(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
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(*   This file completes the theory developed in mxrepresentation.v with the  *)
(* construction and properties of linear representations over finite fields,  *)
(* and in particular the correspondance between internal action on a (normal) *)
(* elementary abelian p-subgroup and a linear representation on an Fp-module. *)
(*   We provide the following next constructions for a finite field F:        *)
(*        'Zm%act == the action of {unit F} on 'M[F]_(m, n).                  *)
(*         rowg A == the additive group of 'rV[F]_n spanned by the row space  *)
(*                   of the matrix A.                                         *)
(*      rowg_mx L == the partial inverse to rowg; for any 'Zm-stable group L  *)
(*                   of 'rV[F]_n we have rowg (rowg_mx L) = L.                *)
(*     GLrepr F n == the natural, faithful representation of 'GL_n[F].        *)
(*     reprGLm rG == the morphism G >-> 'GL_n[F] equivalent to the            *)
(*                   representation r of G (with rG : mx_repr r G).           *)
(*   ('MR rG)%act == the action of G on 'rV[F]_n equivalent to the            *)
(*                   representation r of G (with rG : mx_repr r G).           *)
(* The second set of constructions defines the interpretation of a normal     *)
(* non-trivial elementary abelian p-subgroup as an 'F_p module. We assume     *)
(* abelE : p.-abelem E and ntE : E != 1, throughout, as these are needed to   *)
(* build the isomorphism between E and a nontrivial 'rV['F_p]_n.              *)
(*         'rV(E) == the type of row vectors of the 'F_p module equivalent    *)
(*                   to E when E is a non-trivial p.-abelem group.            *)
(*          'M(E) == the type of matrices corresponding to E.                 *)
(*         'dim E == the width of vectors/matrices in 'rV(E) / 'M(E).         *)
(* abelem_rV abelE ntE == the one-to-one injection of E onto 'rV(E).          *)
(*  rVabelem abelE ntE == the one-to-one projection of 'rV(E) onto E.         *)
(* abelem_repr abelE ntE nEG == the representation of G on 'rV(E) that is     *)
(*                   equivalent to conjugation by G in E; here abelE, ntE are *)
(*                   as above, and G \subset 'N(E).                           *)
(* This file end with basic results on p-modular representations of p-groups, *)
(* and theorems giving the structure of the representation of extraspecial    *)
(* groups; these results use somewhat more advanced group theory than the     *)
(* rest of mxrepresentation, in particular, results of sylow.v and maximal.v. *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
(* Special results for representations on a finite field. In this case, the   *)
(* representation is equivalent to a morphism into the general linear group   *)
(* 'GL_n[F]. It is furthermore equivalent to a group action on the finite     *)
(* additive group of the corresponding row space 'rV_n. In addition, row      *)
(* spaces of matrices in 'M[F]_n correspond to subgroups of that vector group *)
(* (this is only surjective when F is a prime field 'F_p), with moduleules    *)
(* corresponding to subgroups stabilized by the external action.              *)
Section FinRingRepr.
Variable (R : finComUnitRingType) (gT : finGroupType).
Variables (G : {group gT}) (n : nat) (rG : mx_representation R G n).
Definition mx_repr_act (u : 'rV_n) x := u *m rG (val (subg G x)).

Lemma mx_repr_actE u x : x \in G -> mx_repr_act u x = u *m rG x.
Canonical Structure mx_repr_action := Action mx_repr_is_action.