : {in S, forall phi, '[phi] = 1}. | Proof. by move=> phi Sphi; case/orthonormalP: onS => _ -> //; rewrite eqxx. Qed. | Let | nS1 | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"eqxx",
"orthonormalP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_orthonormal : orthonormal (map nu S). | Proof.
rewrite !orthonormalE map_pairwise_orthogonal // andbT.
by apply/allP=> _ /mapP[xi Sxi ->]; rewrite /= Inu ?nS1 // mem_zchar.
Qed. | Lemma | map_orthonormal | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Inu",
"allP",
"apply",
"map",
"mapP",
"map_pairwise_orthogonal",
"mem_zchar",
"nS1",
"orthonormal",
"orthonormalE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfproj_sum_orthonormal z phi :
phi \in S -> '[\sum_(xi <- S) z xi *: nu xi, nu phi] = z phi. | Proof. by move=> Sphi; rewrite cfproj_sum_orthogonal // nS1 // mulr1. Qed. | Lemma | cfproj_sum_orthonormal | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfproj_sum_orthogonal",
"mulr1",
"nS1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_sum_orthonormal z1 z2 :
'[\sum_(xi <- S) z1 xi *: xi, \sum_(xi <- S) z2 xi *: xi]
= \sum_(xi <- S) z1 xi * (z2 xi)^*. | Proof.
rewrite cfdot_sum_orthogonal //; apply: eq_big_seq => phi /nS1->.
by rewrite mulr1.
Qed. | Lemma | cfdot_sum_orthonormal | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfdot_sum_orthogonal",
"eq_big_seq",
"mulr1",
"nS1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm_sum_orthonormal z :
'[\sum_(xi <- S) z xi *: nu xi] = \sum_(xi <- S) `|z xi| ^+ 2. | Proof.
rewrite cfnorm_sum_orthogonal //.
by apply: eq_big_seq => xi /nS1->; rewrite mulr1.
Qed. | Lemma | cfnorm_sum_orthonormal | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfnorm_sum_orthogonal",
"eq_big_seq",
"mulr1",
"nS1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm_map_orthonormal : '[\sum_(xi <- S) nu xi] = (size S)%:R. | Proof.
by rewrite cfnorm_orthogonal // (eq_big_seq _ nS1) big_tnth sumr_const card_ord.
Qed. | Lemma | cfnorm_map_orthonormal | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"big_tnth",
"card_ord",
"cfnorm_orthogonal",
"eq_big_seq",
"nS1",
"size",
"sumr_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthonormal_span phi :
phi \in <<S>>%VS ->
{z | z = fun xi => '[phi, xi] & phi = \sum_(xi <- S) z xi *: xi}. | Proof.
case/orthogonal_span=> // _ -> {2}->; set z := fun _ => _ : algC.
by exists z => //; apply: eq_big_seq => xi /nS1->; rewrite divr1.
Qed. | Lemma | orthonormal_span | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"algC",
"apply",
"divr1",
"eq_big_seq",
"nS1",
"orthogonal_span"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm_orthonormal S :
orthonormal S -> '[\sum_(xi <- S) xi] = (size S)%:R. | Proof. exact: cfnorm_map_orthonormal. Qed. | Lemma | cfnorm_orthonormal | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfnorm_map_orthonormal",
"orthonormal",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vchar_orthonormalP S :
{subset S <= 'Z[irr G]} ->
reflect (exists I : {set Iirr G}, exists b : Iirr G -> bool,
perm_eq S [seq (-1) ^+ b i *: 'chi_i | i in I])
(orthonormal S). | Proof.
move=> vcS; apply: (equivP orthonormalP).
split=> [[uniqS oSS] | [I [b defS]]]; last first.
split=> [|xi1 xi2]; rewrite ?(perm_mem defS).
rewrite (perm_uniq defS) map_inj_uniq ?enum_uniq // => i j /eqP.
by rewrite eq_signed_irr => /andP[_ /eqP].
