fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
sort_bseqr s := Bseq (sort_bseqP r s). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | sort_bseq | |
bseq0: all_equal_to ([bseq] : 0.-bseq T).
Proof. by move=> s; apply: val_inj; case: s => [[]]. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | bseq0 | |
Definitionbseq_hasDecEq n (T : eqType) :=
[Equality of n.-bseq T by <:]. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | Definition | |
bseq_predTypen (T : eqType) :=
Eval hnf in PredType (fun t : n.-bseq T => mem_seq t). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | bseq_predType | |
membsEn (T : eqType) (bs : n.-bseq T) : mem bs = mem (bseqval bs).
Proof. by []. Qed.
HB.instance Definition bseq_hasChoice n (T : choiceType) :=
[Choice of n.-bseq T by <:].
HB.instance Definition bseq_isCountable n (T : countType) :=
[Countable of n.-bseq T by <:]. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | membsE | |
bseq_tagged_tuplen T (s : n.-bseq T) : {k : 'I_n.+1 & k.-tuple T} :=
Tagged _ (in_tuple s : (Ordinal (size_bseq s : size s < n.+1)).-tuple _).
Arguments bseq_tagged_tuple {n T}. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | bseq_tagged_tuple | |
tagged_tuple_bseqn T (t : {k : 'I_n.+1 & k.-tuple T}) : n.-bseq T :=
widen_bseq (leq_ord (tag t)) (tagged t).
Arguments tagged_tuple_bseq {n T}. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tagged_tuple_bseq | |
bseq_tagged_tupleK{n T} :
cancel (@bseq_tagged_tuple n T) tagged_tuple_bseq.
Proof. by move=> bs; apply/val_inj. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | bseq_tagged_tupleK | |
tagged_tuple_bseqK{n T} :
cancel (@tagged_tuple_bseq n T) bseq_tagged_tuple.
Proof.
move=> [[k lt_kn] t]; apply: eq_existT_curried => [|k_eq]; apply/val_inj.
by rewrite /= size_tuple.
by refine (let: erefl := k_eq in _).
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tagged_tuple_bseqK | |
bseq_tagged_tuple_bij{n T} : bijective (@bseq_tagged_tuple n T).
Proof. exact/Bijective/tagged_tuple_bseqK/bseq_tagged_tupleK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | bseq_tagged_tuple_bij | |
tagged_tuple_bseq_bij{n T} : bijective (@tagged_tuple_bseq n T).
Proof. exact/Bijective/bseq_tagged_tupleK/tagged_tuple_bseqK. Qed.
#[global] Hint Resolve bseq_tagged_tuple_bij tagged_tuple_bseq_bij : core.
#[non_forgetful_inheritance]
HB.instance Definition _ n (T : finType) := isFinite.Build (n.-bseq T)
(pcan_enumP (can_pcan (@bseq_tagged_tupleK n T))). | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat",
"From mathcomp Require Import seq choice fintype path"
] | boot/tuple.v | tagged_tuple_bseq_bij | |
groupC: group_closure_field algC gT.
Proof. exact: group_closure_closed_field. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | groupC | |
trow(n1 : nat) :
forall (A : 'rV[F]_n1) m2 n2 (B : 'M[F]_(m2,n2)), 'M[F]_(m2,n1 * n2) :=
if n1 is n'1.+1
then
fun (A : 'M[F]_(1,(1 + n'1))) m2 n2 (B : 'M[F]_(m2,n2)) =>
(row_mx (lsubmx A 0 0 *: B) (trow (rsubmx A) B))
else (fun _ _ _ _ => 0). | Fixpoint | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | trow | |
trow0n1 m2 n2 B : @trow n1 0 m2 n2 B = 0.
Proof.
elim: n1=> //= n1 IH.
rewrite !mxE scale0r linear0.
rewrite IH //; apply/matrixP=> i j; rewrite !mxE.
by case: split=> *; rewrite mxE.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | trow0 | |
trowbn1 m2 n2 B A := @trow n1 A m2 n2 B. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | trowb | |
trowbEn1 m2 n2 A B : trowb B A = @trow n1 A m2 n2 B.
Proof. by []. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | trowbE | |
trowb_is_linearn1 m2 n2 (B : 'M_(m2,n2)) : linear (@trowb n1 m2 n2 B).
Proof.
elim: n1=> [|n1 IH] //= k A1 A2 /=; first by rewrite scaler0 add0r.
rewrite !linearD /= !linearZ /= IH 2!mxE.
by rewrite scalerDl -scalerA -add_row_mx -scale_row_mx.
Qed.
