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_induced_Frobenius_ker i : i != 0 -> 'Ind[G, K] 'chi_i \in irr G. | Proof.
move/inertia_Frobenius_ker/group_inj=> defK.
have [_ _ nsKG _] := Frobenius_kerP frobGK.
have [] := constt_Inertia_bijection i nsKG; rewrite defK cfInd_id => -> //.
by rewrite constt_irr !inE.
Qed. | Theorem | irr_induced_Frobenius_ker | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Frobenius_kerP",
"cfInd_id",
"constt_Inertia_bijection",
"constt_irr",
"frobGK",
"group_inj",
"inE",
"inertia_Frobenius_ker",
"irr",
"nsKG"
] | This is Isaacs, Theorem 6.34(a2) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Frobenius_Ind_irrP j :
reflect (exists2 i, i != 0 & 'chi_j = 'Ind[G, K] 'chi_i)
(~~ (K \subset cfker 'chi_j)). | Proof.
have [_ _ nsKG _] := Frobenius_kerP frobGK; have [sKG nKG] := andP nsKG.
apply: (iffP idP) => [not_chijK1 | [i nzi ->]]; last first.
by rewrite cfker_Ind_irr ?sub_gcore // subGcfker.
have /neq0_has_constt[i chijKi]: 'Res[K] 'chi_j != 0 by apply: Res_irr_neq0.
have nz_i: i != 0.
by apply: contraNneq not_chijK... | Theorem | Frobenius_Ind_irrP | group_representation | group_representation/inertia.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"ssrnum",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Frobenius_kerP",
"Res_irr_neq0",
"apply",
"cfdot_Res_l",
"cfdot_irr",
"cfker",
"cfker_Ind_irr",
"constt0_Res_cfker",
"contraNneq",
"frobGK",
"i0",
"irrP",
"irr_induced_Frobenius_ker",
"last",
"nKG",
"neq0_has_constt",
"nsKG",
"pnatr_eq0",
"sKG",
"subGcfker",
"sub_gcore"
] | This is Isaacs, Theorem 6.34(b) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
group_num_field_exists (gT : finGroupType) (G : {group gT}) :
{Qn : splittingFieldType rat & galois 1 {:Qn} &
{QnC : {rmorphism Qn -> algC}
& forall nuQn : argumentType [in 'Gal({:Qn} / 1)],
{nu : {rmorphism algC -> algC} |
{morph QnC: a / nuQn a >-> nu a}}
& {w : ... | Proof.
have [z prim_z] := C_prim_root_exists (cardG_gt0 G); set n := #|G| in prim_z *.
have [Qn [QnC [[|w []] // [Dz] genQn]]] := num_field_exists [:: z].
have prim_w: n.-primitive_root w by rewrite -Dz fmorph_primitive_root in prim_z.
have Q_Xn1: ('X^n - 1 : {poly Qn}) \is a polyOver 1%AS.
by rewrite rpredB ?rpred1 ... | Lemma | group_num_field_exists | group_representation | group_representation/integral_char.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",
"... | [
"C_prim_root_exists",
"adjoin_seqSr",
"algC",
"allP",
"apply",
"big_image",
"big_mkord",
"cardG_gt0",
"cfun0",
"char_reprP",
"character",
"codom",
"dvdnP",
"eqEsubv",
"eq_bigr",
"eqpxx",
"expr1n",
"exprM",
"extend_algC_subfield_aut",
"factor_Xn_sub_1",
"fin_all_exists",
"fm... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_classM_coef_set (Ki Kj : {set gT}) g | :=
[set xy in [predX Ki & Kj] | let: (x, y) := xy in x * y == g]%g. | Definition | gring_classM_coef_set | group_representation | group_representation/integral_char.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",
"predX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_classM_coef (i j k : 'I_#|classes G|) | :=
#|gring_classM_coef_set (enum_val i) (enum_val j) (repr (enum_val k))|. | Definition | gring_classM_coef | group_representation | group_representation/integral_char.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",
"... | [
"classes",
"enum_val",
"gring_classM_coef_set",
"repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_class_sum (i : 'I_#|classes G|) | := gset_mx F G (enum_val i). | Definition | gring_class_sum | group_representation | group_representation/integral_char.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",
"... | [
"classes",
"enum_val",
"gset_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''K_' i" | := (gring_class_sum i)
(at level 8, i at level 2, format "''K_' i") : ring_scope. | Notation | ''K_' i | group_representation | group_representation/integral_char.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",
"... | [
"gring_class_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
a | := gring_classM_coef. | Notation | a | group_representation | group_representation/integral_char.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",
"... | [
"gring_classM_coef"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_class_sum_central i : ('K_i \in 'Z(group_ring F G))%MS. | Proof. by rewrite -classg_base_center (eq_row_sub i) // rowK. Qed. | Lemma | gring_class_sum_central | group_representation | group_representation/integral_char.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",
"... | [
"classg_base_center",
"eq_row_sub",
"group_ring",
"rowK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_gring_classM_coef (i j k : 'I_#|classes G|) g :
g \in enum_val k ->
a i j k = #|gring_classM_coef_set (enum_val i) (enum_val j) g|. | Proof.
rewrite /a; have /repr_classesP[] := enum_valP k; move: (repr _) => g1 Gg1 ->.
have [/imsetP[zi Gzi ->] /imsetP[zj Gzj ->]] := (enum_valP i, enum_valP j).
move=> g1Gg; have Gg := subsetP (class_subG Gg1 (subxx _)) _ g1Gg.
set Aij := gring_classM_coef_set _ _.
without loss suffices IH: g g1 Gg Gg1 g1Gg / (#|Aij g... | Lemma | set_gring_classM_coef | group_representation | group_representation/integral_char.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",
"... | [
"Gg",
"apply",
"card_imset",
"classGidl",
"class_subG",
"class_sym",
"classes",
"conjMg",
"conjgK",
"enum_val",
"enum_valP",
"eqn_leq",
"gT",
"gring_classM_coef_set",
"imsetP",
"inE",
"repr",
"repr_classesP",
"setIdP",
"subsetP",
"subset_leq_card",
"subxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_classM_expansion i j : 'K_i *m 'K_j = \sum_k (a i j k)%:R *: 'K_k. | Proof.
have [/imsetP[zi Gzi dKi] /imsetP[zj Gzj dKj]] := (enum_valP i, enum_valP j).
pose aG := regular_repr F G; have sKG := subsetP (class_subG _ (subxx G)).
transitivity (\sum_(x in zi ^: G) \sum_(y in zj ^: G) aG (x * y)%g).
rewrite mulmx_suml -/aG dKi; apply: eq_bigr => x /sKG Gx.
