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 |
|---|---|---|---|---|---|---|---|---|---|---|
last_mod U (compU : mx_series U) : modG (last 0 U). | Proof.
by case: compU => modU _; rewrite (last_nth 0) (mx_subseries_module' _ modU).
Qed. | Let | last_mod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"compU",
"last",
"last_nth",
"modG",
"mx_series",
"mx_subseries_module'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rsim_last U V modUm modV compUV :
mx_rsim (@section_repr (last 0 U) V modUm modV)
(@series_repr (rcons U V) (size U) compUV). | Proof.
apply: section_eqmx; last by rewrite nth_rcons ltnn eqxx.
by rewrite -rcons_cons nth_rcons leqnn -last_nth.
Qed. | Let | rsim_last | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqxx",
"last",
"last_nth",
"leqnn",
"ltnn",
"mx_rsim",
"nth_rcons",
"rcons",
"rcons_cons",
"section_eqmx",
"section_repr",
"series_repr",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rsimT | := mx_rsim_trans. | Notation | rsimT | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_rsim_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rsimC | := mx_rsim_sym. | Notation | rsimC | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_rsim_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_JordanHolder U V compU compV :
let m := size U in (last 0 U :=: last 0 V)%MS ->
m = size V /\ (exists p : 'S_m, forall i : 'I_m,
mx_rsim (@series_repr U i compU) (@series_repr V (p i) compV)). | Proof.
move Dr: {-}(size U) => r; move/eqP in Dr.
elim: r U V Dr compU compV => /= [|r IHr] U V.
move/nilP->; case/lastP: V => [|V Vm] /= ? compVm; rewrite ?last_rcons => Vm0.
by split=> //; exists 1%g; case.
by case/mx_series_rcons: (compVm) => _ _ []; rewrite -(lt_eqmx Vm0) ltmx0.
case/lastP: U => // [U Um]; ... | Lemma | mx_JordanHolder | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"bool_irrelevance",
"capmxC",
"capmx_module",
"compU",
"eqSS",
"eqmxP",
"eqmx_sym",
"inj_eq",
"last",
"lastP",
"last_mod",
"last_rcons",
"lift",
"lift_inj",
"lift_max",
"lift_perm",
"lift_perm_id",
"lift_perm_lift",
"lt_eqmx",
"ltmx0",
"ltn_ord",
"max_submod",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_JordanHolder_max U (m := size U) V compU modV :
(last 0 U :=: 1%:M)%MS -> mx_irreducible (@factmod_repr _ G n rG V modV) ->
exists i : 'I_m, mx_rsim (factmod_repr modV) (@series_repr U i compU). | Proof.
rewrite {}/m; set Um := last 0 U => Um1 irrV.
have modUm: modG Um := last_mod compU; have simV := rsimC (mx_factmod_sub modV).
have maxV: max_submod V Um.
move/max_submodP: (mx_rsim_irr simV irrV) => /(_ (submx1 _)).
by apply: max_submod_eqmx; last apply: eqmx_sym.
have [W compW lastW] := mx_JordanHolder_exi... | Lemma | mx_JordanHolder_max | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"compU",
"eqmx_sym",
"factmod_repr",
"last",
"last_mod",
"last_rcons",
"max_submod",
"max_submodP",
"max_submod_eqmx",
"modG",
"mx_JordanHolder",
"mx_JordanHolder_exists",
"mx_factmod_sub",
"mx_irreducible",
"mx_rsim",
"mx_rsim_irr",
"mx_series",
"mx_series_rcons",
"pe... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nG | := #|pred_of_set (gval G)|. | Notation | nG | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"pred_of_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aG | := (regular_repr F G). | Notation | aG | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"regular_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R_G | := (group_ring F G). | Notation | R_G | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"group_ring"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_free : row_free R_G. | Proof.
apply/row_freeP; exists (lin1_mx (row (gring_index G 1) \o vec_mx)).
apply/row_matrixP=> i; rewrite row_mul rowK mul_rV_lin1 /= mxvecK rowK row1.
by rewrite gring_indexK // mul1g gring_valK.
Qed. | Lemma | gring_free | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"apply",
"gring_index",
"gring_indexK",
"gring_valK",
"lin1_mx",
"mul1g",
"mul_rV_lin1",
"mxvecK",
"row",
"row1",
"rowK",
"row_free",
"row_freeP",
"row_matrixP",
"row_mul",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_op_id A : (A \in R_G)%MS -> gring_op aG A = A. | Proof.
case/envelop_mxP=> a ->{A}; rewrite linear_sum.
by apply: eq_bigr => x Gx; rewrite linearZ /= gring_opG.
Qed. | Lemma | gring_op_id | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"aG",
"apply",
"envelop_mxP",
"eq_bigr",
"gring_op",
"gring_opG",
"linearZ",
"linear_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_rowK A : (A \in R_G)%MS -> gring_mx aG (gring_row A) = A. | Proof. exact: gring_op_id. Qed. | Lemma | gring_rowK | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"aG",
"gring_mx",
"gring_op_id",
"gring_row"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_gring_mx m a (M : 'M_(m, nG)) :
(gring_mx aG a \in M *m R_G)%MS = (a <= M)%MS. | Proof. by rewrite vec_mxK submxMfree ?gring_free. Qed. | Lemma | mem_gring_mx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"aG",
"gring_free",
"gring_mx",
"nG",
"submxMfree",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_sub_gring m A (M : 'M_(m, nG)) :
(A \in M *m R_G)%MS = (A \in R_G)%MS && (gring_row A <= M)%MS. | Proof.
rewrite -(andb_idl (memmx_subP (submxMl _ _) A)); apply: andb_id2l => R_A.
by rewrite -mem_gring_mx gring_rowK.
