(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
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(******************************************************************************)
(*  This file provides linkage between classic Group Theory and commutative   *)
(* algebra -- representation theory. Since general abstract linear algebra is *)
(* still being sorted out, we develop the required theory here on the         *)
(* assumption that all vector spaces are matrix spaces, indeed that most are  *)
(* row matrix spaces; our representation theory is specialized to the latter  *)
(* case. We provide many definitions and results of representation theory:    *)
(* enveloping algebras, reducible, irreducible and absolutely irreducible     *)
(* representations, representation centralisers, submodules and kernels,      *)
(* simple and semisimple modules, the Schur lemmas, Maschke's theorem,        *)
(* components, socles, homomorphisms and isomorphisms, the Jacobson density   *)
(* theorem, similar representations, the Jordan-Holder theorem, Clifford's    *)
(* theorem and Wedderburn components, regular representations and the         *)
(* Wedderburn structure theorem for semisimple group rings, and the           *)
(* construction of a splitting field of an irreducible representation, and of *)
(* reduced, tensored, and factored representations.                           *)
(* mx_representation F G n == the Structure type for representations of G     *)
(*                 with n x n matrices with coefficients in F. Note that      *)
(*                 rG : mx_representation F G n coerces to a function from    *)
(*                 the element type of G to 'M_n, and conversely all such     *)
(*                 functions have a Canonical mx_representation.              *)
(*  mx_repr G r <-> r : gT -> 'M_n defines a (matrix) group representation    *)
(*                 on G : {set gT} (Prop predicate).                          *)
(* enveloping_algebra_mx rG == a #|G| x (n ^ 2) matrix whose rows are the     *)
(*                 mxvec encodings of the image of G under rG, and whose      *)
(*                 row space therefore encodes the enveloping algebra of      *)
(*                 the representation of G.                                   *)
(*      rker rG == the kernel of the representation of r on G, i.e., the      *)
(*                 subgroup of elements of G mapped to the identity by rG.    *)
(* mx_faithful rG == the representation rG of G is faithful (its kernel is    *)
(*                 trivial).                                                  *)
(* rfix_mx rG H == an n x n matrix whose row space is the set of vectors      *)
(*                 fixed (centralised) by the representation of H by rG.      *)
(*   rcent rG A == the subgroup of G whose representation via rG commutes     *)
(*                 with the square matrix A.                                  *)
(*   rcenter rG == the subgroup of G whose representation via rG consists of  *)
(*                 scalar matrices.                                           *)
(* centgmx rG f <=> f commutes with every matrix in the representation of G   *)
(*                 (i.e., f is a total rG-homomorphism).                      *)
(*   rstab rG U == the subgroup of G whose representation via r fixes all     *)
(*                 vectors in U, pointwise.                                   *)
(*  rstabs rG U == the subgroup of G whose representation via r fixes the row *)
(*                 space of U globally.                                       *)
(* mxmodule rG U <=> the row-space of the matrix U is a module (globally      *)
(*                 invariant) under the representation rG of G.               *)
(* max_submod rG U V <-> U < V is not a proper is a proper subset of any      *)
(*                 proper rG-submodule of V (if both U and V are modules,     *)
(*                 then U is a maximal proper submodule of V).                *)
(* mx_subseries rG Us <=> Us : seq 'M_n is a list of rG-modules               *)
(* mx_composition_series rG Us <-> Us is an increasing composition series     *)
(*                 for an rG-module (namely, last 0 Us).                      *)
(* mxsimple rG M <-> M is a simple rG-module (i.e., minimal and nontrivial)   *)
(*                 This is a Prop predicate on square matrices.               *)
(* mxnonsimple rG U <-> U is constructively not a submodule, that is, U       *)
(*                 contains a proper nontrivial submodule.                    *)