HB.instance Definition _ n1 m2 n2 B :=
GRing.isSemilinear.Build _ _ _ _ (trowb B)
(GRing.semilinear_linear (@trowb_is_linear n1 m2 n2 B)). | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | trowb_is_linear | |
trow_is_linearn1 m2 n2 (A : 'rV_n1) : linear (@trow n1 A m2 n2).
Proof.
elim: n1 A => [|n1 IH] //= A k A1 A2 /=; first by rewrite scaler0 add0r.
rewrite linearP /=; apply/matrixP=> i j; rewrite !mxE.
by case: split=> a; rewrite ?IH !mxE.
Qed.
HB.instance Definition _ n1 m2 n2 A :=
GRing.isSemilinear.Build _ _ _ _ (@trow n1 A m2 n2)
(GRing.semilinear_linear (@trow_is_linear n1 m2 n2 A)). | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | trow_is_linear | |
tprod(m1 : nat) :
forall n1 (A : 'M[F]_(m1,n1)) m2 n2 (B : 'M[F]_(m2,n2)),
'M[F]_(m1 * m2,n1 * n2) :=
if m1 is m'1.+1
return forall n1 (A : 'M[F]_(m1,n1)) m2 n2 (B : 'M[F]_(m2,n2)),
'M[F]_(m1 * m2,n1 * n2)
then
fun n1 (A : 'M[F]_(1 + m'1,n1)) m2 n2 B =>
(col_mx (trow (usubmx A) B) (tprod (dsubmx A) B))
else (fun _ _ _ _ _ => 0). | Fixpoint | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | tprod | |
dsumx_mulm1 m2 n p A B :
dsubmx ((A *m B) : 'M[F]_(m1 + m2, n)) = dsubmx (A : 'M_(m1 + m2, p)) *m B.
Proof.
apply/matrixP=> i j /[!mxE]; apply: eq_bigr=> k _.
by rewrite !mxE.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | dsumx_mul | |
usumx_mulm1 m2 n p A B :
usubmx ((A *m B) : 'M[F]_(m1 + m2, n)) = usubmx (A : 'M_(m1 + m2, p)) *m B.
Proof.
by apply/matrixP=> i j /[!mxE]; apply: eq_bigr=> k _ /[!mxE].
Qed.
Let trow_mul (m1 m2 n2 p2 : nat)
(A : 'rV_m1) (B1: 'M[F]_(m2,n2)) (B2 :'M[F]_(n2,p2)) :
trow A (B1 *m B2) = B1 *m trow A B2.
Proof.
elim: m1 A => [|m1 IH] A /=; first by rewrite mulmx0.
by rewrite IH mul_mx_row -scalemxAr.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | usumx_mul | |
tprodEm1 n1 p1 (A1 :'M[F]_(m1,n1)) (A2 :'M[F]_(n1,p1))
m2 n2 p2 (B1 :'M[F]_(m2,n2)) (B2 :'M[F]_(n2,p2)) :
tprod (A1 *m A2) (B1 *m B2) = (tprod A1 B1) *m (tprod A2 B2).
Proof.
elim: m1 n1 p1 A1 A2 m2 n2 p2 B1 B2 => /= [|m1 IH].
by move=> *; rewrite mul0mx.
move=> n1 p1 A1 A2 m2 n2 p2 B1 B2.
rewrite mul_col_mx -IH.
congr col_mx; last by rewrite dsumx_mul.
rewrite usumx_mul.
elim: n1 {A1}(usubmx (A1: 'M_(1 + m1, n1))) p1 A2=> //= [u p1 A2|].
by rewrite [A2](flatmx0) !mulmx0 -trowbE linear0.
move=> n1 IH1 A p1 A2 //=.
set Al := lsubmx _; set Ar := rsubmx _.
set Su := usubmx _; set Sd := dsubmx _.
rewrite mul_row_col -IH1.
rewrite -{1}(@hsubmxK F 1 1 n1 A).
rewrite -{1}(@vsubmxK F 1 n1 p1 A2).
rewrite (@mul_row_col F 1 1 n1 p1).
rewrite -trowbE linearD /= trowbE -/Al.
congr (_ + _).
rewrite {1}[Al]mx11_scalar mul_scalar_mx.
by rewrite -trowbE linearZ /= trowbE -/Su trow_mul scalemxAl.
Qed.
Let tprod_tr m1 n1 (A :'M[F]_(m1, 1 + n1)) m2 n2 (B :'M[F]_(m2, n2)) :
tprod A B = row_mx (trow (lsubmx A)^T B^T)^T (tprod (rsubmx A) B).