Qed. | Lemma | mem_sub_gring | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"apply",
"gring_row",
"gring_rowK",
"mem_gring_mx",
"memmx_subP",
"nG",
"submxMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_mxP a : (gring_mx rG a \in enveloping_algebra_mx rG)%MS. | Proof. by rewrite vec_mxK submxMl. Qed. | Lemma | gring_mxP | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"enveloping_algebra_mx",
"gring_mx",
"rG",
"submxMl",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gring_opM A B :
(B \in R_G)%MS -> gring_op rG (A *m B) = gring_op rG A *m gring_op rG B. | Proof. by move=> R_B; rewrite -gring_opJ gring_rowK. Qed. | Lemma | gring_opM | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"gring_op",
"gring_opJ",
"gring_rowK",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rsim_regular_factmod :
{U : 'M_nG & {modU : mxmodule aG U & mx_rsim rG (factmod_repr modU)}}. | Proof.
pose v : 'rV[F]_n := nz_row 1%:M.
pose fU := lin1_mx (mulmx v \o gring_mx rG); pose U := kermx fU.
have modU: mxmodule aG U.
apply/mxmoduleP => x Gx; apply/sub_kermxP/row_matrixP=> i.
rewrite 2!row_mul row0; move: (row i U) (sub_kermxP (row_sub i U)) => u.
by rewrite !mul_rV_lin1 /= gring_mxJ // mulmxA => ... | Lemma | rsim_regular_factmod | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"aG",
"add0r",
"addKr",
"add_sub_fact_mod",
"apply",
"cokermx",
"def_n",
"enum_val",
"eq_row_sub",
"eqmx_eq0",
"eqmx_module",
"eqn_leq",
"factmod_repr",
"genmxE",
"gring_index",
"gring_mx",
"gring_mxJ",
"gring_opE",
"gring_opG",
"gring_row",
"in_factmod_eq0",
"irrG",
"ker... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rsim_regular_series U (compU : mx_composition_series aG U) :
(last 0 U :=: 1%:M)%MS ->
exists i : 'I_(size U), mx_rsim rG (series_repr i compU). | Proof.
move=> lastU; have [V [modV simGV]] := rsim_regular_factmod.
have irrV := mx_rsim_irr simGV irrG.
have [i simVU] := mx_JordanHolder_max compU lastU irrV.
by exists i; apply: mx_rsim_trans simGV simVU.
Qed. | Lemma | rsim_regular_series | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"aG",
"apply",
"compU",
"irrG",
"last",
"mx_JordanHolder_max",
"mx_composition_series",
"mx_rsim",
"mx_rsim_irr",
"mx_rsim_trans",
"rG",
"rsim_regular_factmod",
"series_repr",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F'G : [pchar F]^'.-group G. | Hypothesis | F'G | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"group",
"pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
rsim_regular_submod_pchar :
{U : 'M_nG & {modU : mxmodule aG U & mx_rsim rG (submod_repr modU)}}. | Proof.
have [V [modV eqG'V]] := rsim_regular_factmod.
have [U modU defVU dxVU] := mx_Maschke_pchar F'G modV (submx1 V).
exists U; exists modU; apply: mx_rsim_trans eqG'V _.
by apply: mx_rsim_factmod; rewrite ?mxdirectE /= addsmxC // addnC.
Qed. | Lemma | rsim_regular_submod_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"F'G",
"aG",
"addnC",
"addsmxC",
"apply",
"mx_Maschke_pchar",
"mx_rsim",
"mx_rsim_factmod",
"mx_rsim_trans",
"mxdirectE",
"mxmodule",
"rG",
"rsim_regular_factmod",
"submod_repr",
"submx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gset_mx (A : {set gT}) | := \sum_(x in A) aG x. | Definition | gset_mx | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"aG",
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tG | := #|pred_of_set (classes (gval G))|. | Notation | tG | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"classes",
"pred_of_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
classg_base | := \matrix_(k < tG) mxvec (gset_mx (enum_val k)). | Definition | classg_base | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"enum_val",
"gset_mx",
"mxvec",
"tG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
groupCl : {in G, forall x, {subset x ^: G <= G}}. | Proof. by move=> x Gx; apply: subsetP; apply: class_subG. Qed. | Let | groupCl | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"class_subG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
classg_base_free : row_free classg_base. | Proof.
rewrite -kermx_eq0; apply/rowV0P=> v /sub_kermxP; rewrite mulmx_sum_row => v0.
apply/rowP=> k /[1!mxE].
have [x Gx def_k] := imsetP (enum_valP k).
transitivity (@gring_proj F _ G x (vec_mx 0) 0 0); last first.
by rewrite !linear0 !mxE.
rewrite -{}v0 !linear_sum (bigD1 k) //= 2!linearZ /= rowK mxvecK def_k.
rew... | Lemma | classg_base_free | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addr0",
"apply",
"big1",
"bigD1",
"class_eqP",
"class_refl",
"classg_base",
"enum_valP",
"enum_val_inj",
"eq_sym",
"eqxx",
"gring_proj",
"gring_projE",
"groupCl",
"imsetP",
"inj_eq",
"kermx_eq0",
"last",
"linear0",
"linearZ",
"linear_sum",
"mulmx_sum_row",
"mulr1",
"mx... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
classg_base_center : (classg_base :=: 'Z(R_G))%MS. | Proof.
apply/eqmxP/andP; split.
apply/row_subP=> k; rewrite rowK /gset_mx sub_capmx {1}linear_sum.
have [x Gx ->{k}] := imsetP (enum_valP k); have sxGG := groupCl Gx.
rewrite summx_sub => [y xGy|]; first by rewrite envelop_mx_id ?sxGG.
rewrite memmx_cent_envelop; apply/centgmxP=> y Gy.
rewrite {2}(reindex_act... | Lemma | classg_base_center | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"aG",
"addr0",
"apply",
"astabsJ",
"big1",
"bigD1",
"can_eq",
"centgmxP",
"class",
"classGidl",
"class_norm",
"class_refl",
"classes",
"classg_base",
"conjgC",
"conjgE",
"conjgK",
"conjgKV",
"enum_rankK_in",
"enum_rank_in",
"enum_valP",
"envelop_mxP",
"envelop_mx... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
regular_module_ideal m (M : 'M_(m, nG)) :
mxmodule aG M = right_mx_ideal R_G (M *m R_G). | Proof.
apply/idP/idP=> modM.
apply/mulsmx_subP=> A B; rewrite !mem_sub_gring => /andP[R_A M_A] R_B.
by rewrite envelop_mxM // gring_row_mul (mxmodule_envelop modM).
apply/mxmoduleP=> x Gx; apply/row_subP=> i; rewrite row_mul -mem_gring_mx.
rewrite gring_mxJ // (mulsmx_subP modM) ?envelop_mx_id //.