(* mxnonsimple_sat rG U == U is not a simple as an rG-module.                 *)
(*                 This is a bool predicate, which requires a decField        *)
(*                 structure on the scalar field.                             *)
(* mxsemisimple rG W <-> W is constructively a direct sum of simple modules.  *)
(* mxsplits rG V U <-> V splits over U in rG, i.e., U has an rG-invariant     *)
(*                 complement in V.                                           *)
(* mx_completely_reducible rG V <-> V splits over all its submodules; note    *)
(*                 that this is only classically equivalent to stating that   *)
(*                 V is semisimple.                                           *)
(* mx_irreducible rG <-> the representation rG is irreducible, i.e., the full *)
(*                 module 1%:M of rG is simple.                               *)
(* mx_absolutely_irreducible rG == the representation rG of G is absolutely   *)
(*                 irreducible: its enveloping algebra is the full matrix     *)
(*                 ring. This is only classically equivalent to the more      *)
(*                 standard ``rG does not reduce in any field extension''.    *)
(* group_splitting_field F G <-> F is a splitting field for the group G:      *)
(*                 every irreducible representation of G is absolutely        *)
(*                 irreducible. Any field can be embedded classically into a  *)
(*                 splitting field.                                           *)
(* group_closure_field F gT <-> F is a splitting field for every group        *)
(*                 G : {group gT}, and indeed for any section of such a       *)
(*                 group. This is a convenient constructive substitute for    *)
(*                 algebraic closures, that can be constructed classically.   *)
(* dom_hom_mx rG f == a square matrix encoding the set of vectors for which   *)
(*                 multiplication by the n x n matrix f commutes with the     *)
(*                 representation of G, i.e., the largest domain on which     *)
(*                 f is an rG homomorphism.                                   *)
(*   mx_iso rG U V <-> U and V are (constructively) rG-isomorphic; this is    *)
(*                 a Prop predicate.                                          *)
(* mx_simple_iso rG U V == U and V are rG-isomorphic if one of them is        *)
(*                 simple; this is a bool predicate.                          *)
(*  cyclic_mx rG u == the cyclic rG-module generated by the row vector u      *)
(* annihilator_mx rG u == the annihilator of the row vector u in the          *)
(*                 enveloping algebra the representation rG.                  *)
(* row_hom_mx rG u == the image of u by the set of all rG-homomorphisms on    *)
(*                 its cyclic module, or, equivalently, the null-space of the *)
(*                 annihilator of u.                                          *)
(* component_mx rG M == when M is a simple rG-module, the component of M in   *)
(*                 the representation rG, i.e. the module generated by all    *)
(*                 the (simple) modules rG-isomorphic to M.                   *)
(*    socleType rG == a Structure that represents the type of all components  *)
(*                 of rG (more precisely, it coerces to such a type via       *)
(*                 socle_sort). For sG : socleType, values of type sG (to be  *)
(*                 exact, socle_sort sG) coerce to square matrices. For any   *)
(*                 representation rG we can construct sG : socleType rG       *)
(*                 classically; the socleType structure encapsulates this     *)
(*                 use of classical logic.                                    *)
(* DecSocleType rG == a socleType rG structure, for a representation over a   *)
(*                 decidable field type. DecSocleType rG is opaque.           *)
(*    socle_base W == for W : (sG : socleType), a simple module whose         *)
(*                 component is W; socle_simple W and socle_module W are      *)
(*                 proofs that socle_base W is a simple module.               *)
(*    socle_mult W == the multiplicity of socle_base W in W : sG.             *)
(*                 := \rank W %/ \rank (socle_base W)                         *)
(*        Socle sG == the Socle of rG, given sG : socleType rG, i.e., the     *)
(*                 (direct) sum of all the components of rG.                  *)