rewrite mulmx_sumr -/aG dKj; ... | Theorem | gring_classM_expansion | group_representation | group_representation/integral_char.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",
"... | [
"aG",
"apply",
"class_eqP",
"class_refl",
"class_subG",
"classes1",
"enum_rankK_in",
"enum_rank_in",
"enum_val",
"enum_valK_in",
"enum_valP",
"eq_big",
"eq_bigr",
"eqxx",
"gT",
"groupM",
"imsetP",
"inE",
"mem_classes",
"mulmx_suml",
"mulmx_sumr",
"pair_big",
"partition_bi... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_irr_mode_unlockable | := Unlockable gring_irr_mode.unlock. | Canonical | gring_irr_mode_unlockable | group_representation | group_representation/integral_char.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"order",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''K_' i" | := (gring_class_sum _ i)
(at level 8, i at level 2, format "''K_' i") : ring_scope. | Notation | ''K_' i | group_representation | group_representation/integral_char.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",
"... | [
"gring_class_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''omega_' i [ A ]" | := (xcfun (gring_irr_mode i) A)
(i at level 2, format "''omega_' i [ A ]") : ring_scope. | Notation | ''omega_' i [ A ] | group_representation | group_representation/integral_char.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",
"... | [
"xcfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aint_char (chi : 'CF(G)) x : chi \is a character -> chi x \in Aint. | Proof.
have [Gx /char_reprP[rG ->] {chi} | /cfun0->//] := boolP (x \in G).
have [e [_ [unit_e _] [-> _] _]] := repr_rsim_diag rG Gx.
rewrite rpred_sum // => i _; apply: (@Aint_unity_root #[x]) => //.
exact/unity_rootP.
Qed. | Lemma | Aint_char | group_representation | group_representation/integral_char.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_unity_root",
"apply",
"cfun0",
"char_reprP",
"character",
"chi",
"rG",
"repr_rsim_diag",
"rpred_sum",
"unity_rootP"
] | This is Isaacs, Corollary (3.6). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Aint_irr i x : 'chi[G]_i x \in Aint. | Proof. exact/Aint_char/irr_char. Qed. | Lemma | Aint_irr | group_representation | group_representation/integral_char.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",
"chi",
"irr_char"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R_G | := (group_ring algCfield G). | Notation | R_G | group_representation | group_representation/integral_char.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",
"... | [
"algCfield",
"group_ring"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_irr_gring_op_center_scalar n (rG : mx_representation algCfield G n) A :
mx_irreducible rG -> (A \in 'Z(R_G))%MS -> is_scalar_mx (gring_op rG A). | Proof.
move/groupC=> irrG /center_mxP[R_A cGA].
apply: mx_abs_irr_cent_scalar irrG _ _; apply/centgmxP => x Gx.
by rewrite -(gring_opG rG Gx) -!gring_opM ?cGA // envelop_mx_id.
Qed. | Lemma | mx_irr_gring_op_center_scalar | group_representation | group_representation/integral_char.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",
"... | [
"R_G",
"algCfield",
"apply",
"center_mxP",
"centgmxP",
"envelop_mx_id",
"gring_op",
"gring_opG",
"gring_opM",
"groupC",
"irrG",
"is_scalar_mx",
"mx_abs_irr_cent_scalar",
"mx_irreducible",
"mx_representation",
"rG"
] | This is Isaacs (2.25). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
n | := irr_degree (socle_of_Iirr i). | Let | n | group_representation | group_representation/integral_char.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_degree",
"socle_of_Iirr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxZn_inj: injective (@scalar_mx algCfield n). | Proof. by rewrite -[n]prednK ?irr_degree_gt0 //; apply: fmorph_inj. Qed. | Let | mxZn_inj | group_representation | group_representation/integral_char.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",
"... | [
"algCfield",
"apply",
"fmorph_inj",
"irr_degree_gt0",
"prednK",
"scalar_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cfRepr_gring_center n1 (rG : mx_representation algCfield G n1) A :
cfRepr rG = 'chi_i -> (A \in 'Z(R_G))%MS -> gring_op rG A = 'omega_i[A]%:M. | Proof.
move=> def_rG Z_A; rewrite unlock xcfunZl -{2}def_rG xcfun_repr.
have irr_rG: mx_irreducible rG.
have sim_rG: mx_rsim 'Chi_i rG by apply: cfRepr_inj; rewrite irrRepr.
exact: mx_rsim_irr sim_rG (socle_irr _).
have /is_scalar_mxP[e ->] := mx_irr_gring_op_center_scalar irr_rG Z_A.
congr _%:M; apply: (canRL (mul... | Lemma | cfRepr_gring_center | group_representation | group_representation/integral_char.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",
"... | [
"R_G",
"algCfield",
"apply",
"cfRepr",
"cfRepr_inj",
"cfunE",
"gring_op",
"group1",
"irr1_neq0",
"irrRepr",
"is_scalar_mxP",
"mulKf",
"mulrC",
"mx_irr_gring_op_center_scalar",
"mx_irreducible",
"mx_representation",
"mx_rsim",
"mx_rsim_irr",
"mxtraceZ",
"rG",
"repr_mx1",
"sc... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_gring_center A :
(A \in 'Z(R_G))%MS -> gring_op 'Chi_i A = 'omega_i[A]%:M. | Proof. exact: cfRepr_gring_center (irrRepr i). Qed. | Lemma | irr_gring_center | group_representation | group_representation/integral_char.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",
"... | [
"R_G",
"cfRepr_gring_center",
"gring_op",
"irrRepr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_irr_modeM A B :
(A \in 'Z(R_G))%MS -> (B \in 'Z(R_G))%MS ->
'omega_i[A *m B] = 'omega_i[A] * 'omega_i[B]. | Proof.
move=> Z_A Z_B; have [[R_A cRA] [R_B cRB]] := (center_mxP Z_A, center_mxP Z_B).
apply: mxZn_inj; rewrite scalar_mxM -!irr_gring_center ?gring_opM //.
apply/center_mxP; split=> [|C R_C]; first exact: envelop_mxM.
by rewrite mulmxA cRA // -!mulmxA cRB.
Qed. | Lemma | gring_irr_modeM | group_representation | group_representation/integral_char.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",
"... | [
"R_G",
"apply",
"center_mxP",
"envelop_mxM",
"gring_opM",
"irr_gring_center",
"mulmxA",
"mxZn_inj",
"scalar_mxM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_mode_class_sum_eq (k : 'I_#|classes G|) g :
g \in enum_val k -> 'omega_i['K_k] = #|g ^: G|%:R * 'chi_i g / 'chi_i 1%g. | Proof.
have /imsetP[x Gx DxG] := enum_valP k; rewrite DxG => /imsetP[u Gu ->{g}].
rewrite unlock classGidl ?cfunJ {u Gu}// mulrC mulr_natl.
rewrite xcfunZl raddf_sum DxG -sumr_const /=; congr (_ * _).
by apply: eq_bigr => _ /imsetP[u Gu ->]; rewrite xcfunG ?groupJ ?cfunJ.