by rewrite mem_gr... | Lemma | regular_module_ideal | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"aG",
"apply",
"envelop_mxM",
"envelop_mx_id",
"gring_mxJ",
"gring_row_mul",
"mem_gring_mx",
"mem_sub_gring",
"mulsmx_subP",
"mxmodule",
"mxmoduleP",
"mxmodule_envelop",
"nG",
"right_mx_ideal",
"row_mul",
"row_sub",
"row_subP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irrType | := socleType aG. | Definition | irrType | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"aG",
"socleType"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_degree (i : sG) | := \rank (socle_base i). | Definition | irr_degree | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"rank",
"sG",
"socle_base"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'n_ i" | := (irr_degree i) : group_ring_scope. | Notation | 'n_ i | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_degree"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_degreeE i : 'n_i = \rank (socle_base i). | Proof. by []. Qed. | Lemma | irr_degreeE | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"rank",
"socle_base"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_degree_gt0 i : 'n_i > 0. | Proof. by rewrite lt0n mxrank_eq0; case: (socle_simple i). Qed. | Lemma | irr_degree_gt0 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"lt0n",
"mxrank_eq0",
"socle_simple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_repr i : mx_representation F G 'n_i | := socle_repr i. | Definition | irr_repr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mx_representation",
"socle_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_reprE i x : irr_repr i x = submod_mx (socle_module i) x. | Proof. by []. Qed. | Lemma | irr_reprE | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_repr",
"socle_module",
"submod_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_regular : (rfix_mx aG G :=: gring_row (gset_mx G))%MS. | Proof.
apply/eqmxP/andP; split; last first.
apply/rfix_mxP => x Gx; rewrite -gring_row_mul; congr gring_row.
rewrite {2}/gset_mx (reindex_astabs 'R x) ?astabsR //= mulmx_suml.
by apply: eq_bigr => y Gy; rewrite repr_mxM.
apply/rV_subP=> v /rfix_mxP cGv.
have /envelop_mxP[a def_v]: (gring_mx aG v \in R_G)%MS.
by... | Lemma | rfix_regular | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"aG",
"addr0",
"apply",
"astabsR",
"big1",
"bigD1",
"envelop_mxP",
"eq_bigr",
"eq_mulgV1",
"eq_sym",
"eqmxP",
"eqxx",
"gring_mx",
"gring_mxJ",
"gring_mxK",
"gring_proj",
"gring_projE",
"gring_row",
"gring_row_mul",
"groupM",
"gset_mx",
"last",
"linearZ",
"linea... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
principal_comp_subproof : mxsimple aG (rfix_mx aG G). | Proof.
apply: linear_mxsimple; first exact: rfix_mx_module.
apply/eqP; rewrite rfix_regular eqn_leq rank_leq_row lt0n mxrank_eq0.
apply/eqP => /(congr1 (gring_proj 1 \o gring_mx aG)); apply/eqP.
rewrite /= -[gring_mx _ _]/(gring_op _ _) !linear0 !linear_sum (bigD1 1%g) //=.
rewrite gring_opG ?gring_projE // eqxx big1 ?... | Lemma | principal_comp_subproof | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"aG",
"addr0",
"apply",
"big1",
"bigD1",
"eq_sym",
"eqn_leq",
"eqxx",
"gring_mx",
"gring_op",
"gring_opG",
"gring_proj",
"gring_projE",
"linear0",
"linear_mxsimple",
"linear_sum",
"lt0n",
"mxrank_eq0",
"mxsimple",
"oner_eq0",
"rank_leq_row",
"rfix_mx",
"rfix_mx_module",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
principal_comp_key : unit. | Proof. by []. Qed. | Fact | principal_comp_key | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
principal_comp_def | :=
PackSocle (component_socle sG principal_comp_subproof). | Definition | principal_comp_def | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"component_socle",
"principal_comp_subproof",
"sG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
principal_comp | := locked_with principal_comp_key principal_comp_def. | Definition | principal_comp | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"principal_comp_def",
"principal_comp_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"1" | := principal_comp : irrType_scope. | Notation | 1 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"principal_comp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr1_rfix : (1%irr :=: rfix_mx aG G)%MS. | Proof.
rewrite [1%irr]unlock PackSocleK; apply/eqmxP.
rewrite (component_mx_id principal_comp_subproof) andbT.
have [I [W isoW ->]] := component_mx_def principal_comp_subproof.
apply/sumsmx_subP=> i _; have [f _ hom_f <-]:= isoW i.
(* FIX ME : this takes time *)
by apply/rfix_mxP=> x Gx; rewrite -(hom_mxP hom_f) // (rf... | Lemma | irr1_rfix | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"PackSocleK",
"aG",
"apply",
"component_mx_def",
"component_mx_id",
"eqmxP",
"hom_mxP",
"irr",
"principal_comp_subproof",
"rfix_mx",
"rfix_mxP",
"sumsmx_subP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_irr1 : \rank 1%irr = 1. | Proof.
apply/eqP; rewrite eqn_leq lt0n mxrank_eq0 nz_socle andbT.
by rewrite irr1_rfix rfix_regular rank_leq_row.
Qed. | Lemma | rank_irr1 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eqn_leq",
"irr",
"irr1_rfix",
"lt0n",
"mxrank_eq0",
"nz_socle",
"rank",
"rank_leq_row",
"rfix_regular"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
degree_irr1 : 'n_1 = 1. | Proof.
apply/eqP; rewrite eqn_leq irr_degree_gt0 -rank_irr1.
by rewrite mxrankS ?component_mx_id //; apply: socle_simple.
Qed. | Lemma | degree_irr1 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"component_mx_id",
"eqn_leq",
"irr_degree_gt0",
"mxrankS",
"rank_irr1",
"socle_simple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_subring (i : sG) | := <<i *m R_G>>%MS. | Definition | Wedderburn_subring | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"sG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''R_' i" | := (Wedderburn_subring i) : group_ring_scope. | Notation | ''R_' i | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_subring"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sums_R : (\sum_i 'R_i :=: Socle sG *m R_G)%MS. | Proof.
apply/eqmxP; set R_S := (_ <= _)%MS.
have sRS: R_S by apply/sumsmx_subP=> i; rewrite genmxE submxMr ?(sumsmx_sup i).
rewrite sRS -(mulmxKpV sRS) mulmxA submxMr //; apply/sumsmx_subP=> i _.
rewrite -(submxMfree _ _ gring_free) -(mulmxA _ _ R_G) mulmxKpV //.
by rewrite (sumsmx_sup i) ?genmxE.