(* mx_rsim rG rG' <-> rG and rG' are similar representations of the same      *)
(*                 group G. Note that rG and rG' must then have equal, but    *)
(*                 not necessarily convertible, degree.                       *)
(* submod_repr modU == a representation of G on 'rV_(\rank U) equivalent to   *)
(*                 the restriction of rG to U (here modU : mxmodule rG U).    *)
(*    socle_repr W := submod_repr (socle_module W)                            *)
(* val/in_submod rG U == the projections resp. from/onto 'rV_(\rank U),       *)
(*                 that correspond to submod_repr r G U (these work both on   *)
(*                 vectors and row spaces).                                   *)
(* factmod_repr modV == a representation of G on 'rV_(\rank (cokermx V)) that *)
(*                 is equivalent to the factor module 'rV_n / V induced by V  *)
(*                 and rG (here modV : mxmodule rG V).                        *)
(* val/in_factmod rG U == the projections for factmod_repr r G U.             *)
(* section_repr modU modV == the restriction to in_factmod V U of the factor  *)
(*                 representation factmod_repr modV (for modU : mxmodule rG U *)
(*                 and modV : mxmodule rG V); section_repr modU modV is       *)
(*                 irreducible iff max_submod rG U V.                         *)
(* subseries_repr modUs i == the representation for the section module        *)
(*                 in_factmod (0 :: Us)`_i Us`_i, where                       *)
(*                 modUs : mx_subseries rG Us.                                *)
(* series_repr compUs i == the representation for the section module          *)
(*                 in_factmod (0 :: Us)`_i Us`_i, where                       *)
(*                 compUs : mx_composition_series rG Us. The Jordan-Holder    *)
(*                 theorem asserts the uniqueness of the set of such          *)
(*                 representations, up to similarity and permutation.         *)
(* regular_repr F G == the regular F-representation of the group G.           *)
(*   group_ring F G == a #|G| x #|G|^2 matrix that encodes the free group     *)
(*                 ring of G -- that is, the enveloping algebra of the        *)
(*                 regular F-representation of G.                             *)
(*   gring_index x == the index corresponding to x \in G in the matrix        *)
(*                 encoding of regular_repr and group_ring.                   *)
(*     gring_row A == the row vector corresponding to A \in group_ring F G in *)
(*                 the regular FG-module.                                     *)
(*  gring_proj x A == the 1 x 1 matrix holding the coefficient of x \in G in  *)
(*                 (A \in group_ring F G)%MS.                                 *)
(*   gring_mx rG u == the image of a row vector u of the regular FG-module,   *)
(*                 in the enveloping algebra of another representation rG.    *)
(*   gring_op rG A == the image of a matrix of the free group ring of G,      *)
(*                 in the enveloping algebra of rG.                           *)
(*   gset_mx F G C == the group sum of C in the free group ring of G -- the   *)
(*                 sum of the images of all the x \in C in group_ring F G.    *)
(* classg_base F G == a #|classes G| x #|G|^2 matrix whose rows encode the    *)
(*                 group sums of the conjugacy classes of G -- this is a      *)
(*                 basis of 'Z(group_ring F G)%MS.                            *)
(*     irrType F G == a type indexing irreducible representations of G over a *)
(*                 field F, provided its characteristic does not divide the   *)
(*                 order of G; it also indexes Wedderburn subrings.           *)
(*                 :=  socleType (regular_repr F G)                           *)
(*      irr_repr i == the irreducible representation corresponding to the     *)
(*                 index i : irrType sG                                       *)
(*                 := socle_repr i as i coerces to a component matrix.        *)
(* 'n_i, irr_degree i == the degree of irr_repr i; the notation is only       *)
(*                 active after Open Scope group_ring_scope.                  *)
(*   linear_irr sG == the set of sG-indices of linear irreducible             *)
(*                 representations of G.                                      *)
(*  irr_comp sG rG == the sG-index of the unique irreducible representation   *)