Proof.
elim: m1 n1 A m2 n2 B=> [|m1 IH] n1 A m2 n2 B //=.
by rewrite trmx0 row_mx0.
rewrite !IH.
pose A1 := A : 'M_(1 + m1, 1 + n1).
have F1: dsubmx (rsubmx A1) = rsubmx (dsubmx A1).
by apply/matrixP=> i j; rewrite !mxE.
have F2: rsubmx (usubmx A1) = usubmx (rsubmx A1).
by apply/matrixP=> i j; rewrite !mxE.
have F3: lsubmx (dsubmx A1) = dsubmx (lsubmx A1).
by apply/matrixP=> i j; rewrite !mxE.
rewrite tr_row_mx -block_mxEv -block_mxEh
... | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | tprodE | |
tprod1m n : tprod (1%:M : 'M[F]_(m,m)) (1%:M : 'M[F]_(n,n)) = 1%:M.
Proof.
elim: m n => [|m IH] n //=; first by rewrite [1%:M]flatmx0.
rewrite tprod_tr.
set u := rsubmx _; have->: u = 0.
apply/matrixP=> i j; rewrite !mxE.
by case: i; case: j=> /= j Hj; case.
set v := lsubmx (dsubmx _); have->: v = 0.
apply/matrixP=> i j; rewrite !mxE.
by case: i; case: j; case.
set w := rsubmx _; have->: w = 1%:M.
apply/matrixP=> i j; rewrite !mxE.
by case: i; case: j; case.
rewrite IH -!trowbE !linear0.
rewrite -block_mxEv.
set z := (lsubmx _) 0 0; have->: z = 1.
by rewrite /z !mxE eqxx.
by rewrite scale1r scalar_mx_block.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | tprod1 | |
mxtrace_prodm n (A :'M[F]_(m)) (B :'M[F]_(n)) :
\tr (tprod A B) = \tr A * \tr B.
Proof.
elim: m n A B => [|m IH] n A B //=.
by rewrite [A]flatmx0 mxtrace0 mul0r.
rewrite tprod_tr -block_mxEv mxtrace_block IH.
rewrite linearZ/= -mulrDl -trace_mx11; congr (_ * _).
pose A1 := A : 'M_(1 + m).
rewrite -[A in RHS](@submxK _ 1 m 1 m A1).
by rewrite (@mxtrace_block _ _ _ (ulsubmx A1)).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | mxtrace_prod | |
representation:=
Representation {rdegree; mx_repr_of_repr :> reprG rdegree}. | Record | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | representation | |
mx_repr0: mx_repr G (fun _ : gT => 1%:M : 'M[R]_0).
Proof. by split=> // g h Hg Hx; rewrite mulmx1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | mx_repr0 | |
grepr0:= Representation (MxRepresentation mx_repr0). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | grepr0 | |
add_mx_repr(rG1 rG2 : representation) :
mx_repr G (fun g => block_mx (rG1 g) 0 0 (rG2 g)).
Proof.
split=> [|x y Hx Hy]; first by rewrite !repr_mx1 -scalar_mx_block.
by rewrite mulmx_block !(mulmx0, mul0mx, addr0, add0r, repr_mxM).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | add_mx_repr | |
dadd_greprrG1 rG2 :=
Representation (MxRepresentation (add_mx_repr rG1 rG2)). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | dadd_grepr | |
mx_rsim_dadd(U V W : 'M_n) (rU rV : representation)
(modU : mxmodule rG U) (modV : mxmodule rG V) (modW : mxmodule rG W) :
(U + V :=: W)%MS -> mxdirect (U + V) ->
mx_rsim (submod_repr modU) rU -> mx_rsim (submod_repr modV) rV ->
mx_rsim (submod_repr modW) (dadd_grepr rU rV).
Proof.
case: rU; case: rV=> nV rV nU rU defW dxUV /=.
have tiUV := mxdirect_addsP dxUV.
move=> [fU def_nU]; rewrite -{nU}def_nU in rU fU * => inv_fU hom_fU.
move=> [fV def_nV]; rewrite -{nV}def_nV in rV fV * => inv_fV hom_fV.
pose pU := in_submod U (proj_mx U V) *m fU.
pose pV := in_submod V (proj_mx V U) *m fV.
exists (val_submod 1%:M *m row_mx pU pV) => [||g Gg].
- by rewrite -defW (mxdirectP dxUV).