Qed. | Lemma | gring_mode_class_sum_eq | group_representation | group_representation/integral_char.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",
"cfunJ",
"classGidl",
"classes",
"enum_val",
"enum_valP",
"eq_bigr",
"groupJ",
"imsetP",
"mulrC",
"mulr_natl",
"raddf_sum",
"sumr_const",
"xcfunG",
"xcfunZl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aint_gring_mode_class_sum k : 'omega_i['K_k] \in Aint. | Proof.
move: k; pose X := [tuple 'omega_i['K_k] | k < #|classes G| ].
have memX k: 'omega_i['K_k] \in X by apply: image_f.
have S_P := Cint_spanP X; set S := Cint_span X in S_P.
have S_X: {subset X <= S} by apply: mem_Cint_span.
have S_1: 1 \in S.
apply: S_X; apply/codomP; exists (enum_rank_in (classes1 G) 1%g).
re... | Lemma | Aint_gring_mode_class_sum | group_representation | group_representation/integral_char.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",
"Cint_span",
"Cint_spanP",
"apply",
"cards1",
"class1G",
"classes",
"classes1",
"codomP",
"enum_rankK_in",
"enum_rank_in",
"fin_Csubring_Aint",
"gring_classM_expansion",
"gring_class_sum_central",
"gring_irr_modeM",
"gring_mode_class_sum_eq",
"image_f",
"irr1_neq0",
"mem_... | This is Isaacs, Theorem (3.7). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Aint_class_div_irr1 x :
x \in G -> #|x ^: G|%:R * 'chi_i x / 'chi_i 1%g \in Aint. | Proof.
move=> Gx; have clGxG := mem_classes Gx; pose k := enum_rank_in clGxG (x ^: G).
have k_x: x \in enum_val k by rewrite enum_rankK_in // class_refl.
by rewrite -(gring_mode_class_sum_eq k_x) Aint_gring_mode_class_sum.
Qed. | Corollary | Aint_class_div_irr1 | group_representation | group_representation/integral_char.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_gring_mode_class_sum",
"class_refl",
"enum_rankK_in",
"enum_rank_in",
"enum_val",
"gring_mode_class_sum_eq",
"mem_classes"
] | A more usable reformulation that does not involve the class sums. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
coprime_degree_support_cfcenter g :
coprime (Num.truncn ('chi_i 1%g)) #|g ^: G| -> g \notin ('Z('chi_i))%CF ->
'chi_i g = 0. | Proof.
set m := Num.truncn _ => co_m_gG notZg.
have [Gg | /cfun0-> //] := boolP (g \in G).
have Dm: 'chi_i 1%g = m%:R by rewrite truncnK ?Cnat_irr1.
have m_gt0: (0 < m)%N by rewrite -ltC_nat -Dm irr1_gt0.
have nz_m: m%:R != 0 :> algC by rewrite pnatr_eq0 -lt0n.
pose alpha := 'chi_i g / m%:R.
have a_lt1: `|alpha| < 1.
... | Theorem | coprime_degree_support_cfcenter | group_representation | group_representation/integral_char.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_aut",
"Aint_class_div_irr1",
"Aint_irr",
"Bezoutl",
"Cint_rat_Aint",
"Cnat_irr1",
"Da",
"Gg",
"addrK",
"algC",
"alg_num_field",
"alpha",
"apply",
"aut_Iirr",
"aut_IirrE",
"bigD1",
"big_rec",
"can2_eq",
"can_eq",
"cfun0",
"cfunE",
"char1_ge_norm",
"coprime"... | This is Isaacs, Theorem (3.8). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
primes_class_simple_gt1 C :
simple G -> ~~ abelian G -> C \in (classes G)^# -> (size (primes #|C|) > 1)%N. | Proof.
move=> simpleG not_cGG /setD1P[ntC /imsetP[g Gg defC]].
have{ntC} nt_g: g != 1%g by rewrite defC classG_eq1 in ntC.
rewrite ltnNge {C}defC; set m := #|_|; apply/negP=> p_natC.
have{p_natC} [p p_pr [a Dm]]: {p : nat & prime p & {a | m = p ^ a}%N}.
have /prod_prime_decomp->: (0 < m)%N by rewrite /m -index_cent1.... | Theorem | primes_class_simple_gt1 | group_representation | group_representation/integral_char.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_Cint",
"Aint_irr",
"Cint_rat_Aint",
"Gg",
"abelian",
"add0r",
"addrC",
"addr_eq0",
"algC",
"alpha",
"apply",
"big1",
"bigD1",
"bigID",
"big_andbC",
"big_nil",
"big_seq1",
"center_idP",
"center_normal",
"cfReg",
"cfRegE",
"cfReg_sum",
"cfcenter_eq_center",
"cfker",... | This is Isaacs, Theorem (3.9). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Burnside_p_a_q_b gT (G : {group gT}) :
(size (primes #|G|) <= 2)%N -> solvable G. | Proof.
move: {2}_.+1 (ltnSn #|G|) => n; elim: n => // n IHn in gT G *.
rewrite ltnS => leGn piGle2; have [simpleG | ] := boolP (simple G); last first.
rewrite negb_forall_in => /exists_inP[N sNG]; rewrite eq_sym.
have [->|] := eqVneq N G.
rewrite groupP /= genGid normG andbT eqb_id negbK => /eqP->.
exact: s... | Theorem | Burnside_p_a_q_b | group_representation | group_representation/integral_char.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",
"... | [
"Sylow_exists",
"abelian_sol",
"apply",
"cardG_gt0",
"cardSg",
"center_nil_eq1",
"classG_eq1",
"dvdn_indexg",
"dvdn_quotient",
"eqEproper",
"eqVneq",
"eq_sym",
"eqbF_neg",
"eqb_id",
"exists_inP",
"gT",
"genGid",
"group",
"groupP",
"inE",
"index_cent1",
"indexgS",
"isgroup... | This is Burnside's famous p^a.q^b theorem (Isaacs, Theorem (3.10)). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
dvd_irr1_cardG gT (G : {group gT}) i : ('chi[G]_i 1%g %| #|G|)%C. | Proof.
rewrite unfold_in -if_neg irr1_neq0 Cint_rat_Aint //=.
by rewrite rpred_div ?rpred_nat // rpred_nat_num ?Cnat_irr1.
rewrite -[n in n / _]/(_ *+ true) -(eqxx i) -mulr_natr.
rewrite -first_orthogonality_relation mulVKf ?neq0CG //.
rewrite sum_by_classes => [x y Gx Gy|]; rewrite -?conjVg ?cfunJ //.