Qed. | Let | sums_R | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"Socle",
"apply",
"eqmxP",
"genmxE",
"gring_free",
"mulmxA",
"mulmxKpV",
"sG",
"submxMfree",
"submxMr",
"sumsmx_subP",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_ideal i : mx_ideal R_G 'R_i. | Proof.
apply/andP; split; last first.
rewrite /right_mx_ideal genmxE (muls_eqmx (genmxE _) (eqmx_refl _)).
by rewrite -[(_ <= _)%MS]regular_module_ideal component_mx_module.
apply/mulsmx_subP=> A B R_A; rewrite !genmxE !mem_sub_gring => /andP[R_B SiB].
rewrite envelop_mxM {R_A}// gring_row_mul -{R_B}(gring_rowK R_B... | Lemma | Wedderburn_ideal | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"aG",
"apply",
"component_mx_module",
"envelop_mxM",
"eqmx_refl",
"genmxE",
"gring_mx",
"gring_mxJ",
"gring_row",
"gring_rowK",
"gring_row_mul",
"hom_component_mx",
"hom_mxP",
"last",
"lin1_mx",
"mem_sub_gring",
"mul_rV_lin1",
"mulmx",
"mulmxA",
"muls_eqmx",
"mulsmx_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_direct : mxdirect (\sum_i 'R_i)%MS. | Proof.
apply/mxdirectP; rewrite /= sums_R mxrankMfree ?gring_free //.
rewrite (mxdirectP (Socle_direct sG)); apply: eq_bigr=> i _ /=.
by rewrite genmxE mxrankMfree ?gring_free.
Qed. | Lemma | Wedderburn_direct | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Socle_direct",
"apply",
"eq_bigr",
"genmxE",
"gring_free",
"mxdirect",
"mxdirectP",
"mxrankMfree",
"sG",
"sums_R"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_disjoint i j : i != j -> ('R_i :&: 'R_j)%MS = 0. | Proof.
move=> ne_ij; apply/eqP; rewrite -submx0 capmxC.
by rewrite -(mxdirect_sumsP Wedderburn_direct j) // capmxS // (sumsmx_sup i).
Qed. | Lemma | Wedderburn_disjoint | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_direct",
"apply",
"capmxC",
"capmxS",
"mxdirect_sumsP",
"submx0",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_annihilate i j : i != j -> ('R_i * 'R_j)%MS = 0. | Proof.
move=> ne_ij; apply/eqP; rewrite -submx0 -(Wedderburn_disjoint ne_ij).
rewrite sub_capmx; apply/andP; split.
case/andP: (Wedderburn_ideal i) => _; apply: submx_trans.
by rewrite mulsmxS // genmxE submxMl.
case/andP: (Wedderburn_ideal j) => idlRj _; apply: submx_trans idlRj.
by rewrite mulsmxS // genmxE submx... | Lemma | Wedderburn_annihilate | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_disjoint",
"Wedderburn_ideal",
"apply",
"genmxE",
"mulsmxS",
"split",
"sub_capmx",
"submx0",
"submxMl",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_mulmx0 i j A B :
i != j -> (A \in 'R_i)%MS -> (B \in 'R_j)%MS -> A *m B = 0. | Proof.
move=> ne_ij RiA RjB; apply: memmx0.
by rewrite -(Wedderburn_annihilate ne_ij) mem_mulsmx.
Qed. | Lemma | Wedderburn_mulmx0 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_annihilate",
"apply",
"mem_mulsmx",
"memmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_mx_sum_pchar : (\sum_(i : sG) i = 1%:M)%MS. | Proof. by apply: reducible_Socle1; apply: mx_Maschke_pchar. Qed. | Lemma | irr_mx_sum_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"mx_Maschke_pchar",
"reducible_Socle1",
"sG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_sum_pchar : (\sum_i 'R_i :=: R_G)%MS. | Proof.
by apply: eqmx_trans sums_R _; rewrite /Socle irr_mx_sum_pchar mul1mx.
Qed. | Lemma | Wedderburn_sum_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"Socle",
"apply",
"eqmx_trans",
"irr_mx_sum_pchar",
"mul1mx",
"sums_R"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_id i | :=
vec_mx (mxvec 1%:M *m proj_mx 'R_i (\sum_(j | j != i) 'R_j)%MS). | Definition | Wedderburn_id | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"mxvec",
"proj_mx",
"vec_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''e_' i" | := (Wedderburn_id i) : group_ring_scope. | Notation | ''e_' i | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_sum_id_pchar : \sum_i 'e_i = 1%:M. | Proof.
rewrite -linear_sum; apply: canLR mxvecK _.
have: (1%:M \in R_G)%MS := envelop_mx1 aG.
rewrite -Wedderburn_sum_pchar.
case/(sub_dsumsmx Wedderburn_direct) => e Re -> _.
apply: eq_bigr => i _; have dxR := mxdirect_sumsP Wedderburn_direct i (erefl _).
rewrite (bigD1 i) // mulmxDl proj_mx_id ?Re // proj_mx_0 ?addr0... | Lemma | Wedderburn_sum_id_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"Re",
"Wedderburn_direct",
"Wedderburn_sum_pchar",
"aG",
"addr0",
"apply",
"bigD1",
"envelop_mx1",
"eq_bigr",
"linear_sum",
"mulmxDl",
"mxdirect_sumsP",
"mxvecK",
"proj_mx_0",
"proj_mx_id",
"sub_dsumsmx",
"summx_sub",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_id_mem i : ('e_i \in 'R_i)%MS. | Proof. by rewrite vec_mxK proj_mx_sub. Qed. | Lemma | Wedderburn_id_mem | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"proj_mx_sub",
"vec_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_is_id_pchar i : mxring_id 'R_i 'e_i. | Proof.
have ideRi A: (A \in 'R_i)%MS -> 'e_i *m A = A.
move=> RiA; rewrite -{2}[A]mul1mx -Wedderburn_sum_id_pchar mulmx_suml.
rewrite (bigD1 i) //= big1 ?addr0 // => j ne_ji.
by rewrite (Wedderburn_mulmx0 ne_ji) ?Wedderburn_id_mem.
split=> // [||A RiA]; first 2 [exact: Wedderburn_id_mem].
apply: contraNneq (n... | Lemma | Wedderburn_is_id_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_id_mem",
"Wedderburn_mulmx0",
"Wedderburn_sum_id_pchar",
"addr0",
"apply",
"big1",
"bigD1",
"contraNneq",
"e0",
"eq_sym",
"genmxE",
"gring_mxK",
"linear0",
"mem_gring_mx",
"mul0mx",
"mul1mx",
"mulmx1",
"mulmx_suml",
"mulmx_sumr",
"mxring_id",
"nz_socle",
"rowV0P... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_closed_pchar i : ('R_i * 'R_i = 'R_i)%MS. | Proof.
rewrite -{3}['R_i]genmx_id -/'R_i -genmx_muls; apply/genmxP.
have [idlRi idrRi] := andP (Wedderburn_ideal i).
apply/andP; split.
by apply: submx_trans idrRi; rewrite mulsmxS // genmxE submxMl.
have [_ Ri_e ideRi _] := Wedderburn_is_id_pchar i.
by apply/memmx_subP=> A RiA; rewrite -[A]ideRi ?mem_mulsmx.