- apply/row_freeP.
pose pU' := invmx fU *m val_submod 1%:M.
pose pV' := invmx fV *m val_submod 1%:M.
exists (in_submod _ (col_mx pU' pV')).
rewrite in_submodE mulmxA -in_submodE -mulmxA mul_row_col mulmxDr.
rewrite -[pU *m _]mulmxA -[pV *m _]mulmxA !mulKVmx -?row_free_unit //.
rewrite addrC (in_submodE V) 2![val_submod 1%:M *m _]mulmxA -in_submodE.
rewrite addrC (in_submodE U) 2![val_submod 1%:M *m _ in X in X + _]mulmxA.
rewrite -in_submodE -!val_submodE !in_submodK ?proj_mx_sub //.
by rewrite add_proj_mx ?val_submodK // val_submod1 defW.
rewrite mulmxA -val_submodE -[submod_repr _ g]mul1mx val_submodJ //.
rewrite -(mulmxA _ (rG g)) mul_mx_row -[in RHS]mulmxA mul_row_block.
rewrite !mulmx0 addr0 add0r !mul_mx_row.
set W' := val_submod 1%:M; congr (row_mx _ _).
rewrite 3!mulmxA in_submo
... | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | mx_rsim_dadd | |
mx_rsim_dsum(I : finType) (P : pred I) U rU (W : 'M_n)
(modU : forall i, mxmodule rG (U i)) (modW : mxmodule rG W) :
let S := (\sum_(i | P i) U i)%MS in (S :=: W)%MS -> mxdirect S ->
(forall i, mx_rsim (submod_repr (modU i)) (rU i : representation)) ->
mx_rsim (submod_repr modW) (\big[dadd_grepr/grepr0]_(i | P i) rU i).
Proof.
move=> /= defW dxW rsimU.
rewrite mxdirectE /= -!(big_filter _ P) in dxW defW *.
elim: {P}(filter P _) => [|i e IHe] in W modW dxW defW *.
rewrite !big_nil /= in defW *.
by exists 0 => [||? _]; rewrite ?mul0mx ?mulmx0 // /row_free -defW !mxrank0.
rewrite !big_cons /= in dxW defW *.
rewrite 2!(big_nth i) !big_mkord /= in IHe dxW defW.
set Wi := (\sum_i _)%MS in defW dxW IHe.
rewrite -mxdirectE mxdirect_addsE !mxdirectE eqxx /= -/Wi in dxW.
have modWi: mxmodule rG Wi by apply: sumsmx_module.
case/andP: dxW; move/(IHe Wi modWi) {IHe}; move/(_ (eqmx_refl _))=> rsimWi.
by move/eqP; move/mxdirect_addsP=> dxUiWi; apply: mx_rsim_dadd (rsimU i) rsimWi.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | mx_rsim_dsum | |
muln_greprrW k := \big[dadd_grepr/grepr0]_(i < k) rW. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | muln_grepr | |
mx_rsim_socle(sG : socleType rG) (W : sG) (rW : representation) :
let modW : mxmodule rG W := component_mx_module rG (socle_base W) in
mx_rsim (socle_repr W) rW ->
mx_rsim (submod_repr modW) (muln_grepr rW (socle_mult W)).
Proof.
set M := socle_base W => modW rsimM.
have simM: mxsimple rG M := socle_simple W.
have rankM_gt0: (\rank M > 0)%N by rewrite lt0n mxrank_eq0; case: simM.
have [I /= U_I simU]: mxsemisimple rG W by apply: component_mx_semisimple.
pose U (i : 'I_#|I|) := U_I (enum_val i).
have reindexI := reindex _ (onW_bij I (enum_val_bij I)).
rewrite mxdirectE /= !reindexI -mxdirectE /= => defW dxW.
have isoU: forall i, mx_iso rG M (U i).
move=> i; have sUiW: (U i <= W)%MS by rewrite -defW (sumsmx_sup i).
exact: component_mx_iso (simU _) sUiW.
have ->: socle_mult W = #|I|.
rewrite -(mulnK #|I| rankM_gt0); congr (_ %/ _)%N.
rewrite -defW (mxdirectP dxW) /= -sum_nat_const reindexI /=.
by apply: eq_bigr => i _; rewrite -(mxrank_iso (isoU i)).
have modU: mxmodule rG (U _) := mxsimple_module (simU _).
suff: mx_rsim (submod_repr (modU _)) rW by apply: mx_rsim_dsum defW dxW.
by move=> i; apply: mx_rsim_trans (mx_rsim_sym _) rsimM; apply/mx_rsim_iso.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | mx_rsim_socle | |
prod_mx_repr: mx_repr G (fun g => tprod (rG1 g) (rG2 g)).