rewrite mulr_s... | Theorem | dvd_irr1_cardG | group_representation | group_representation/integral_char.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_class_div_irr1",
"Aint_irr",
"Cint_rat_Aint",
"Cnat_irr1",
"cfunJ",
"chi",
"conjVg",
"eqxx",
"first_orthogonality_relation",
"gT",
"group",
"irr1_neq0",
"mulVKf",
"mulrA",
"mulrAC",
"mulr_natr",
"mulr_suml",
"neq0CG",
"repr_classesP",
"rpredM",
"rpred_div",
"rpred_nat... | This is Isaacs, Theorem (3.11). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
dvd_irr1_index_center gT (G : {group gT}) i :
('chi[G]_i 1%g %| #|G : 'Z('chi_i)%CF|)%C. | Proof.
without loss fful: gT G i / cfaithful 'chi_i.
rewrite -{2}[i](quo_IirrK _ (subxx _)) 1?mod_IirrE ?cfModE ?cfker_normal //.
rewrite morph1; set i1 := quo_Iirr _ i => /(_ _ _ i1) IH.
have fful_i1: cfaithful 'chi_i1.
by rewrite quo_IirrE ?cfker_normal ?cfaithful_quo.
have:= IH fful_i1; rewrite cfcenter_... | Theorem | dvd_irr1_index_center | group_representation | group_representation/integral_char.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_class_div_irr1",
"Aint_irr",
"Cint_rat_Aint",
"Cnat_irr1",
"add0r",
"apply",
"big1",
"big_setID",
"can_eq",
"cardX",
"card_in_image",
"cardsE",
"centerP",
"center_sub",
"cfModE",
"cfResE",
"cfaithful",
"cfaithful_quo",
"cfcenter_Res",
"cfcenter_eq_center",
"cf... | This is Isaacs, Theorem (3.12). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
gring_classM_coef_sum_eq gT (G : {group gT}) j1 j2 k g1 g2 g :
let a := @gring_classM_coef gT G j1 j2 in let a_k := a k in
g1 \in enum_val j1 -> g2 \in enum_val j2 -> g \in enum_val k ->
let sum12g := \sum_i 'chi[G]_i g1 * 'chi_i g2 * ('chi_i g)^* / 'chi_i 1%g in
a_k%:R = (#|enum_val j1| * #|enum_val j2|)%:R... | Proof.
move=> a /= Kg1 Kg2 Kg; rewrite mulrAC; apply: canRL (mulfK (neq0CG G)) _.
transitivity (\sum_j (#|G| * a j)%:R *+ (j == k) : algC).
by rewrite (bigD1 k) //= eqxx -natrM mulnC big1 ?addr0 // => j /negPf->.
have defK (j : 'I_#|classes G|) x: x \in enum_val j -> enum_val j = x ^: G.
by have /imsetP[y Gy ->] :=... | Lemma | gring_classM_coef_sum_eq | group_representation | group_representation/integral_char.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",
"... | [
"Gg",
"Lagrange",
"addr0",
"algC",
"apply",
"big1",
"bigD1",
"chi",
"class_eqP",
"class_refl",
"classes",
"divfK",
"enum_val",
"enum_valP",
"enum_val_inj",
"eq_bigr",
"eqxx",
"exchange_big",
"gT",
"gring_classM_coef",
"gring_classM_expansion",
"gring_class_sum_central",
"... | This is Isaacs, Problem (3.7). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
index_support_dvd_degree gT (G H : {group gT}) chi :
H \subset G -> chi \is a character -> chi \in 'CF(G, H) ->
(H :==: 1%g) || abelian G ->
(#|G : H| %| chi 1%g)%C. | Proof.
move=> sHG Nchi Hchi ZHG.
suffices: (#|G : H| %| 'Res[H] chi 1%g)%C by rewrite cfResE ?group1.
rewrite ['Res _]cfun_sum_cfdot sum_cfunE rpred_sum // => i _.
rewrite cfunE dvdC_mulr ?intr_nat ?Cnat_irr1 //.
have [j ->]: exists j, 'chi_i = 'Res 'chi[G]_j.
case/predU1P: ZHG => [-> | cGG] in i *.
suffices ->: ... | Lemma | index_support_dvd_degree | group_representation | group_representation/integral_char.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",
"Cnat_irr1",
"Iirr",
"Lagrange",
"NirrE",
"abelian",
"abelianS",
"apply",
"cGG",
"card_Iirr_abelian",
"card_imset",
"card_quotient",
"cfModK",
"cfRes1",
"cfResE",
"cfRes_cfun1",
"cfRes_char",
"cfdotEl",
"cfker1",
"cfkerEirr",
"cfker_mod",
"cfun1E",
... | This is Isaacs, Problem (2.16). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
faithful_degree_p_part gT (p : nat) (G P : {group gT}) i :
cfaithful 'chi[G]_i -> p.-nat (Num.truncn ('chi_i 1%g)) ->
p.-Sylow(G) P -> abelian P ->
'chi_i 1%g = (#|G : 'Z(G)|`_p)%:R. | Proof.
have [p_pr | pr'p] := boolP (prime p); last first.
have p'n n: (n > 0)%N -> p^'.-nat n.
by move/p'natEpi->; rewrite mem_primes (negPf pr'p).
rewrite irr1_degree natrK => _ /pnat_1-> => [|_ _].
by rewrite p'n ?irr_degree_gt0.
by rewrite part_p'nat ?p'n.
move=> fful_i /p_natP[a Dchi1] sylP cPP.
have ... | Theorem | faithful_degree_p_part | group_representation | group_representation/integral_char.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",
"... | [
"Px",
"Sylow",
"abelian",
"apply",
"card_Hall",
"card_quotient",
"center_normal",
"cfResE",
"cfRes_char",
"cfaithful",
"cfcenter_fful_irr",
"cfun0",
"cfun_onP",
"chi",
"coprimeXl",
"coprime_degree_support_cfcenter",
"dvdC_nat",
"dvd_irr1_index_center",
"eqn_dvd",
"eqr_nat",
"... | This is Isaacs, Theorem (3.13). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
sum_norm2_char_generators gT (G : {group gT}) (chi : 'CF(G)) :
let S := [pred s | generator G s] in
chi \is a character -> {in S, forall s, chi s != 0} ->
\sum_(s in S) `|chi s| ^+ 2 >= #|S|%:R. | Proof.
move=> S Nchi nz_chi_S; pose n := #|G|.
have [g Sg | S_0] := pickP (generator G); last first.
by rewrite eq_card0 // big_pred0 ?lerr.
have defG: <[g]> = G by apply/esym/eqP.
have [cycG Gg]: cyclic G /\ g \in G by rewrite -defG cycle_cyclic cycle_id.