Qed. | Lemma | Wedderburn_closed_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_ideal",
"Wedderburn_is_id_pchar",
"apply",
"genmxE",
"genmxP",
"genmx_id",
"genmx_muls",
"mem_mulsmx",
"memmx_subP",
"mulsmxS",
"split",
"submxMl",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_is_ring_pchar i : mxring 'R_i. | Proof.
rewrite /mxring /left_mx_ideal Wedderburn_closed_pchar submx_refl.
by apply/mxring_idP; exists 'e_i; apply: Wedderburn_is_id_pchar.
Qed. | Lemma | Wedderburn_is_ring_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_closed_pchar",
"Wedderburn_is_id_pchar",
"apply",
"left_mx_ideal",
"mxring",
"mxring_idP",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_min_ideal_pchar m i (E : 'A_(m, nG)) :
E != 0 -> (E <= 'R_i)%MS -> mx_ideal R_G E -> (E :=: 'R_i)%MS. | Proof.
move=> nzE sE_Ri /andP[idlE idrE]; apply/eqmxP; rewrite sE_Ri.
pose M := E *m pinvmx R_G; have defE: E = M *m R_G.
by rewrite mulmxKpV // (submx_trans sE_Ri) // genmxE submxMl.
have modM: mxmodule aG M by rewrite regular_module_ideal -defE.
have simSi := socle_simple i; set Si := socle_base i in simSi.
have [I... | Lemma | Wedderburn_min_ideal_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"Re",
"Wedderburn_is_id_pchar",
"aG",
"apply",
"capmxSl",
"capmx_idPl",
"capmx_module",
"component_mx_def",
"eqVneq",
"eqmxMr",
"eqmxP",
"genmxE",
"gring_mxA",
"gring_mxP",
"gring_rowK",
"hom_envelop_mxC",
"last",
"linear0",
"linear_sum",
"mem_gring_mx",
"mem_mulsmx"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
not_rsim_op0_pchar (iG j : sG) A :
mx_rsim rG (socle_repr iG) -> iG != j -> (A \in 'R_j)%MS ->
gring_op rG A = 0. | Proof.
case/mx_rsim_def=> f [f' _ hom_f] ne_iG_j RjA.
transitivity (f *m in_submod _ (val_submod 1%:M *m A) *m f').
have{RjA}: (A \in R_G)%MS by rewrite -Wedderburn_sum_pchar (sumsmx_sup j).
case/envelop_mxP=> a ->{A}; rewrite !(linear_sum, mulmx_suml).
by apply: eq_bigr => x Gx; rewrite 4!linearZ /= -scalemxAl -... | Let | not_rsim_op0_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"Wedderburn_mulmx0",
"Wedderburn_sum_pchar",
"apply",
"component_mx_id",
"envelop_mxP",
"eq_bigr",
"genmxE",
"gring_mxK",
"gring_op",
"gring_opG",
"gring_row_mul",
"iG",
"in_submod",
"linear0",
"linearZ",
"linear_sum",
"mem_gring_mx",
"mul0mx",
"mulmx_suml",
"mx_rsim",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_comp | := odflt 1%irr [pick i | gring_op rG 'e_i != 0]. | Definition | irr_comp | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"gring_op",
"irr",
"pick",
"rG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iG | := irr_comp. | Notation | iG | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_comp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rsim_irr_comp_pchar : mx_rsim rG (irr_repr iG). | Proof.
have [M [modM rsimM]] := rsim_regular_submod_pchar irrG F'G.
have simM: mxsimple aG M.
case/mx_irrP: irrG => n_gt0 minG.
have [f def_n injf homf] := mx_rsim_sym rsimM.
apply/(submod_mx_irr modM)/mx_irrP.
split=> [|U modU nzU]; first by rewrite def_n.
rewrite /row_full -(mxrankMfree _ injf) -genmxE {4}d... | Lemma | rsim_irr_comp_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"F'G",
"PackSocleK",
"Wedderburn_id_mem",
"Wedderburn_sum_id_pchar",
"aG",
"apply",
"big1",
"component_mx",
"component_mx_id",
"component_mx_iso",
"component_socle",
"def_n",
"eqVneq",
"eqmx_module",
"eqxx",
"genmxE",
"gring_op",
"gring_opG",
"iG",
"injf",
"irrG",
"irr_repr... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_comp'_op0_pchar j A : j != iG -> (A \in 'R_j)%MS -> gring_op rG A = 0. | Proof. by rewrite eq_sym; apply: not_rsim_op0_pchar rsim_irr_comp_pchar. Qed. | Lemma | irr_comp'_op0_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"eq_sym",
"gring_op",
"iG",
"not_rsim_op0_pchar",
"rG",
"rsim_irr_comp_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_comp_envelop_pchar : ('R_iG *m lin_mx (gring_op rG) :=: E_G)%MS. | Proof.
apply/eqmxP/andP; split; apply/row_subP=> i.
by rewrite row_mul mul_rV_lin gring_mxP.
rewrite rowK /= -gring_opG ?enum_valP // -mul_vec_lin -gring_opG ?enum_valP //.
rewrite vec_mxK /= -mulmxA mulmx_sub {i}//= -(eqmxMr _ Wedderburn_sum_pchar).
rewrite (bigD1 iG) //= addsmxMr addsmxC [_ *m _](sub_kermxP _) ?add... | Lemma | irr_comp_envelop_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"Wedderburn_sum_pchar",
"adds0mx",
"addsmxC",
"addsmxMr",
"apply",
"bigD1",
"enum_valP",
"eqmxMr",
"eqmxP",
"gring_mxP",
"gring_op",
"gring_opG",
"iG",
"irr_comp'_op0_pchar",
"lin_mx",
"linear0",
"memmx_subP",
"mul_rV_lin",
"mul_vec_lin",
"mulmxA",
"mulmx_sub",
"rG... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ker_irr_comp_op_pchar : ('R_iG :&: kermx (lin_mx (gring_op rG)))%MS = 0. | Proof.
apply/eqP; rewrite -submx0; apply/memmx_subP=> A.
rewrite sub_capmx /= submx0 mxvec_eq0 => /andP[R_A].
rewrite (sameP sub_kermxP eqP) mul_vec_lin mxvec_eq0 /= => opA0.
have [_ Re ideR _] := Wedderburn_is_id_pchar iG; rewrite -[A]ideR {ideR}//.
move: Re; rewrite genmxE mem_sub_gring /socle_val => /andP[Re].
rewri... | Lemma | ker_irr_comp_op_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"Re",
"Wedderburn_is_id_pchar",
"Wedderburn_sum_pchar",
"aG",
"apply",
"component_mx_def",
"envelop_mxP",
"eq_bigr",
"genmxE",
"gring_mx",
"gring_mxJ",
"gring_op",
"gring_opG",
"gring_rowK",
"iG",
"in_submod",
"in_submodK",
"kermx",
"lin_mx",
"linear0",
"linearZ",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
regular_op_inj_pchar :
{in [pred A | (A \in 'R_iG)%MS] &, injective (gring_op rG)}. | Proof.
move=> A B RnA RnB /= eqAB; apply/eqP; rewrite -subr_eq0 -mxvec_eq0 -submx0.
rewrite -ker_irr_comp_op_pchar sub_capmx (sameP sub_kermxP eqP) mul_vec_lin.
by rewrite 2!raddfB /= eqAB subrr linear0 addmx_sub ?eqmx_opp /=.