Proof.
split=>[|i j InG JnG]; first by rewrite !repr_mx1 tprod1.
by rewrite !repr_mxM // tprodE.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | prod_mx_repr | |
prod_repr:= MxRepresentation prod_mx_repr. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | prod_repr | |
prod_repr_linn2 (rG1 : reprG 1) (rG2 : reprG n2) :
{in G, forall x, let cast_n2 := esym (mul1n n2) in
prod_repr rG1 rG2 x = castmx (cast_n2, cast_n2) (rG1 x 0 0 *: rG2 x)}.
Proof.
move=> x Gx /=; set cast_n2 := esym _; rewrite /prod_repr /= !mxE !lshift0.
apply/matrixP=> i j; rewrite castmxE /=.
do 2![rewrite mxE; case: splitP => [? ? | []//]].
by congr ((_ *: rG2 x) _ _); apply: val_inj.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | prod_repr_lin | |
cfReprn rG := Cfun 0 (@cfRepr_subproof n rG). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfRepr | |
cfRepr1n rG : @cfRepr n rG 1%g = n%:R.
Proof. by rewrite cfunE group1 repr_mx1 mxtrace1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfRepr1 | |
cfRepr_simn1 n2 rG1 rG2 :
mx_rsim rG1 rG2 -> @cfRepr n1 rG1 = @cfRepr n2 rG2.
Proof.
case/mx_rsim_def=> f12 [f21] fK def_rG1; apply/cfun_inP=> x Gx.
by rewrite !cfunE def_rG1 // mxtrace_mulC mulmxA fK mul1mx.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfRepr_sim | |
cfRepr0: cfRepr grepr0 = 0.
Proof. by apply/cfun_inP=> x Gx; rewrite !cfunE Gx mxtrace1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfRepr0 | |
cfRepr_daddrG1 rG2 :
cfRepr (dadd_grepr rG1 rG2) = cfRepr rG1 + cfRepr rG2.
Proof. by apply/cfun_inP=> x Gx; rewrite !cfunE Gx mxtrace_block. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfRepr_dadd | |
cfRepr_dsumI r (P : pred I) rG :
cfRepr (\big[dadd_grepr/grepr0]_(i <- r | P i) rG i)
= \sum_(i <- r | P i) cfRepr (rG i).
Proof. exact: (big_morph _ cfRepr_dadd cfRepr0). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfRepr_dsum | |
cfRepr_mulnrG k : cfRepr (muln_grepr rG k) = cfRepr rG *+ k.
Proof. by rewrite cfRepr_dsum /= sumr_const card_ord. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfRepr_muln | |
standard_irr(W : sG) := irr_comp iG (socle_repr W). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | standard_irr | |
standard_soclei := pick [pred W | standard_irr W == i].
Local Notation soc := standard_socle. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | standard_socle | |
standard_irr_coefi := oapp (fun W => socle_mult W) 0 (soc i). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | standard_irr_coef | |
standard_grepr:=
\big[dadd_grepr/grepr0]_i
muln_grepr (Representation (socle_repr i)) (standard_irr_coef i). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | standard_grepr | |
mx_rsim_standard: mx_rsim rG standard_grepr.
Proof.
pose W i := oapp val 0 (soc i); pose S := (\sum_i W i)%MS.
have C'G: [pchar algC]^'.-group G := algC'G_pchar G.
have [defS dxS]: (S :=: 1%:M)%MS /\ mxdirect S.
rewrite /S mxdirectE /= !(bigID soc xpredT) /=.
rewrite addsmxC big1 => [|i]; last by rewrite /W; case (soc i).
rewrite adds0mx_id addnC (@big1 nat) ?add0n => [|i]; last first.
by rewrite /W; case: (soc i); rewrite ?mxrank0.
have <-: Socle sG = 1%:M := reducible_Socle1 sG (mx_Maschke_pchar rG C'G).
have [W0 _ | noW] := pickP sG; last first.
suff no_i: (soc : pred iG) =1 xpred0 by rewrite /Socle !big_pred0 ?mxrank0.
by move=> i; rewrite /soc; case: pickP => // W0; have:= noW W0.
have irrK Wi: soc (standard_irr Wi) = Some Wi.
rewrite /soc; case: pickP => [W' | /(_ Wi)] /= /eqP // eqWi.
apply/eqP/socle_rsimP.
apply: mx_rsim_trans
(rsim_irr_comp_pchar iG C'G (socle_irr _)) (mx_rsim_sym _).
by rewrite [irr_comp _ _]eqWi; apply: rsim_irr_comp_pchar (socle_irr _).
have bij_irr: {on [pred i | soc i], bijective standard_irr}.
exists (odflt W0 \o soc) => [Wi _ | i]; first by rewrite /= irrK.
by rewrite inE /soc /=; case: pickP => //= Wi; move/eqP.
rewrite !(reindex standard_irr) {bij_irr}//=.
have all_soc Wi: soc (standard_irr Wi) by rewrite irrK.
rewrite (eq_bigr val) => [|Wi _]; last by rewrite /W irrK.
rewrite !(eq_bigl _ _ all_soc); split=> //.
rewrite (eq_bigr (mxrank \o val)) => [|Wi _]; last by rewrite /W irrK.
by rewri
... | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | mx_rsim_standard | |
cfReg(B : {set gT}) : 'CF(B) := #|B|%:R *: '1_[1]. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfReg | |
cfRegEx : @cfReg G x = #|G|%:R *+ (x == 1%g).