case/mapP=> [i _ ->] /mapP[j _ ->]; rewrite eq_signed_i... | Lemma | vchar_orthonormalP | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cint_cfdot_vchar_irr",
"Iirr",
"addr0",
"addr_eq0",
"apply",
"aut_intr",
"big1",
"bigD1",
"cfdotNr",
"cfdotZl",
"cfdotZr",
"cfdot_irr",
"cfdot_sum_irr",
"cfun_sum_cfdot",
"enum_uniq",
"eq_signed_irr",
"eqxx",
"expf_eq0",
"inE",
"irr",
"last",
"mapP",
"map_inj_uniq",
"m... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vchar_norm1P phi :
phi \in 'Z[irr G] -> '[phi] = 1 ->
exists b : bool, exists i : Iirr G, phi = (-1) ^+ b *: 'chi_i. | Proof.
move=> Zphi phiN1.
have: orthonormal phi by rewrite /orthonormal/= phiN1 eqxx.
case/vchar_orthonormalP=> [xi /predU1P[->|] // | I [b def_phi]].
have: phi \in (phi : seq _) := mem_head _ _.
by rewrite (perm_mem def_phi) => /mapP[i _ ->]; exists (b i), i.
Qed. | Lemma | vchar_norm1P | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Iirr",
"eqxx",
"irr",
"mapP",
"mem_head",
"orthonormal",
"perm_mem",
"predU1P",
"seq",
"vchar_orthonormalP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zchar_small_norm phi n :
phi \in 'Z[irr G] -> '[phi] = n%:R -> (n < 4)%N ->
{S : n.-tuple 'CF(G) |
[/\ orthonormal S, {subset S <= 'Z[irr G]} & phi = \sum_(xi <- S) xi]}. | Proof.
move=> Zphi def_n lt_n_4.
pose S := [seq '[phi, 'chi_i] *: 'chi_i | i in irr_constt phi].
have def_phi: phi = \sum_(xi <- S) xi.
rewrite big_image big_mkcond {1}[phi]cfun_sum_cfdot.
by apply: eq_bigr => i _; rewrite if_neg; case: eqP => // ->; rewrite scale0r.
have orthS: orthonormal S.
apply/orthonormalP;... | Lemma | zchar_small_norm | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cint_cfdot_vchar_irr",
"addrC",
"addrK",
"apply",
"bigD1",
"big_image",
"big_mkcond",
"cfdotZl",
"cfdotZr",
"cfdot_irr",
"cfdot_sum_irr",
"cfnorm_orthonormal",
"cfun_sum_cfdot",
"def_n",
"enum_uniq",
"eqC_nat",
"eq_bigr",
"eq_scaled_irr",
"eqn_leq",
"eqr_nat",
"eqxx",
"exp... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vchar_norm2 phi :
phi \in 'Z[irr G, G^#] -> '[phi] = 2 ->
exists i, exists2 j, j != i & phi = 'chi_i - 'chi_j. | Proof.
rewrite zchar_split cfunD1E => /andP[Zphi phi1_0].
case/zchar_small_norm => // [[[|chi [|xi [|?]]] //= S2]].
case=> /andP[/and3P[Nchi Nxi _] /= ochi] /allP/and3P[Zchi Zxi _].
rewrite big_cons big_seq1 => def_phi.
have [b [i def_chi]] := vchar_norm1P Zchi (eqP Nchi).
have [c [j def_xi]] := vchar_norm1P Zxi (eqP N... | Lemma | vchar_norm2 | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Nxi",
"S2",
"addrC",
"allP",
"apply",
"big_cons",
"big_seq1",
"cfdotZl",
"cfdotZr",
"cfnorm_irr",
"cfunD1E",
"cfunE",
"chi",
"contraTneq",
"eq_le",
"eq_sym",
"irr",
"irr1_gt0",
"last",
"lt_geF",
"ltr_pDl",
"mulf_eq0",
"mulr1",
"rmorph_sign",
"scalerDr",
"scaler_sig... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zisometry_of_cfnorm (tauS : seq 'CF(G)) :
pairwise_orthogonal S -> pairwise_orthogonal tauS ->
map cfnorm tauS = map cfnorm S -> {subset tauS <= 'Z[irr G]} ->
{tau : {linear 'CF(L) -> 'CF(G)} | map tau S = tauS
& {in 'Z[S], isometry tau, to 'Z[irr G]}}. | Proof.
move=> oSS oTT /isometry_of_cfnorm[||tau defT Itau] // Z_T; exists tau => //.
split=> [|_ /zchar_nth_expansion[u Zu ->]].
by apply: sub_in2 Itau; apply: zchar_span.
rewrite big_seq linear_sum rpred_sum // => xi Sxi.
by rewrite linearZ scale_zchar ?Z_T // -defT map_f ?mem_nth.