Qed. | Lemma | regular_op_inj_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"addmx_sub",
"apply",
"eqmx_opp",
"gring_op",
"ker_irr_comp_op_pchar",
"linear0",
"mul_vec_lin",
"mxvec_eq0",
"rG",
"raddfB",
"sub_capmx",
"sub_kermxP",
"submx0",
"subr_eq0",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_irr_comp_pchar : \rank 'R_iG = \rank E_G. | Proof.
rewrite -irr_comp_envelop_pchar; apply/esym/mxrank_injP.
by rewrite ker_irr_comp_op_pchar.
Qed. | Lemma | rank_irr_comp_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"apply",
"irr_comp_envelop_pchar",
"ker_irr_comp_op_pchar",
"mxrank_injP",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_comp_rsim_pchar n1 n2 rG1 rG2 :
@mx_rsim _ G n1 rG1 n2 rG2 -> irr_comp rG1 = irr_comp rG2. | Proof.
case=> f eq_n12; rewrite -eq_n12 in rG2 f * => inj_f hom_f.
rewrite /irr_comp; apply/f_equal/eq_pick => i; rewrite -!mxrank_eq0.
(* [congr (odflt 1%irr _)] works but is very slow *)
rewrite -(mxrankMfree _ inj_f); symmetry; rewrite -(eqmxMfull _ inj_f).
have /envelop_mxP[e ->{i}]: ('e_i \in R_G)%MS.
by rewrite... | Lemma | irr_comp_rsim_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"Wedderburn_id_mem",
"Wedderburn_sum_pchar",
"apply",
"envelop_mxP",
"eq_bigr",
"eq_pick",
"eqmxMfull",
"gring_opG",
"inj_f",
"irr_comp",
"linearZ",
"linear_sum",
"mulmx_suml",
"mx_rsim",
"mxrankMfree",
"mxrank_eq0",
"rank",
"scalemxAl",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_reprK_pchar i : irr_comp (irr_repr i) = i. | Proof.
apply/eqP; apply/component_mx_isoP; try exact: socle_simple.
by move/mx_rsim_iso: (rsim_irr_comp_pchar (socle_irr i)); apply: mx_iso_sym.
Qed. | Lemma | irr_reprK_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"component_mx_isoP",
"irr_comp",
"irr_repr",
"mx_iso_sym",
"mx_rsim_iso",
"rsim_irr_comp_pchar",
"socle_irr",
"socle_simple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_repr'_op0_pchar i j A :
j != i -> (A \in 'R_j)%MS -> gring_op (irr_repr i) A = 0. | Proof.
move=> neq_ij /(irr_comp'_op0_pchar _).
by move=> ->; [apply: socle_irr|rewrite irr_reprK_pchar|].
Qed. | Lemma | irr_repr'_op0_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"gring_op",
"irr_comp'_op0_pchar",
"irr_repr",
"irr_reprK_pchar",
"socle_irr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
op_Wedderburn_id_pchar i : gring_op (irr_repr i) 'e_i = 1%:M. | Proof.
rewrite -(gring_op1 (irr_repr i)) -Wedderburn_sum_id_pchar.
rewrite linear_sum (bigD1 i) //= addrC big1 ?add0r // => j neq_ji.
exact: irr_repr'_op0_pchar (Wedderburn_id_mem j).
Qed. | Lemma | op_Wedderburn_id_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_id_mem",
"Wedderburn_sum_id_pchar",
"add0r",
"addrC",
"big1",
"bigD1",
"gring_op",
"gring_op1",
"irr_repr",
"irr_repr'_op0_pchar",
"linear_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_comp_id_pchar (M : 'M_nG) (modM : mxmodule aG M) (iM : sG) :
mxsimple aG M -> (M <= iM)%MS -> irr_comp (submod_repr modM) = iM. | Proof.
move=> simM sMiM; rewrite -[iM]irr_reprK_pchar.
apply/esym/irr_comp_rsim_pchar/mx_rsim_iso/component_mx_iso => //.
exact: socle_simple.
Qed. | Lemma | irr_comp_id_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"aG",
"apply",
"component_mx_iso",
"irr_comp",
"irr_comp_rsim_pchar",
"irr_reprK_pchar",
"mx_rsim_iso",
"mxmodule",
"mxsimple",
"sG",
"socle_simple",
"submod_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr1_repr x : x \in G -> irr_repr 1 x = 1%:M. | Proof.
move=> Gx; suffices: x \in rker (irr_repr 1) by case/rkerP.
apply: subsetP x Gx; rewrite rker_submod rfix_mx_rstabC // -irr1_rfix.
by apply: component_mx_id; apply: socle_simple.
Qed. | Lemma | irr1_repr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"component_mx_id",
"irr1_rfix",
"irr_repr",
"rfix_mx_rstabC",
"rker",
"rkerP",
"rker_submod",
"socle_simple",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_Wedderburn_subring_pchar i : \rank 'R_i = ('n_i ^ 2)%N. | Proof.
apply/eqP; rewrite -{1}[i]irr_reprK_pchar; have irrSi := socle_irr i.
by case/andP: (splitG irrSi) => _; rewrite rank_irr_comp_pchar.
Qed. | Lemma | rank_Wedderburn_subring_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"irr_reprK_pchar",
"rank",
"rank_irr_comp_pchar",
"socle_irr",
"splitG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_irr_degree_pchar : (\sum_i 'n_i ^ 2 = nG)%N. | Proof.
apply: etrans (eqnP gring_free).
rewrite -Wedderburn_sum_pchar (mxdirectP Wedderburn_direct) /=.
by apply: eq_bigr => i _; rewrite rank_Wedderburn_subring_pchar.