Proof. by rewrite cfunE cfuniE ?normal1 // inE mulr_natr. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfRegE | |
cfReprReg: cfRepr (regular_repr algC G) = cfReg G.
Proof.
apply/cfun_inP=> x Gx; rewrite cfRegE.
have [-> | ntx] := eqVneq x 1%g; first by rewrite cfRepr1.
rewrite cfunE Gx [\tr _]big1 // => i _; rewrite 2!mxE /=.
rewrite -(inj_eq enum_val_inj) gring_indexK ?groupM ?enum_valP //.
by rewrite eq_mulVg1 mulKg (negbTE ntx).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfReprReg | |
xcfun(chi : 'CF(G)) A :=
(gring_row A *m (\col_(i < #|G|) chi (enum_val i))) 0 0. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | xcfun | |
xcfun_is_zmod_morphismphi : zmod_morphism (xcfun phi).
Proof. by move=> A B; rewrite /xcfun [gring_row _]linearB mulmxBl !mxE. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `xcfun_is_zmod_morphism` instead")] | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | xcfun_is_zmod_morphism | |
xcfun_is_additive:= xcfun_is_zmod_morphism.
HB.instance Definition _ phi :=
GRing.isZmodMorphism.Build 'M_(gcard G) _ (xcfun phi) (xcfun_is_zmod_morphism phi). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | xcfun_is_additive | |
xcfunZra phi A : xcfun phi (a *: A) = a * xcfun phi A.
Proof. by rewrite /xcfun linearZ -scalemxAl mxE. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | xcfunZr | |
xcfun_rA phi := xcfun phi A.
Arguments xcfun_r A phi /. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | xcfun_r | |
xcfun_rEA chi : xcfun_r A chi = xcfun chi A. Proof. by []. Qed.
Fact xcfun_r_is_zmod_morphism A : zmod_morphism (xcfun_r A).
Proof.
move=> phi psi; rewrite /= /xcfun !mxE -sumrB; apply: eq_bigr => i _.
by rewrite !mxE !cfunE mulrBr.
Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `xcfun_r_is_zmod_morphism` instead")] | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | xcfun_rE | |
xcfun_r_is_additive:= xcfun_r_is_zmod_morphism.
HB.instance Definition _ A := GRing.isZmodMorphism.Build _ _ (xcfun_r A)
(xcfun_r_is_zmod_morphism A). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | xcfun_r_is_additive | |
xcfunZla phi A : xcfun (a *: phi) A = a * xcfun phi A.
Proof.
rewrite /xcfun !mxE big_distrr; apply: eq_bigr => i _ /=.
by rewrite !mxE cfunE mulrCA.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | xcfunZl | |
xcfun_reprn rG A : xcfun (@cfRepr n rG) A = \tr (gring_op rG A).
Proof.
rewrite gring_opE [gring_row A]row_sum_delta !linear_sum /xcfun !mxE.
apply: eq_bigr => i _; rewrite !mxE /= !linearZ cfunE enum_valP /=.
by congr (_ * \tr _); rewrite {A}/gring_mx /= -rowE rowK mxvecK.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | xcfun_repr | |
pred_NirrgT B := #|@classes gT B|.-1.
Arguments pred_Nirr {gT} B%_g. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | pred_Nirr | |
NirrG := (pred_Nirr G).+1. | Notation | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | Nirr | |
IirrG := 'I_(Nirr G). | Notation | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | Iirr | |
NirrE: Nirr G = #|classes G|.
Proof. by rewrite /pred_Nirr (cardD1 [1]) classes1. Qed.
Fact Iirr_cast : Nirr G = #|sG|.
Proof. by rewrite NirrE ?card_irr_pchar ?algC'G_pchar //; apply: groupC. Qed.