Qed. | Lemma | Zisometry_of_cfnorm | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"big_seq",
"cfnorm",
"irr",
"isometry",
"isometry_of_cfnorm",
"linear",
"linearZ",
"linear_sum",
"map",
"map_f",
"mem_nth",
"oSS",
"pairwise_orthogonal",
"rpred_sum",
"scale_zchar",
"seq",
"split",
"to",
"zchar_nth_expansion",
"zchar_span"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zisometry_of_iso f :
free S -> {in S, isometry f, to 'Z[irr G]} ->
{tau : {linear 'CF(L) -> 'CF(G)} | {in S, tau =1 f}
& {in 'Z[S], isometry tau, to 'Z[irr G]}}. | Proof.
move=> freeS [If Zf]; have [tau Dtau Itau] := isometry_of_free freeS If.
exists tau => //; split; first by apply: sub_in2 Itau; apply: zchar_span.
move=> _ /zchar_nth_expansion[a Za ->]; rewrite linear_sum rpred_sum // => i _.
by rewrite linearZ rpredZ_int ?Dtau ?Zf ?mem_nth.
Qed. | Lemma | Zisometry_of_iso | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"If",
"apply",
"free",
"freeS",
"irr",
"isometry",
"isometry_of_free",
"linear",
"linearZ",
"linear_sum",
"mem_nth",
"rpredZ_int",
"rpred_sum",
"split",
"to",
"zchar_nth_expansion",
"zchar_span"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zisometry_inj A nu :
{in 'Z[S, A] &, isometry nu} -> {in 'Z[S, A] &, injective nu}. | Proof. by move/isometry_raddf_inj; apply; apply: rpredB. Qed. | Lemma | Zisometry_inj | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"isometry",
"isometry_raddf_inj",
"rpredB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isometry_in_zchar nu : {in S &, isometry nu} -> {in 'Z[S] &, isometry nu}. | Proof.
move=> Inu _ _ /zchar_nth_expansion[u Zu ->] /zchar_nth_expansion[v Zv ->].
rewrite !raddf_sum; apply: eq_bigr => j _ /=.
rewrite !cfdot_suml; apply: eq_bigr => i _.
by rewrite !raddfZ_int //= !cfdotZl !cfdotZr Inu ?mem_nth.
Qed. | Lemma | isometry_in_zchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Inu",
"apply",
"cfdotZl",
"cfdotZr",
"cfdot_suml",
"eq_bigr",
"isometry",
"mem_nth",
"raddfZ_int",
"raddf_sum",
"zchar_nth_expansion"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"alpha ^u" | := (cfAut u alpha). | Notation | alpha ^u | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"alpha",
"cfAut"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_zchar S A psi :
cfAut_closed u S -> psi \in 'Z[S, A] -> psi^u \in 'Z[S, A]. | Proof.
rewrite zchar_split => SuS /andP[/zchar_nth_expansion[z Zz Dpsi] Apsi].
rewrite zchar_split cfAut_on {}Apsi {psi}Dpsi rmorph_sum rpred_sum //= => i _.
by rewrite cfAutZ_Cint // scale_zchar // mem_zchar ?SuS ?mem_nth.
Qed. | Lemma | cfAut_zchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfAutZ_Cint",
"cfAut_closed",
"cfAut_on",
"mem_nth",
"mem_zchar",
"rmorph_sum",
"rpred_sum",
"scale_zchar",
"zchar_nth_expansion",
"zchar_split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfAut_vchar A psi : psi \in 'Z[irr G, A] -> psi^u \in 'Z[irr G, A]. | Proof. by apply: cfAut_zchar; apply: irr_aut_closed. Qed. | Lemma | cfAut_vchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfAut_zchar",
"irr",
"irr_aut_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_aut_zchar S A psi :
{subset S <= 'Z[irr G]} -> psi \in 'Z[S, A] -> psi^u \in 'Z[S, A] ->
psi - psi^u \in 'Z[S, A^#]. | Proof.
move=> Z_S Spsi Spsi_u; rewrite zcharD1 !cfunE subr_eq0 rpredB //=.
by rewrite aut_intr // Cint_vchar1 // (zchar_trans Z_S) ?(zcharW Spsi).