pose I := {k : 'I_n | coprime n k}; pose ItoS (k : I) := (g ^... | Lemma | sum_norm2_char_generators | group_representation | group_representation/integral_char.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_char",
"Cint_rat_Aint",
"Crat",
"Crat_rat",
"Da",
"Gg",
"Qn_aut_exists",
"Sub",
"ahom_inP",
"apply",
"big_image",
"big_imset",
"big_morph",
"big_pred0",
"big_uniq",
"cardD1",
"char_sum_irr",
"character",
"chi",
"codom",
"codomP",
"contraTeq",
"coprime",
"coprime_s... | empty if this is not the case. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
nonlinear_irr_vanish gT (G : {group gT}) i :
'chi[G]_i 1%g > 1 -> exists2 x, x \in G & 'chi_i x = 0. | Proof.
move=> chi1gt1; apply/exists_eq_inP; apply: contraFT (lt_geF chi1gt1).
move=> /exists_inPn-nz_chi.
rewrite -(norm_natr (Cnat_irr1 i)) -(@expr_le1 _ 2)//.
rewrite -(lerD2r (#|G|%:R * '['chi_i])) {1}cfnorm_irr mulr1.
rewrite (cfnormE (cfun_onG _)) mulVKf ?neq0CG // (big_setD1 1%g) //=.
rewrite addrCA lerD2l (cards... | Theorem | nonlinear_irr_vanish | group_representation | group_representation/integral_char.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",
"Gg",
"addrCA",
"apply",
"big_setD1",
"cardsD1",
"cfResE",
"cfRes_char",
"cfnormE",
"cfnorm_irr",
"cfun_onG",
"chi",
"cycle_eq1",
"cycle_generator",
"cycle_subG",
"eq_bigl",
"eq_bigr",
"eq_sym",
"exists_eq_inP",
"exists_inPn",
"expr_le1",
"gT",
"generator",
... | This is Burnside's vanishing theorem (Isaacs, Theorem (3.15)). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mx_repr_act (u : 'rV_n) x | := u *m rG (val (subg G x)). | Definition | mx_repr_act | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"rG",
"subg",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_repr_actE u x : x \in G -> mx_repr_act u x = u *m rG x. | Proof. by move=> Gx; rewrite /mx_repr_act /= subgK. Qed. | Lemma | mx_repr_actE | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"mx_repr_act",
"rG",
"subgK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_repr_is_action : is_action G mx_repr_act. | Proof.
split=> [x | u x y Gx Gy]; first exact: can_inj (repr_mxK _ (subgP _)).
by rewrite !mx_repr_actE ?groupM // -mulmxA repr_mxM.
Qed. | Fact | mx_repr_is_action | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"groupM",
"is_action",
"mulmxA",
"mx_repr_act",
"mx_repr_actE",
"repr_mxK",
"repr_mxM",
"split",
"subgP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_repr_action | := Action mx_repr_is_action. | Canonical | mx_repr_action | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"mx_repr_is_action"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_repr_is_groupAction : is_groupAction [set: 'rV[R]_n] mx_repr_action. | Proof.
move=> x Gx /[!inE]; apply/andP; split; first by apply/subsetP=> u /[!inE].
by apply/morphicP=> /= u v _ _; rewrite !actpermE /= /mx_repr_act mulmxDl.
Qed. | Fact | mx_repr_is_groupAction | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"actpermE",
"apply",
"inE",
"is_groupAction",
"morphicP",
"mulmxDl",
"mx_repr_act",
"mx_repr_action",
"split",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_repr_groupAction | := GroupAction mx_repr_is_groupAction. | Canonical | mx_repr_groupAction | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"mx_repr_is_groupAction"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''MR' rG" | := (mx_repr_action rG)
(at level 10, rG at level 8) : action_scope. | Notation | ''MR' rG | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"mx_repr_action",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''MR' rG" | := (mx_repr_groupAction rG) : groupAction_scope. | Notation | ''MR' rG | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"mx_repr_groupAction",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale_act (A : 'M[F]_(m, n)) (a : {unit F}) | := val a *: A. | Definition | scale_act | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"unit",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale_actE A a : scale_act A a = val a *: A. | Proof. by []. Qed. | Lemma | scale_actE | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"scale_act",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale_is_action : is_action setT scale_act. | Proof.
apply: is_total_action=> [A | A a b]; rewrite /scale_act ?scale1r //.
by rewrite ?scalerA mulrC.
Qed. | Fact | scale_is_action | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"is_action",
"is_total_action",
"mulrC",
"scale1r",
"scale_act",
"scalerA",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale_action | := Action scale_is_action. | Canonical | scale_action | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"scale_is_action"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale_is_groupAction : is_groupAction setT scale_action. | Proof.
move=> a _ /[1!inE]; apply/andP; split; first by apply/subsetP=> A /[!inE].
by apply/morphicP=> u A _ _ /=; rewrite !actpermE /= /scale_act scalerDr.
Qed. | Fact | scale_is_groupAction | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"actpermE",
"apply",
"inE",
"is_groupAction",
"morphicP",
"scale_act",
"scale_action",
"scalerDr",
"setT",
"split",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
scale_groupAction | := GroupAction scale_is_groupAction. | Canonical | scale_groupAction | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"scale_is_groupAction"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
astab1_scale_act A : A != 0 -> 'C[A | scale_action] = 1%g. | Proof.
rewrite -mxrank_eq0=> nzA; apply/trivgP/subsetP=> a; apply: contraLR.
rewrite !inE -val_eqE -subr_eq0 sub1set !inE => nz_a1.
by rewrite -subr_eq0 -scaleN1r -scalerDl -mxrank_eq0 eqmx_scale.
Qed. | Lemma | astab1_scale_act | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"eqmx_scale",
"inE",
"mxrank_eq0",
"scaleN1r",
"scale_action",
"scalerDl",
"sub1set",
"subr_eq0",
"subsetP",
"trivgP",
"val_eqE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'Zm" | := (scale_action _ _) : action_scope. | Notation | 'Zm | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"scale_action"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVn | := 'rV[F]_n. | Notation | rVn | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowg m (A : 'M[F]_(m, n)) : {set rVn} | := [set u | u <= A]%MS. | Definition | rowg | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"rVn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_rowg m A v : (v \in @rowg m A) = (v <= A)%MS. | Proof. by rewrite inE. Qed. | Lemma | mem_rowg | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"inE",
"rowg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowg_group_set m A : group_set (@rowg m A). | Proof.
by apply/group_setP; split=> [|u v]; rewrite !inE ?sub0mx //; apply: addmx_sub.