Let offset := cast_ord (esym Iirr_cast) (enum_rank [1 sG]%irr). | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | NirrE | |
socle_of_Iirr(i : Iirr G) : sG :=
enum_val (cast_ord Iirr_cast (i + offset)). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | socle_of_Iirr | |
irr_of_socle(Wi : sG) : Iirr G :=
cast_ord (esym Iirr_cast) (enum_rank Wi) - offset.
Local Notation W := socle_of_Iirr. | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr_of_socle | |
socle_Iirr0: W 0 = [1 sG]%irr.
Proof. by rewrite /W add0r cast_ordKV enum_rankK. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | socle_Iirr0 | |
socle_of_IirrK: cancel W irr_of_socle.
Proof. by move=> i; rewrite /irr_of_socle enum_valK cast_ordK addrK. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | socle_of_IirrK | |
irr_of_socleK: cancel irr_of_socle W.
Proof. by move=> Wi; rewrite /W subrK cast_ordKV enum_rankK. Qed.
Hint Resolve socle_of_IirrK irr_of_socleK : core. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr_of_socleK | |
irr_of_socle_bij(A : {pred (Iirr G)}) : {on A, bijective irr_of_socle}.
Proof. by apply: onW_bij; exists W. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr_of_socle_bij | |
socle_of_Iirr_bij(A : {pred sG}) : {on A, bijective W}.
Proof. by apply: onW_bij; exists irr_of_socle. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | socle_of_Iirr_bij | |
congr_irri1 i2 : i1 = i2 -> 'chi_i1 = 'chi_i2. Proof. by move->. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | congr_irr | |
Iirr1_neq0: G :!=: 1%g -> inord 1 != 0 :> Iirr G.
Proof. by rewrite -classes_gt1 -NirrE -val_eqE /= => /inordK->. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | Iirr1_neq0 | |
has_nonprincipal_irr: G :!=: 1%g -> {i : Iirr G | i != 0}.
Proof. by move/Iirr1_neq0; exists (inord 1). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | has_nonprincipal_irr | |
irrRepri : cfRepr 'Chi_i = 'chi_i.
Proof.
rewrite irr.unlock (tnth_nth 0) nth_mkseq // -[<<G>>]/(gval _) genGidG.
by rewrite cfRes_id inord_val.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irrRepr | |
irr0: 'chi[G]_0 = 1.
Proof.
apply/cfun_inP=> x Gx; rewrite -irrRepr cfun1E cfunE Gx.
by rewrite socle_Iirr0 irr1_repr // mxtrace1 degree_irr1.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr0 | |
cfun1_irr: 1 \in irr G.
Proof. by rewrite -irr0 mem_tnth. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfun1_irr | |
mem_irri : 'chi_i \in irr G.
Proof. exact: mem_tnth. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | mem_irr | |
irrPxi : reflect (exists i, xi = 'chi_i) (xi \in irr G).
Proof.
apply: (iffP idP) => [/(nthP 0)[i] | [i ->]]; last exact: mem_irr.
rewrite size_tuple => lt_i_G <-.
by exists (Ordinal lt_i_G); rewrite (tnth_nth 0).
Qed.
Let sG := DecSocleType (regular_repr algC G).
Let C'G := algC'G_pchar G.
Let closG := @groupC _ G.
Local Notation W i := (@socle_of_Iirr _ G i).
Local Notation "''n_' i" := 'n_(W i).
Local Notation "''R_' i" := 'R_(W i).
Local Notation "''e_' i" := 'e_(W i). | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irrP | |
irr1_degreei : 'chi_i 1%g = ('n_i)%:R.
Proof. by rewrite -irrRepr cfRepr1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr1_degree | |
Cnat_irr1i : 'chi_i 1%g \in Num.nat.
Proof. by rewrite irr1_degree rpred_nat. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | Cnat_irr1 | |
irr1_gt0i : 0 < 'chi_i 1%g.
Proof. by rewrite irr1_degree ltr0n irr_degree_gt0. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr1_gt0 | |
irr1_neq0i : 'chi_i 1%g != 0.
Proof. by rewrite eq_le lt_geF ?irr1_gt0. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr1_neq0 | |
irr_neq0i : 'chi_i != 0.
Proof. by apply: contraNneq (irr1_neq0 i) => ->; rewrite cfunE. Qed.
Local Remark cfIirr_key : unit. Proof. by []. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr_neq0 | |
cfIirr: forall B, 'CF(B) -> Iirr B :=
locked_with cfIirr_key (fun B chi => inord (index chi (irr B))). | Definition | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfIirr | |
cfIirrEchi : chi \in irr G -> 'chi_(cfIirr chi) = chi.