Qed. | Lemma | sub_aut_zchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cint_vchar1",
"Z_S",
"aut_intr",
"cfunE",
"irr",
"rpredB",
"subr_eq0",
"zcharD1",
"zcharW",
"zchar_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC_vcharAut chi x : chi \in 'Z[irr G] -> (u (chi x))^* = u (chi x)^*. | Proof.
case/vcharP=> chi1 Nchi1 [chi2 Nchi2 ->].
by rewrite !cfunE !rmorphB /= !conjC_charAut.
Qed. | Lemma | conjC_vcharAut | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfunE",
"chi",
"conjC_charAut",
"irr",
"rmorphB",
"vcharP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_aut_vchar phi chi :
chi \in 'Z[irr G] -> '[phi^u , chi^u] = u '[phi, chi]. | Proof.
by case/vcharP=> chi1 Nchi1 [chi2 Nchi2 ->]; rewrite !raddfB /= !cfdot_aut_char.
Qed. | Lemma | cfdot_aut_vchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdot_aut_char",
"chi",
"irr",
"raddfB",
"vcharP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vchar_aut A chi : (chi^u \in 'Z[irr G, A]) = (chi \in 'Z[irr G, A]). | Proof.
rewrite !(zchar_split _ A) cfAut_on; congr (_ && _).
apply/idP/idP=> [Zuchi|]; last exact: cfAut_vchar.
rewrite [chi]cfun_sum_cfdot rpred_sum // => i _.
rewrite scale_zchar ?irr_vchar //.
by rewrite -(intr_aut u) -cfdot_aut_irr -aut_IirrE Cint_cfdot_vchar_irr.
Qed. | Lemma | vchar_aut | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cint_cfdot_vchar_irr",
"apply",
"aut_IirrE",
"cfAut_on",
"cfAut_vchar",
"cfdot_aut_irr",
"cfun_sum_cfdot",
"chi",
"intr_aut",
"irr",
"irr_vchar",
"last",
"rpred_sum",
"scale_zchar",
"zchar_split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfConjC_vchar | := cfAut_vchar Num.conj. | Definition | cfConjC_vchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfAut_vchar",
"conj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_vchar phi : phi \in 'Z[irr G] -> 'Res[H] phi \in 'Z[irr H]. | Proof.
case/vcharP=> xi1 Nx1 [xi2 Nxi2 ->].
by rewrite raddfB rpredB ?char_vchar ?cfRes_char.
Qed. | Lemma | cfRes_vchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfRes_char",
"char_vchar",
"irr",
"raddfB",
"rpredB",
"vcharP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRes_vchar_on A phi :
H \subset G -> phi \in 'Z[irr G, A] -> 'Res[H] phi \in 'Z[irr H, A]. | Proof.
rewrite zchar_split => sHG /andP[Zphi Aphi]; rewrite zchar_split cfRes_vchar //.
apply/cfun_onP=> x /(cfun_onP Aphi); rewrite !cfunElock !genGid sHG => ->.
exact: mul0rn.
Qed. | Lemma | cfRes_vchar_on | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfRes_vchar",
"cfunElock",
"cfun_onP",
"genGid",
"irr",
"mul0rn",
"sHG",
"zchar_split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfInd_vchar phi : phi \in 'Z[irr H] -> 'Ind[G] phi \in 'Z[irr G]. | Proof.
move=> /vcharP[xi1 Nx1 [xi2 Nxi2 ->]].
by rewrite raddfB rpredB ?char_vchar ?cfInd_char.
Qed. | Lemma | cfInd_vchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfInd_char",
"char_vchar",
"irr",
"raddfB",
"rpredB",
"vcharP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_conjC_vchar A phi :
phi \in 'Z[irr G, A] -> phi - (phi^*)%CF \in 'Z[irr G, A^#]. | Proof.
move=> Zphi; rewrite sub_aut_zchar ?cfAut_zchar // => _ /irrP[i ->].
exact: irr_vchar.
exact: cfConjC_irr.