(*                 similar to rG, at least when rG is irreducible and the     *)
(*                 characteristic is coprime.                                 *)
(*    irr_mode i z == the unique eigenvalue of irr_repr i z, at least when    *)
(*                 irr_repr i z is scalar (e.g., when z \in 'Z(G)).           *)
(*      [1 sG]%irr == the index of the principal representation of G, in      *)
(*                 sG : irrType F G. The i argument of irr_repr, irr_degree   *)
(*                 and irr_mode is in the %irr scope. This notation may be    *)
(*                 replaced locally by an interpretation of 1%irr as [1 sG]   *)
(*                 for some specific irrType sG.                              *)
(* 'R_i, Wedderburn_subring i == the subring (indeed, the component) of the   *)
(*                 free group ring of G corresponding to the component i : sG *)
(*                 of the regular FG-module, where sG : irrType F g. In       *)
(*                 coprime characteristic the Wedderburn structure theorem    *)
(*                 asserts that the free group ring is the direct sum of      *)
(*                 these subrings; as with 'n_i above, the notation is only   *)
(*                 active in group_ring_scope.                                *)
(* 'e_i, Wedderburn_id i == the projection of the identity matrix 1%:M on the *)
(*                 Wedderburn subring of i : sG (with sG a socleType). In     *)
(*                 coprime characteristic this is the identity element of     *)
(*                 the subring, and the basis of its center if the field F is *)
(*                 a splitting field. As 'R_i, 'e_i is in group_ring_scope.   *)
(* subg_repr rG sHG == the restriction to H of the representation rG of G;    *)
(*                 here sHG : H \subset G.                                    *)
(* eqg_repr rG eqHG == the representation rG of G viewed a a representation   *)
(*                 of H; here eqHG : G == H.                                  *)
(* morphpre_repr f rG == the representation of f @*^-1 G obtained by          *)
(*                 composing the group morphism f with rG.                    *)
(* morphim_repr rGf sGD == the representation of G induced by a               *)
(*                 representation rGf of f @* G; here sGD : G \subset D where *)
(*                 D is the domain of the group morphism f.                   *)
(* rconj_repr rG uB == the conjugate representation x |-> B * rG x * B^-1;    *)
(*                 here uB : B \in unitmx.                                    *)
(* quo_repr sHK nHG == the representation of G / H induced by rG, given       *)
(*                 sHK : H \subset rker rG, and nHG : G \subset 'N(H).        *)
(* kquo_repr rG == the representation induced on G / rker rG by rG.           *)
(* map_repr f rG == the representation f \o rG, whose module is the tensor    *)
(*                 product of the module of rG with the extension field into  *)
(*                 which f : {rmorphism F -> Fstar} embeds F.                 *)
(*      'Cl%act == the transitive action of G on the Wedderburn components of *)
(*                 H, with nsGH : H <| G, given by Clifford's theorem. More   *)
(*                 precisely this is a total action of G on socle_sort sH,    *)
(*                 where sH : socleType (subg_repr rG (normal_sub sGH)).      *)
(* We build on the MatrixFormula toolkit to define decision procedures for    *)
(* the reducibility property:                                                 *)
(*  mxmodule_form rG U == a formula asserting that the interpretation of U is *)
(*                 a module of the representation rG.                         *)
(*  mxnonsimple_form rG U == a formula asserting that the interpretation of U *)
(*                 contains a proper nontrivial rG-module.                    *)
(*  mxnonsimple_sat rG U <=> mxnonsimple_form rG U is satisfied.              *)
(* More involved constructions are encapsulated in two Coq submodules:        *)
(* MatrixGenField == a module that encapsulates the lengthy details of the    *)
(*                 construction of appropriate extension fields. We assume we *)
(*                 have an irreducible representation rG of a group G, and a  *)
(*                 non-scalar matrix A that centralises rG(G), as this data   *)
(*                 is readily extracted from the Jacobson density theorem. It *)
(*                 then follows from Schur's lemma that the ring generated by *)