Proof.
move=> chi_irr; rewrite (tnth_nth 0) [cfIirr]unlock inordK ?nth_index //.
by rewrite -index_mem size_tuple in chi_irr.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfIirrE | |
cfIirrPEJ (f : J -> 'CF(G)) (P : pred J) :
(forall j, P j -> f j \in irr G) ->
forall j, P j -> 'chi_(cfIirr (f j)) = f j.
Proof. by move=> irr_f j /irr_f; apply: cfIirrE. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfIirrPE | |
cfReg_sum: cfReg G = \sum_i 'chi_i 1%g *: 'chi_i.
Proof.
apply/cfun_inP=> x Gx.
rewrite -cfReprReg cfunE Gx (mxtrace_regular_pchar sG) //=.
rewrite sum_cfunE (reindex _ (socle_of_Iirr_bij _)); apply: eq_bigr => i _.
by rewrite -irrRepr cfRepr1 !cfunE Gx mulr_natl.
Qed.
Let aG := regular_repr algC G.
Let R_G := group_ring algC G. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfReg_sum | |
xcfun_annihilatei j A : i != j -> (A \in 'R_j)%MS -> ('chi_i).[A]%CF = 0.
Proof.
move=> neq_ij RjA; rewrite -irrRepr xcfun_repr.
rewrite (irr_repr'_op0_pchar _ _ RjA) ?raddf0 //.
by rewrite eq_sym (can_eq socle_of_IirrK).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | xcfun_annihilate | |
xcfunGphi x : x \in G -> phi.[aG x]%CF = phi x.
Proof.
by move=> Gx; rewrite /xcfun /gring_row rowK -rowE !mxE !(gring_indexK, mul1g).
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | xcfunG | |
xcfun_mul_idi A :
(A \in R_G)%MS -> ('chi_i).['e_i *m A]%CF = ('chi_i).[A]%CF.
Proof.
move=> RG_A; rewrite -irrRepr !xcfun_repr gring_opM //.
by rewrite op_Wedderburn_id_pchar ?mul1mx.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | xcfun_mul_id | |
xcfun_idi j : ('chi_i).['e_j]%CF = 'chi_i 1%g *+ (i == j).
Proof.
have [<-{j} | /xcfun_annihilate->//] := eqVneq; last exact: Wedderburn_id_mem.
by rewrite -xcfunG // repr_mx1 -(xcfun_mul_id _ (envelop_mx1 _)) mulmx1.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | xcfun_id | |
irr_free: free (irr G).
Proof.
apply/freeP=> s s0 i; apply: (mulIf (irr1_neq0 i)).
rewrite mul0r -(raddf0 (xcfun_r 'e_i)) -{}s0 raddf_sum /=.
rewrite (bigD1 i)//= -tnth_nth xcfunZl xcfun_id eqxx big1 ?addr0 // => j ne_ji.
by rewrite -tnth_nth xcfunZl xcfun_id (negbTE ne_ji) mulr0.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr_free | |
irr_inj: injective (tnth (irr G)).
Proof. by apply/injectiveP/free_uniq; rewrite map_tnth_enum irr_free. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr_inj | |
irrK: cancel (tnth (irr G)) (@cfIirr G).
Proof. by move=> i; apply: irr_inj; rewrite cfIirrE ?mem_irr. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irrK | |
irr_eq1i : ('chi_i == 1) = (i == 0).
Proof. by rewrite -irr0 (inj_eq irr_inj). Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr_eq1 | |
cforder_irr_eq1i : (#['chi_i]%CF == 1) = (i == 0).
Proof. by rewrite -dvdn1 dvdn_cforder irr_eq1. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cforder_irr_eq1 | |
irr_basis: basis_of 'CF(G)%VS (irr G).
Proof.
rewrite /basis_of irr_free andbT -dimv_leqif_eq ?subvf //.
by rewrite dim_cfun (eqnP irr_free) size_tuple NirrE.
Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | irr_basis | |
eq_sum_nth_irra : \sum_i a i *: 'chi[G]_i = \sum_i a i *: (irr G)`_i.
Proof. by apply: eq_bigr => i; rewrite -tnth_nth. Qed. | Lemma | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | eq_sum_nth_irr | |
cfun_irr_sumphi : {a | phi = \sum_i a i *: 'chi[G]_i}.
Proof.
rewrite (coord_basis irr_basis (memvf phi)) -eq_sum_nth_irr.
by exists ((coord (irr G))^~ phi).
Qed. | Theorem | character | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype choice ssrnat seq",
"From mathcomp Require Import path div fintype tuple finfun bigop prime order",
"From mathcomp Require Import ssralg poly finset gproduct fingroup morphism",
"From mathcomp Require Import p... | character/character.v | cfun_irr_sum |
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