Qed. | Lemma | sum_irr_degree_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_direct",
"Wedderburn_sum_pchar",
"apply",
"eq_bigr",
"eqnP",
"gring_free",
"mxdirectP",
"nG",
"rank_Wedderburn_subring_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_mx_mult_pchar i : socle_mult i = 'n_i. | Proof.
rewrite /socle_mult -(mxrankMfree _ gring_free) -genmxE.
by rewrite rank_Wedderburn_subring_pchar mulKn ?irr_degree_gt0.
Qed. | Lemma | irr_mx_mult_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"genmxE",
"gring_free",
"irr_degree_gt0",
"mulKn",
"mxrankMfree",
"rank_Wedderburn_subring_pchar",
"socle_mult"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxtrace_regular_pchar :
{in G, forall x, \tr (aG x) = \sum_i \tr (socle_repr i x) *+ 'n_i}. | Proof.
move=> x Gx; have soc1: (Socle sG :=: 1%:M)%MS by rewrite -irr_mx_sum_pchar.
rewrite -(mxtrace_submod1 (Socle_module sG) soc1) // mxtrace_Socle //.
by apply: eq_bigr => i _; rewrite irr_mx_mult_pchar.
Qed. | Lemma | mxtrace_regular_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Socle",
"Socle_module",
"aG",
"apply",
"eq_bigr",
"irr_mx_mult_pchar",
"irr_mx_sum_pchar",
"mxtrace_Socle",
"mxtrace_submod1",
"sG",
"socle_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linear_irr | := [set i | 'n_i == 1]. | Definition | linear_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_degree_abelian : abelian G -> forall i, 'n_i = 1. | Proof. by move=> cGG i; apply: mxsimple_abelian_linear (socle_simple i). Qed. | Lemma | irr_degree_abelian | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"abelian",
"apply",
"cGG",
"mxsimple_abelian_linear",
"socle_simple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
linear_irr_comp_pchar i : 'n_i = 1 -> (i :=: socle_base i)%MS. | Proof.
move=> ni1; apply/eqmxP; rewrite andbC -mxrank_leqif_eq -/'n_i.
exact: component_mx_id (socle_simple i).
rewrite -(mxrankMfree _ gring_free) -genmxE.
by rewrite rank_Wedderburn_subring_pchar ni1.
Qed. | Lemma | linear_irr_comp_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"component_mx_id",
"eqmxP",
"genmxE",
"gring_free",
"mxrankMfree",
"mxrank_leqif_eq",
"rank_Wedderburn_subring_pchar",
"socle_base",
"socle_simple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_subring_center_pchar i : ('Z('R_i) :=: mxvec 'e_i)%MS. | Proof.
have [nz_e Re ideR idRe] := Wedderburn_is_id_pchar i.
have Ze: (mxvec 'e_i <= 'Z('R_i))%MS.
rewrite sub_capmx [(_ <= _)%MS]Re.
by apply/cent_mxP=> A R_A; rewrite ideR // idRe.
pose irrG := socle_irr i; set rG := socle_repr i in irrG.
pose E_G := enveloping_algebra_mx rG; have absG := splitG irrG.
apply/eqmxP... | Lemma | Wedderburn_subring_center_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"E_G",
"R_G",
"Re",
"Wedderburn_is_id_pchar",
"Wedderburn_sum_pchar",
"addnC",
"apply",
"capmxS",
"capmxSl",
"cent_mxP",
"enveloping_algebra_mx",
"eqmxP",
"eqn_leq",
"geq_leqif",
"gring_op",
"gring_opM",
"irrG",
"irr_comp_envelop_pchar",
"irr_reprK_pchar",
"ker_irr_comp_op_pcha... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Wedderburn_center_pchar :
('Z(R_G) :=: \matrix_(i < #|sG|) mxvec 'e_(enum_val i))%MS. | Proof.
have:= mxdirect_sums_center
Wedderburn_sum_pchar Wedderburn_direct Wedderburn_ideal.
move/eqmx_trans; apply; apply/eqmxP/andP; split.
apply/sumsmx_subP=> i _; rewrite Wedderburn_subring_center_pchar.
by apply: (eq_row_sub (enum_rank i)); rewrite rowK enum_rankK.
apply/row_subP=> i; rewrite rowK -Wedderburn... | Lemma | Wedderburn_center_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"R_G",
"Wedderburn_direct",
"Wedderburn_ideal",
"Wedderburn_subring_center_pchar",
"Wedderburn_sum_pchar",
"apply",
"enum_rank",
"enum_rankK",
"enum_val",
"eq_row_sub",
"eqmxP",
"eqmx_trans",
"mxdirect_sums_center",
"mxvec",
"rowK",
"row_subP",
"sG",
"split",
"sumsmx_subP",
"su... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_irr_pchar : #|sG| = tG. | Proof.
rewrite -(eqnP classg_base_free) classg_base_center.
have:= mxdirect_sums_center
Wedderburn_sum_pchar Wedderburn_direct Wedderburn_ideal.
move->; rewrite (mxdirectP _) /=.
apply/mxdirect_sumsP=> i _; apply/eqP; rewrite -submx0.
rewrite -{2}(mxdirect_sumsP Wedderburn_direct i) // capmxS ?capmxSl //=.
by a... | Lemma | card_irr_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"Wedderburn_direct",
"Wedderburn_ideal",
"Wedderburn_is_id_pchar",
"Wedderburn_subring_center_pchar",
"Wedderburn_sum_pchar",
"apply",
"capmxS",
"capmxSl",
"classg_base_center",
"classg_base_free",
"eq_bigr",
"eqnP",
"eqn_leq",
"lt0n",
"mxdirectP",
"mxdirect_sumsP",
"mxdirect_sums_ce... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
i0 | := Ordinal (irr_degree_gt0 i). | Let | i0 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_degree_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_mode x | := irr_repr i x i0 i0. | Definition | irr_mode | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"i0",
"irr_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_mode1 : irr_mode 1 = 1. | Proof. by rewrite /irr_mode repr_mx1 mxE eqxx. Qed. | Lemma | irr_mode1 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"eqxx",
"irr_mode",
"mxE",
"repr_mx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_center_scalar : {in 'Z(G), forall x, irr_repr i x = (irr_mode x)%:M}. | Proof.
rewrite /irr_mode => x /setIP[Gx cGx].
suffices [a ->]: exists a, irr_repr i x = a%:M by rewrite mxE eqxx.
apply/is_scalar_mxP; apply: (mx_abs_irr_cent_scalar (splitG (socle_irr i))).
by apply/centgmxP=> y Gy; rewrite -!{1}repr_mxM 1?(centP cGx).