Qed. | Lemma | sub_conjC_vchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfAut_zchar",
"cfConjC_irr",
"irr",
"irrP",
"irr_vchar",
"sub_aut_zchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Frobenius_kernel_exists :
[Frobenius G with complement H] -> {K : {group gT} | [Frobenius G = K ><| H]}. | Proof.
move=> frobG; have [_ ntiHG] := andP frobG.
have [[_ sHG regGH][_ tiHG /eqP defNH]] := (normedTI_memJ_P ntiHG, and3P ntiHG).
suffices /sigW[K defG]: exists K, gval K ><| H == G by exists K; apply/andP.
pose K1 := G :\: cover (H^# :^: G).
have oK1: #|K1| = #|G : H|.
rewrite cardsD (setIidPr _).
rewrite cove... | Lemma | Frobenius_kernel_exists | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cnat_irr1",
"Iirr",
"Lagrange",
"TI_cardMg",
"TI_cfker_irr",
"add0r",
"addIr",
"addNr",
"addnK",
"addrC",
"addrK",
"addr_eq0",
"apply",
"big1",
"big_setID",
"bigcapP",
"bigcap_min",
"bigcapsP",
"bigcupsP",
"card_support_normedTI",
"cardsD",
"cardsD1",
"cfIndE",
"cfInd_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr (gT : finGroupType) (B : {set gT}) : {pred 'CF(B)} | :=
[pred f | (f \in irr B) || (- f \in irr B)]. | Definition | dirr | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"gT",
"irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_oppr_closed : oppr_closed (dirr G). | Proof. by move=> xi; rewrite !inE opprK orbC. Qed. | Fact | dirr_oppr_closed | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dirr",
"inE",
"opprK",
"oppr_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_opp v : (- v \in dirr G) = (v \in dirr G). | Proof. exact: rpredN. Qed. | Lemma | dirr_opp | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dirr",
"rpredN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_sign n v : ((-1)^+ n *: v \in dirr G) = (v \in dirr G). | Proof. exact: rpredZsign. Qed. | Lemma | dirr_sign | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dirr",
"rpredZsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_dirr i : 'chi_i \in dirr G. | Proof. by rewrite !inE mem_irr. Qed. | Lemma | irr_dirr | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dirr",
"inE",
"mem_irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirrP f :
reflect (exists b : bool, exists i, f = (-1) ^+ b *: 'chi_i) (f \in dirr G). | Proof.
apply: (iffP idP) => [| [b [i ->]]]; last by rewrite dirr_sign irr_dirr.
case/orP=> /irrP[i Hf]; first by exists false, i; rewrite scale1r.
by exists true, i; rewrite scaleN1r -Hf opprK.
Qed. | Lemma | dirrP | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Hf",
"apply",
"dirr",
"dirr_sign",
"irrP",
"irr_dirr",
"last",
"opprK",
"scale1r",
"scaleN1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirrE phi : (phi \in dirr G) = (phi \in 'Z[irr G]) && ('[phi] == 1). | Proof.
apply/dirrP/andP=> [[b [i ->]] | [Zphi /eqP/vchar_norm1P]]; last exact.
by rewrite rpredZsign irr_vchar cfnorm_sign cfnorm_irr.
Qed. | Lemma | dirrE | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfnorm_irr",
"cfnorm_sign",
"dirr",
"dirrP",
"irr",
"irr_vchar",
"last",
"rpredZsign",
"vchar_norm1P"
] | This should perhaps be the definition of dirr. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cfdot_dirr f g : f \in dirr G -> g \in dirr G ->
'[f, g] = (if f == - g then -1 else (f == g)%:R). | Proof.
case/dirrP=> [b1 [i1 ->]] /dirrP[b2 [i2 ->]].
rewrite cfdotZl cfdotZr rmorph_sign mulrA -signr_addb cfdot_irr.
rewrite -scaleNr -signrN !eq_scaled_irr signr_eq0 !(inj_eq signr_inj) /=.
by rewrite -!negb_add addbN mulr_sign -mulNrn mulrb; case: ifP.
Qed. | Lemma | cfdot_dirr | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotZl",
"cfdotZr",
"cfdot_irr",
"dirr",
"dirrP",
"eq_scaled_irr",
"inj_eq",
"mulNrn",
"mulrA",
"mulr_sign",
"mulrb",
"negb_add",
"rmorph_sign",
"scaleNr",
"signrN",
"signr_addb",
"signr_eq0",
"signr_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_norm1 phi : phi \in 'Z[irr G] -> '[phi] = 1 -> phi \in dirr G. | Proof. by rewrite dirrE => -> -> /=. Qed. | Lemma | dirr_norm1 | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dirr",
"dirrE",
"irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_aut u phi : (cfAut u phi \in dirr G) = (phi \in dirr G). | Proof.
rewrite !dirrE vchar_aut; apply: andb_id2l => /cfdot_aut_vchar->.
exact: fmorph_eq1.