Qed. | Fact | rowg_group_set | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"addmx_sub",
"apply",
"group_set",
"group_setP",
"inE",
"rowg",
"split",
"sub0mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowg_group m A | := Group (@rowg_group_set m A). | Canonical | rowg_group | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"rowg_group_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowg_stable m (A : 'M_(m, n)) : [acts setT, on rowg A | 'Zm]. | Proof. by apply/actsP=> a _ v; rewrite !inE eqmx_scale // -unitfE (valP a). Qed. | Lemma | rowg_stable | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"actsP",
"apply",
"eqmx_scale",
"inE",
"on",
"rowg",
"setT",
"unitfE",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowgS m1 m2 (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(rowg A \subset rowg B) = (A <= B)%MS. | Proof.
apply/subsetP/idP=> sAB => [|u /[!inE] suA]; last exact: submx_trans sAB.
by apply/row_subP=> i; have /[!(inE, row_sub)]-> := sAB (row i A).
Qed. | Lemma | rowgS | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"inE",
"last",
"row",
"row_sub",
"row_subP",
"rowg",
"submx_trans",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_rowg m1 m2 (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A :=: B)%MS -> rowg A = rowg B. | Proof. by move=> eqAB; apply/eqP; rewrite eqEsubset !rowgS !eqAB andbb. Qed. | Lemma | eq_rowg | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"eqEsubset",
"rowg",
"rowgS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowg0 m : rowg (0 : 'M_(m, n)) = 1%g. | Proof. by apply/trivgP/subsetP=> v; rewrite !inE eqmx0 submx0. Qed. | Lemma | rowg0 | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"eqmx0",
"inE",
"rowg",
"submx0",
"subsetP",
"trivgP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowg1 : rowg 1%:M = setT. | Proof. by apply/setP=> x; rewrite !inE submx1. Qed. | Lemma | rowg1 | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"inE",
"rowg",
"setP",
"setT",
"submx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trivg_rowg m (A : 'M_(m, n)) : (rowg A == 1%g) = (A == 0). | Proof. by rewrite -submx0 -rowgS rowg0 (sameP trivgP eqP). Qed. | Lemma | trivg_rowg | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"rowg",
"rowg0",
"rowgS",
"submx0",
"trivgP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowg_mx (L : {set rVn}) | := <<\matrix_(i < #|L|) enum_val i>>%MS. | Definition | rowg_mx | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"enum_val",
"rVn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowgK m (A : 'M_(m, n)) : (rowg_mx (rowg A) :=: A)%MS. | Proof.
apply/eqmxP; rewrite !genmxE; apply/andP; split.
by apply/row_subP=> i; rewrite rowK; have /[!inE] := enum_valP i.
apply/row_subP=> i; set v := row i A.
have Av: v \in rowg A by rewrite inE row_sub.
by rewrite (eq_row_sub (enum_rank_in Av v)) // rowK enum_rankK_in.
Qed. | Lemma | rowgK | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"enum_rankK_in",
"enum_rank_in",
"enum_valP",
"eq_row_sub",
"eqmxP",
"genmxE",
"inE",
"row",
"rowK",
"row_sub",
"row_subP",
"rowg",
"rowg_mx",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowg_mxS (L M : {set 'rV[F]_n}) :
L \subset M -> (rowg_mx L <= rowg_mx M)%MS. | Proof.
move/subsetP=> sLM; rewrite !genmxE; apply/row_subP=> i.
rewrite rowK; move: (enum_val i) (sLM _ (enum_valP i)) => v Mv.
by rewrite (eq_row_sub (enum_rank_in Mv v)) // rowK enum_rankK_in.
Qed. | Lemma | rowg_mxS | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"enum_rankK_in",
"enum_rank_in",
"enum_val",
"enum_valP",
"eq_row_sub",
"genmxE",
"rowK",
"row_subP",
"rowg_mx",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_rowg_mx (L : {set rVn}) : L \subset rowg (rowg_mx L). | Proof.
apply/subsetP=> v Lv; rewrite inE genmxE.
by rewrite (eq_row_sub (enum_rank_in Lv v)) // rowK enum_rankK_in.
Qed. | Lemma | sub_rowg_mx | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"enum_rankK_in",
"enum_rank_in",
"eq_row_sub",
"genmxE",
"inE",
"rVn",
"rowK",
"rowg",
"rowg_mx",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stable_rowg_mxK (L : {group rVn}) :
[acts setT, on L | 'Zm] -> rowg (rowg_mx L) = L. | Proof.
move=> linL; apply/eqP; rewrite eqEsubset sub_rowg_mx andbT.
apply/subsetP=> v; rewrite inE genmxE => /submxP[u ->{v}].
rewrite mulmx_sum_row group_prod // => i _.
rewrite rowK; move: (enum_val i) (enum_valP i) => v Lv.
have [->|] := eqVneq (u 0 i) 0; first by rewrite scale0r group1.
by rewrite -unitfE => aP; re... | Lemma | stable_rowg_mxK | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"aP",
"actsP",
"apply",
"enum_val",
"enum_valP",
"eqEsubset",
"eqVneq",
"genmxE",
"group",
"group1",
"group_prod",
"inE",
"mulmx_sum_row",
"on",
"rVn",
"rowK",
"rowg",
"rowg_mx",
"scale0r",
"setT",
"sub_rowg_mx",
"submxP",
"subsetP",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowg_mx1 : rowg_mx 1%g = 0. | Proof. by apply/eqP; rewrite -submx0 -(rowg0 0) rowgK sub0mx. Qed. | Lemma | rowg_mx1 | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"rowg0",
"rowgK",
"rowg_mx",
"sub0mx",
"submx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowg_mx_eq0 (L : {group rVn}) : (rowg_mx L == 0) = (L :==: 1%g). | Proof.
rewrite -trivg_rowg; apply/idP/eqP=> [|->]; last by rewrite rowg_mx1 rowg0.
exact/contraTeq/subG1_contra/sub_rowg_mx.
Qed. | Lemma | rowg_mx_eq0 | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"contraTeq",
"group",
"last",
"rVn",
"rowg0",
"rowg_mx",
"rowg_mx1",
"subG1_contra",
"sub_rowg_mx",
"trivg_rowg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowgI m1 m2 (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
rowg (A :&: B)%MS = rowg A :&: rowg B. | Proof. by apply/setP=> u; rewrite !inE sub_capmx. Qed. | Lemma | rowgI | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"inE",
"rowg",
"setP",
"sub_capmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_rowg m (A : 'M_(m, n)) : #|rowg A| = (#|F| ^ \rank A)%N. | Proof.
rewrite -[\rank A]mul1n -card_mx.
have injA: injective (mulmxr (row_base A)).
have /row_freeP[A' A'K] := row_base_free A.
by move=> ?; apply: can_inj (mulmxr A') _ => u; rewrite /= -mulmxA A'K mulmx1.
rewrite -(card_image (injA _)); apply: eq_card => v.
by rewrite inE -(eq_row_base A) (sameP submxP codomP).