Qed. | Lemma | irr_center_scalar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"centP",
"centgmxP",
"eqxx",
"irr_mode",
"irr_repr",
"is_scalar_mxP",
"mxE",
"mx_abs_irr_cent_scalar",
"repr_mxM",
"setIP",
"socle_irr",
"splitG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_modeM : {in 'Z(G) &, {morph irr_mode : x y / (x * y)%g >-> x * y}}. | Proof.
move=> x y Zx Zy; rewrite {1}/irr_mode repr_mxM ?(subsetP (center_sub G)) //.
by rewrite !irr_center_scalar // -scalar_mxM mxE eqxx.
Qed. | Lemma | irr_modeM | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"center_sub",
"eqxx",
"irr_center_scalar",
"irr_mode",
"mxE",
"repr_mxM",
"scalar_mxM",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_modeX n : {in 'Z(G), {morph irr_mode : x / (x ^+ n)%g >-> x ^+ n}}. | Proof.
elim: n => [|n IHn] x Zx; first exact: irr_mode1.
by rewrite expgS irr_modeM ?groupX // exprS IHn.
Qed. | Lemma | irr_modeX | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"expgS",
"exprS",
"groupX",
"irr_mode",
"irr_mode1",
"irr_modeM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_mode_unit : {in 'Z(G), forall x, irr_mode x \is a GRing.unit}. | Proof.
move=> x Zx /=; have:= unitr1 F.
by rewrite -irr_mode1 -(mulVg x) irr_modeM ?groupV // unitrM; case/andP=> _.
Qed. | Lemma | irr_mode_unit | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"groupV",
"irr_mode",
"irr_mode1",
"irr_modeM",
"mulVg",
"unit",
"unitr1",
"unitrM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_mode_neq0 : {in 'Z(G), forall x, irr_mode x != 0}. | Proof. by move=> x /irr_mode_unit; rewrite unitfE. Qed. | Lemma | irr_mode_neq0 | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr_mode",
"irr_mode_unit",
"unitfE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr_modeV : {in 'Z(G), {morph irr_mode : x / (x^-1)%g >-> x^-1}}. | Proof.
move=> x Zx /=; rewrite -[_^-1]mul1r; apply: canRL (mulrK (irr_mode_unit Zx)) _.
by rewrite -irr_modeM ?groupV // mulVg irr_mode1.
Qed. | Lemma | irr_modeV | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"apply",
"groupV",
"irr_mode",
"irr_mode1",
"irr_modeM",
"irr_mode_unit",
"mul1r",
"mulVg",
"mulrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
irr1_mode x : x \in G -> irr_mode 1 x = 1. | Proof. by move=> Gx; rewrite /irr_mode irr1_repr ?mxE. Qed. | Lemma | irr1_mode | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"irr1_repr",
"irr_mode",
"mxE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 1 sG ]" | := (principal_comp sG) : irrType_scope. | Notation | [ 1 sG ] | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"principal_comp",
"sG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_linear_irr (sG : irrType G) :
[pchar F]^'.-group G -> group_splitting_field G ->
#|linear_irr sG| = #|G : G^`(1)|%g. | Proof.
move=> F'G splitG; apply/eqP.
wlog sGq: / irrType (G / G^`(1))%G by apply: socle_exists.
have [_ nG'G] := andP (der_normal 1 G); apply/eqP; rewrite -card_quotient //.
have cGqGq: abelian (G / G^`(1))%g by apply: sub_der1_abelian.
have F'Gq: [pchar F]^'.-group (G / G^`(1))%g by apply: morphim_pgroup.
have splitGq... | Lemma | card_linear_irr | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"F'G",
"Sub",
"abelian",
"apply",
"card_quotient",
"coset",
"der_normal",
"eq_big",
"group",
"group_splitting_field",
"inE",
"insub",
"insubT",
"irr",
"irrG",
"irrType",
"irr_comp",
"irr_degreeE",
"irr_degree_abelian",
"irr_repr",
"last",
"linear_irr",
"mem_quotient",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
primitive_root_splitting_abelian (z : F) :
#|G|.-primitive_root z -> abelian G -> group_splitting_field G. | Proof.
move=> ozG cGG [|n] rG irrG; first by case/mx_irrP: irrG.
case: (pickP [pred x in G | ~~ is_scalar_mx (rG x)]) => [x | scalG].
case/andP=> Gx nscal_rGx; have: horner_mx (rG x) ('X^#|G| - 1) == 0.
rewrite rmorphB rmorphXn /= horner_mx_C horner_mx_X.
rewrite -repr_mxX ?inE // ((_ ^+ _ =P 1)%g _) ?repr_mx... | Lemma | primitive_root_splitting_abelian | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"abelian",
"addmx_sub",
"apply",
"big_nat_recr",
"big_nil",
"cGG",
"centgmxP",
"centsP",
"contraNneq",
"delta_mx",
"eq_sym",
"eqmx_opp",
"factor_Xn_sub_1",
"genmxE",
"group_splitting_field",
"horner_mx",
"horner_mx_C",
"horner_mx_X",
"inE",
"irrG",
"is_scalar_mx",
"is_scala... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cycle_repr_structure_pchar x (sG : irrType G) :
G :=: <[x]> -> [pchar F]^'.-group G -> group_splitting_field G ->
exists2 w : F, #|G|.-primitive_root w &
exists iphi : 'I_#|G| -> sG,
[/\ bijective iphi,
#|sG| = #|G|,
forall i, irr_mode (iphi i) x = w ^+ i
& forall i, irr_repr (iphi i) x = (w ^... | Proof.
move=> defG; rewrite {defG}(group_inj defG) -/#[x] in sG * => F'X splitF.
have Xx := cycle_id x; have cXX := cycle_abelian x.
have card_sG: #|sG| = #[x].
by rewrite card_irr_pchar //; apply/eqP; rewrite -card_classes_abelian.
have linX := irr_degree_abelian splitF cXX (_ : sG).
pose r (W : sG) := irr_mode W x.... | Lemma | cycle_repr_structure_pchar | group_representation | group_representation/mxrepresentation.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
... | [
"allP",
"apply",
"cardE",
"card_classes_abelian",
"card_image",
"card_irr_pchar",
"card_ord",
"center_idP",
"codom",
"cycleP",
"cycle_abelian",
"cycle_id",
"defG",
"def_r",
"enum",
"enum_uniq",
"expg_order",
"f_iinv",
"group",
"group_inj",
"group_splitting_field",
"has",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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