(*                 A is a field on which the extension of the representation  *)
(*                 rG of G is reducible. Note that this is equivalent to the  *)
(*                 more traditional quotient of the polynomial ring by an     *)
(*                 irreducible polynomial (the minimal polynomial of A), but  *)
(*                 much better suited to our needs.                           *)
(*   Here are the main definitions of MatrixGenField; they all have three     *)
(* proofs as arguments: (implicit) rG : mx_repr n G, irrG : mx_irreducible rG *)
(* and cGA : centgmx rG A. These ensure the validity of the construction and  *)
(* allow us to define Canonical instances; we assume degree_mxminpoly A > 1   *)
(* (which is equivalent to ~~ is_scalar_mx A) only to prove reducibility.     *)
(*  + gen_of irrG cGA == the carrier type of the field generated by A. It is  *)
(*                 at least equipped with a fieldType structure; we also      *)
(*                 propagate any decFieldType/finFieldType structures on the  *)
(*                 original field.                                            *)
(*  + gen irrG cGA == the morphism injecting into gen_of irrG cGA.            *)
(*  + groot irrG cGA == the root of mxminpoly A in the gen_of irrG cGA field. *)
(*  + pval x, rVval x, mxval x == the interpretation of x : gen_of irrG cGA   *)
(*                 as a polynomial, a row vector, and a matrix, respectively. *)
(*                 Both irrG and cGA are implicit arguments here.             *)
(*  + gen_repr irrG cGA == an alternative to the field extension              *)
(*                 representation, which consists in reconsidering the        *)
(*                 original module as a module over the new gen_of field,     *)
(*                 thereby DIVIDING the original dimension n by the degree of *)
(*                 the minimal polynomial of A. This can be simpler than the  *)
(*                 extension method, is actually required by the proof that   *)
(*                 odd groups are p-stable (B & G 6.1-2, and Appendix A), but *)
(*                 is only applicable if G is the LARGEST group represented   *)
(*                 by rG (e.g., NOT for B & G 2.6).                           *)
(*  + gen_dim A == the dimension of gen_repr irrG cGA (only depends on A).    *)
(*  + in_gen irrG cGA W == the ROWWISE image of a matrix W : 'M[F]_(m, n),    *)
(*                 i.e., interpreting W as a sequence of m tow vectors,       *)
(*                 under the bijection from rG to gen_repr irrG cGA.          *)
(*                 The sequence length m is a maximal implicit argument       *)
(*                 passed between the explicit argument cGA and W.            *)
(*  + val_gen W == the ROWWISE image of an 'M[gen_of irrG cGA]_(m, gen_dim A) *)
(*                 matrix W under the bijection from gen_repr irrG cGA to rG. *)
(*  + rowval_gen W == the ROWSPACE image of W under the bijection from        *)
(*                 gen_repr irrG cGA to rG, i.e., a 'M[F]_n matrix whose row  *)
(*                 space is the image of the row space of W.                  *)
(*                 This is the A-ideal generated by val_gen W.                *)
(*  + gen_sat e f <=> f : GRing.formula (gen_of irrG cGA) is satisfied in     *)
(*                 environment e : seq (gen_of irrG cGA), provided F has a    *)
(*                 decFieldType structure.                                    *)
(*  + gen_env e, gen_term t, gen_form f == interpretations of environments,   *)
(*                 terms, and RING formulas over gen_of irrG cGA as row       *)
(*                 vector formulae, used to construct gen_sat.                *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Reserved Notation "''n_' i" (at level 8, i at level 2, format "''n_' i").
Reserved Notation "''R_' i" (at level 8, i at level 2, format "''R_' i").
Reserved Notation "''e_' i" (at level 8, i at level 2, format "''e_' i").
Delimit Scope irrType_scope with irr.
Section RingRepr.
Variable R : comUnitRingType.
Section OneRepresentation.
Variable gT : finGroupType.
Definition mx_repr (G : {set gT}) n (r : gT -> 'M[R]_n) :=
r 1%g = 1%:M /\ {in G &, {morph r : x y / (x * y)%g >-> x *m y}}.
Structure mx_representation G n :=
MxRepresentation { repr_mx :> gT -> 'M_n; _ : mx_repr G repr_mx }.
Variables (G : {group gT}) (n : nat) (rG : mx_representation G n).
Arguments rG _%group_scope : extra scopes.

Lemma repr_mxKV m x :
x \in G -> cancel ((@mulmx R m n n)^~ (rG x^-1)) (mulmx^~ (rG x)).