Qed. | Lemma | dirr_aut | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfAut",
"cfdot_aut_vchar",
"dirr",
"dirrE",
"fmorph_eq1",
"vchar_aut"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dIirr (B : {set gT}) | := (bool * (Iirr B))%type. | Definition | dIirr | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Iirr",
"gT",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr1 (B : {set gT}) : dIirr B | := (false, 0). | Definition | dirr1 | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dIirr",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ndirr (B : {set gT}) (i : dIirr B) : dIirr B | :=
(~~ i.1, i.2). | Definition | ndirr | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dIirr",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ndirr_diff (i : dIirr G) : ndirr i != i. | Proof. by case: i => [] [|] i. Qed. | Lemma | ndirr_diff | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dIirr",
"ndirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ndirrK : involutive (@ndirr G). | Proof. by move=> [b i]; rewrite /ndirr /= negbK. Qed. | Lemma | ndirrK | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"ndirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ndirr_inj : injective (@ndirr G). | Proof. exact: (inv_inj ndirrK). Qed. | Lemma | ndirr_inj | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"ndirr",
"ndirrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dchi (B : {set gT}) (i : dIirr B) : 'CF(B) | := (-1)^+ i.1 *: 'chi_i.2. | Definition | dchi | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dIirr",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dchi1 : dchi (dirr1 G) = 1. | Proof. by rewrite /dchi scale1r irr0. Qed. | Lemma | dchi1 | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dchi",
"dirr1",
"irr0",
"scale1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_dchi i : dchi i \in dirr G. | Proof. by apply/dirrP; exists i.1; exists i.2. Qed. | Lemma | dirr_dchi | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"dchi",
"dirr",
"dirrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dIrrP phi : reflect (exists i, phi = dchi i) (phi \in dirr G). | Proof.
by apply: (iffP idP)=> [/dirrP[b]|] [i ->]; [exists (b, i) | apply: dirr_dchi].
Qed. | Lemma | dIrrP | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"dchi",
"dirr",
"dirrP",
"dirr_dchi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dchi_ndirrE (i : dIirr G) : dchi (ndirr i) = - dchi i. | Proof. by case: i => [b i]; rewrite /ndirr /dchi signrN scaleNr. Qed. | Lemma | dchi_ndirrE | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dIirr",
"dchi",
"ndirr",
"scaleNr",
"signrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfdot_dchi (i j : dIirr G) :
'[dchi i, dchi j] = (i == j)%:R - (i == ndirr j)%:R. | Proof.
case: i => bi i; case: j => bj j; rewrite cfdot_dirr ?dirr_dchi // !xpair_eqE.
rewrite -dchi_ndirrE !eq_scaled_irr signr_eq0 !(inj_eq signr_inj) /=.
by rewrite -!negb_add addbN negbK; case: andP => [[->]|]; rewrite ?subr0 ?add0r.
Qed. | Lemma | cfdot_dchi | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"add0r",
"cfdot_dirr",
"dIirr",
"dchi",
"dchi_ndirrE",
"dirr_dchi",
"eq_scaled_irr",
"inj_eq",
"ndirr",
"negb_add",
"signr_eq0",
"signr_inj",
"subr0",
"xpair_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dchi_vchar i : dchi i \in 'Z[irr G]. | Proof. by case: i => b i; rewrite rpredZsign irr_vchar. Qed. | Lemma | dchi_vchar | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dchi",
"irr",
"irr_vchar",
"rpredZsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfnorm_dchi (i : dIirr G) : '[dchi i] = 1. | Proof. by case: i => b i; rewrite cfnorm_sign cfnorm_irr. Qed. | Lemma | cfnorm_dchi | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfnorm_irr",
"cfnorm_sign",
"dIirr",
"dchi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_inj : injective (@dchi G). | Proof.
case=> b1 i1 [b2 i2] /eqP; rewrite eq_scaled_irr (inj_eq signr_inj) /=.
by rewrite signr_eq0 -xpair_eqE => /eqP.