... | Lemma | card_rowg | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"A'",
"apply",
"card_image",
"card_mx",
"codomP",
"eq_card",
"eq_row_base",
"inE",
"injA",
"mul1n",
"mulmx1",
"mulmxA",
"mulmxr",
"rank",
"row_base",
"row_base_free",
"row_freeP",
"rowg",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowgD m1 m2 (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
rowg (A + B)%MS = (rowg A * rowg B)%g. | Proof.
apply/eqP; rewrite eq_sym eqEcard mulG_subG /= !rowgS.
rewrite addsmxSl addsmxSr -(@leq_pmul2r #|rowg A :&: rowg B|) ?cardG_gt0 //=.
by rewrite -mul_cardG -rowgI !card_rowg -!expnD mxrank_sum_cap.
Qed. | Lemma | rowgD | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"addsmxSl",
"addsmxSr",
"apply",
"cardG_gt0",
"card_rowg",
"eqEcard",
"eq_sym",
"expnD",
"leq_pmul2r",
"mulG_subG",
"mul_cardG",
"mxrank_sum_cap",
"rowg",
"rowgI",
"rowgS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprod_rowg m1 m2 (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(rowg A \* rowg B)%g = rowg (A + B)%MS. | Proof. by rewrite rowgD cprodE // (sub_abelian_cent2 (zmod_abelian setT)). Qed. | Lemma | cprod_rowg | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"cprodE",
"rowg",
"rowgD",
"setT",
"sub_abelian_cent2",
"zmod_abelian"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_rowg m1 m2 (A : 'M[F]_(m1, n)) (B : 'M[F]_(m2, n)) :
mxdirect (A + B) -> rowg A \x rowg B = rowg (A + B)%MS. | Proof.
rewrite (sameP mxdirect_addsP eqP) -trivg_rowg rowgI => /eqP tiAB.
by rewrite -cprod_rowg dprodEcp.
Qed. | Lemma | dprod_rowg | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"cprod_rowg",
"dprodEcp",
"mxdirect",
"mxdirect_addsP",
"rowg",
"rowgI",
"trivg_rowg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcprod_rowg m I r (P : pred I) (A : I -> 'M[F]_n) (B : 'M[F]_(m, n)) :
(\sum_(i <- r | P i) A i :=: B)%MS ->
\big[cprod/1%g]_(i <- r | P i) rowg (A i) = rowg B. | Proof.
by move/eq_rowg <-; apply/esym/big_morph=> [? ?|]; rewrite (rowg0, cprod_rowg).
Qed. | Lemma | bigcprod_rowg | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"big_morph",
"cprod",
"cprod_rowg",
"eq_rowg",
"rowg",
"rowg0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigdprod_rowg m (I : finType) (P : pred I) A (B : 'M[F]_(m, n)) :
let S := (\sum_(i | P i) A i)%MS in (S :=: B)%MS -> mxdirect S ->
\big[dprod/1%g]_(i | P i) rowg (A i) = rowg B. | Proof.
move=> S defS; rewrite mxdirectE defS /= => /eqP rankB.
apply: bigcprod_card_dprod (bigcprod_rowg defS) (eq_leq _).
by rewrite card_rowg rankB expn_sum; apply: eq_bigr => i; rewrite card_rowg.
Qed. | Lemma | bigdprod_rowg | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"bigcprod_card_dprod",
"bigcprod_rowg",
"card_rowg",
"dprod",
"eq_bigr",
"eq_leq",
"expn_sum",
"mxdirect",
"mxdirectE",
"rowg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GL_mx_repr : mx_repr 'GL_n[F] GLval. | Proof. by []. Qed. | Fact | GL_mx_repr | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"GLval",
"mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GLrepr | := MxRepresentation GL_mx_repr. | Canonical | GLrepr | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"GL_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
GLmx_faithful : mx_faithful GLrepr. | Proof. by apply/subsetP=> A; rewrite !inE mul1mx. Qed. | Lemma | GLmx_faithful | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"GLrepr",
"apply",
"inE",
"mul1mx",
"mx_faithful",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reprGLm x : {'GL_n[F]} | := insubd (1%g : {'GL_n[F]}) (rG x). | Definition | reprGLm | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"insubd",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_reprGLm x : x \in G -> val (reprGLm x) = rG x. | Proof. by move=> Gx; rewrite val_insubd (repr_mx_unitr rG). Qed. | Lemma | val_reprGLm | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"rG",
"reprGLm",
"repr_mx_unitr",
"val",
"val_insubd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comp_reprGLm : {in G, GLval \o reprGLm =1 rG}. | Proof. exact: val_reprGLm. Qed. | Lemma | comp_reprGLm | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"GLval",
"rG",
"reprGLm",
"val_reprGLm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reprGLmM : {in G &, {morph reprGLm : x y / x * y}}%g. | Proof.
by move=> x y Gx Gy; apply: val_inj; rewrite /= !val_reprGLm ?groupM ?repr_mxM.
Qed. | Lemma | reprGLmM | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"groupM",
"reprGLm",
"repr_mxM",
"val_inj",
"val_reprGLm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reprGL_morphism | := Morphism reprGLmM. | Canonical | reprGL_morphism | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"reprGLmM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ker_reprGLm : 'ker reprGLm = rker rG. | Proof.
apply/setP=> x; rewrite !inE mul1mx; apply: andb_id2l => Gx.
by rewrite -val_eqE val_reprGLm.
Qed. | Lemma | ker_reprGLm | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"inE",
"ker",
"mul1mx",
"rG",
"reprGLm",
"rker",
"setP",
"val_eqE",
"val_reprGLm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
astab_rowg_repr m (A : 'M_(m, n)) : 'C(rowg A | 'MR rG) = rstab rG A. | Proof.
apply/setP=> x /[!inE]/=; apply: andb_id2l => Gx.
apply/subsetP/eqP=> cAx => [|u]; last first.
by rewrite !inE mx_repr_actE // => /submxP[u' ->]; rewrite -mulmxA cAx.
apply/row_matrixP=> i; apply/eqP; move/implyP: (cAx (row i A)).
by rewrite !inE row_sub mx_repr_actE //= row_mul.
Qed. | Lemma | astab_rowg_repr | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"inE",
"last",
"mulmxA",
"mx_repr_actE",
"rG",
"row",
"row_matrixP",
"row_mul",
"row_sub",
"rowg",
"rstab",
"setP",
"submxP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
astabs_rowg_repr m (A : 'M_(m, n)) : 'N(rowg A | 'MR rG) = rstabs rG A. | Proof.
apply/setP=> x /[!inE]/=; apply: andb_id2l => Gx.
apply/subsetP/idP=> nAx => [|u]; last first.
by rewrite !inE mx_repr_actE // => Au; apply: (submx_trans (submxMr _ Au)).
apply/row_subP=> i; move/implyP: (nAx (row i A)).
by rewrite !inE row_sub mx_repr_actE //= row_mul.