Qed. | Lemma | dirr_inj | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dchi",
"eq_scaled_irr",
"inj_eq",
"signr_eq0",
"signr_inj",
"xpair_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_dIirr (B : {set gT}) J (f : J -> 'CF(B)) j : dIirr B | :=
odflt (dirr1 B) [pick i | dchi i == f j]. | Definition | dirr_dIirr | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dIirr",
"dchi",
"dirr1",
"gT",
"pick"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_dIirrPE J (f : J -> 'CF(G)) (P : pred J) :
(forall j, P j -> f j \in dirr G) ->
forall j, P j -> dchi (dirr_dIirr f j) = f j. | Proof.
rewrite /dirr_dIirr => dirrGf j Pj; case: pickP => [i /eqP //|].
by have /dIrrP[i-> /(_ i)/eqP] := dirrGf j Pj.
Qed. | Lemma | dirr_dIirrPE | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dIrrP",
"dchi",
"dirr",
"dirr_dIirr",
"pickP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_dIirrE J (f : J -> 'CF(G)) :
(forall j, f j \in dirr G) -> forall j, dchi (dirr_dIirr f j) = f j. | Proof. by move=> dirrGf j; apply: (@dirr_dIirrPE _ _ xpredT). Qed. | Lemma | dirr_dIirrE | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"dchi",
"dirr",
"dirr_dIirr",
"dirr_dIirrPE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_constt (B : {set gT}) (phi: 'CF(B)) : {set (dIirr B)} | :=
[set i | 0 < '[phi, dchi i]]. | Definition | dirr_constt | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dIirr",
"dchi",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_consttE (phi : 'CF(G)) (i : dIirr G) :
(i \in dirr_constt phi) = (0 < '[phi, dchi i]). | Proof. by rewrite inE. Qed. | Lemma | dirr_consttE | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"dIirr",
"dchi",
"dirr_constt",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Cnat_dirr (phi : 'CF(G)) i :
phi \in 'Z[irr G] -> i \in dirr_constt phi -> '[phi, dchi i] \in Num.nat. | Proof.
move=> PiZ; rewrite natrEint dirr_consttE andbC => /ltW -> /=.
by case: i => b i; rewrite cfdotZr rmorph_sign rpredMsign Cint_cfdot_vchar_irr.
Qed. | Lemma | Cnat_dirr | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"Cint_cfdot_vchar_irr",
"cfdotZr",
"dchi",
"dirr_constt",
"dirr_consttE",
"irr",
"ltW",
"nat",
"natrEint",
"rmorph_sign",
"rpredMsign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_constt_oppr (i : dIirr G) (phi : 'CF(G)) :
(i \in dirr_constt (-phi)) = (ndirr i \in dirr_constt phi). | Proof. by rewrite !dirr_consttE dchi_ndirrE cfdotNl cfdotNr. Qed. | Lemma | dirr_constt_oppr | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotNl",
"cfdotNr",
"dIirr",
"dchi_ndirrE",
"dirr_constt",
"dirr_consttE",
"ndirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_constt_oppI (phi: 'CF(G)) :
dirr_constt phi :&: dirr_constt (-phi) = set0. | Proof.
apply/setP=> i; rewrite inE !dirr_consttE cfdotNl inE.
apply/idP=> /andP [L1 L2]; have := ltr_pDl L1 L2.
by rewrite subrr lt_def eqxx.
Qed. | Lemma | dirr_constt_oppI | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"apply",
"cfdotNl",
"dirr_constt",
"dirr_consttE",
"eqxx",
"inE",
"lt_def",
"ltr_pDl",
"set0",
"setP",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dirr_constt_oppl (phi: 'CF(G)) i :
i \in dirr_constt phi -> (ndirr i) \notin dirr_constt phi. | Proof.
by rewrite !dirr_consttE dchi_ndirrE cfdotNr oppr_gt0 => /ltW /le_gtF ->.
Qed. | Lemma | dirr_constt_oppl | group_representation | group_representation/vcharacter.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [
"cfdotNr",
"dchi_ndirrE",
"dirr_constt",
"dirr_consttE",
"le_gtF",
"ltW",
"ndirr",
"oppr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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