Qed. | Lemma | astabs_rowg_repr | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"inE",
"last",
"mx_repr_actE",
"rG",
"row",
"row_mul",
"row_sub",
"row_subP",
"rowg",
"rstabs",
"setP",
"submxMr",
"submx_trans",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
acts_rowg (A : 'M_n) : [acts G, on rowg A | 'MR rG] = mxmodule rG A. | Proof. by rewrite astabs_rowg_repr. Qed. | Lemma | acts_rowg | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"astabs_rowg_repr",
"mxmodule",
"on",
"rG",
"rowg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
astab_setT_repr : 'C(setT | 'MR rG) = rker rG. | Proof. by rewrite -rowg1 astab_rowg_repr. Qed. | Lemma | astab_setT_repr | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"astab_rowg_repr",
"rG",
"rker",
"rowg1",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_repr_action_faithful :
[faithful G, on setT | 'MR rG] = mx_faithful rG. | Proof.
by rewrite /faithful astab_setT_repr (setIidPr _) // [rker _]setIdE subsetIl.
Qed. | Lemma | mx_repr_action_faithful | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"astab_setT_repr",
"faithful",
"mx_faithful",
"on",
"rG",
"rker",
"setIdE",
"setIidPr",
"setT",
"subsetIl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
afix_repr (H : {set gT}) :
H \subset G -> 'Fix_('MR rG)(H) = rowg (rfix_mx rG H). | Proof.
move/subsetP=> sHG; apply/setP=> /= u; rewrite !inE.
apply/subsetP/rfix_mxP=> cHu x Hx; have:= cHu x Hx;
by rewrite !inE /= => /eqP; rewrite mx_repr_actE ?sHG.
Qed. | Lemma | afix_repr | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"gT",
"inE",
"mx_repr_actE",
"rG",
"rfix_mx",
"rfix_mxP",
"rowg",
"sHG",
"setP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gacent_repr (H : {set gT}) :
H \subset G -> 'C_(| 'MR rG)(H) = rowg (rfix_mx rG H). | Proof. by move=> sHG; rewrite gacentE // setTI afix_repr. Qed. | Lemma | gacent_repr | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"afix_repr",
"gT",
"gacentE",
"rG",
"rfix_mx",
"rowg",
"sHG",
"setTI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Zm'" | := (scale_action _ _ _) : action_scope. | Notation | ''Zm' | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"scale_action"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Zm'" | := (scale_groupAction _ _ _) : groupAction_scope. | Notation | ''Zm' | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"scale_groupAction"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exponent_mx_group m n q :
m > 0 -> n > 0 -> q > 1 -> exponent [set: 'M['Z_q]_(m, n)] = q. | Proof.
move=> m_gt0 n_gt0 q_gt1; apply/eqP; rewrite eqn_dvd; apply/andP; split.
apply/exponentP=> x _; apply/matrixP=> i j; rewrite mulmxnE !mxE.
by rewrite -mulr_natr -Zp_nat_mod // modnn mulr0.
pose cmx1 := const_mx 1%R : 'M['Z_q]_(m, n).
apply: dvdn_trans (dvdn_exponent (in_setT cmx1)).
have/matrixP/(_ (Ordinal ... | Lemma | exponent_mx_group | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"Zp_cast",
"Zp_nat_mod",
"apply",
"const_mx",
"dvdn_exponent",
"dvdn_trans",
"eqn_dvd",
"expg_order",
"exponent",
"exponentP",
"in_setT",
"matrixP",
"modnn",
"mulmxnE",
"mulr0",
"mulr_natr",
"mxE",
"n_gt0",
"order_Zp1",
"order_dvdn",
"q_gt1",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_mx_group m n q : 'r([set: 'M['Z_q]_(m, n)]) = (m * n)%N. | Proof.
wlog q_gt1: q / q > 1 by case: q => [|[|q -> //]] /(_ 2)->.
set G := setT; have cGG: abelian G := zmod_abelian _.
have [mn0 | ] := posnP (m * n).
by rewrite [G](card1_trivg _) ?rank1 // cardsT card_mx mn0.
rewrite muln_gt0 => /andP[m_gt0 n_gt0].
have expG: exponent G = q := exponent_mx_group m_gt0 n_gt0 q_gt1.... | Lemma | rank_mx_group | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"Zp_cast",
"abelian",
"apply",
"cGG",
"card1_trivg",
"card_mx",
"card_ord",
"card_prod",
"cardsT",
"delta_mx",
"eqn_leq",
"exponent",
"exponent_mx_group",
"grank_abelian",
"grank_min",
"groupX",
"group_prod",
"imsetP",
"inE",
"leq_exp2l",
"leq_imset_card",
"leq_trans",
"m... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_group_homocyclic m n q : homocyclic [set: 'M['Z_q]_(m, n)]. | Proof.
wlog q_gt1: q / q > 1 by case: q => [|[|q -> //]] /(_ 2)->.
set G := setT; have cGG: abelian G := zmod_abelian _.
rewrite -max_card_abelian //= rank_mx_group cardsT card_mx card_ord -/G.
rewrite {1}Zp_cast //; have [-> // | ] := posnP (m * n).
by rewrite muln_gt0 => /andP[m_gt0 n_gt0]; rewrite exponent_mx_group.... | Lemma | mx_group_homocyclic | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"Zp_cast",
"abelian",
"cGG",
"card_mx",
"card_ord",
"cardsT",
"exponent_mx_group",
"homocyclic",
"max_card_abelian",
"muln_gt0",
"n_gt0",
"posnP",
"q_gt1",
"rank_mx_group",
"setT",
"zmod_abelian"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelian_type_mx_group m n q :
q > 1 -> abelian_type [set: 'M['Z_q]_(m, n)] = nseq (m * n) q. | Proof.
rewrite (abelian_type_homocyclic (mx_group_homocyclic m n q)) rank_mx_group.
have [-> // | ] := posnP (m * n); rewrite muln_gt0 => /andP[m_gt0 n_gt0] q_gt1.
by rewrite exponent_mx_group.
Qed. | Lemma | abelian_type_mx_group | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelian_type",
"abelian_type_homocyclic",
"exponent_mx_group",
"muln_gt0",
"mx_group_homocyclic",
"n_gt0",
"nseq",
"posnP",
"q_gt1",
"rank_mx_group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_dim' (gT : finGroupType) (E : {set gT}) | :=
(logn (pdiv #|E|) #|E|).-1. | Definition | abelem_dim' | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"gT",
"logn",
"pdiv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.