(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
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(* This file contains the basic notions of character theory, based on Isaacs. *)
(*         irr G == tuple of the elements of 'CF(G) that are irreducible      *)
(*                  characters of G.                                          *)
(*        Nirr G == number of irreducible characters of G.                    *)
(*        Iirr G == index type for the irreducible characters of G.           *)
(*               := 'I_(Nirr G).                                              *)
(*        'chi_i == the i-th element of irr G, for i : Iirr G.                *)
(*     'chi[G]_i    Note that 'chi_0 = 1, the principal character of G.       *)
(*        'Chi_i == an irreducible representation that affords 'chi_i.        *)
(* socle_of_Iirr i == the Wedderburn component of the regular representation  *)
(*                    of G, corresponding to 'Chi_i.                          *)
(*   Iirr_of_socle == the inverse of socle_of_Iirr (which is one-to-one).     *)
(*    phi.[A]%CF == the image of A \in group_ring G under phi : 'CF(G).       *)
(*     cfRepr rG == the character afforded by the representation rG of G.     *)
(*       cfReg G == the regular character, afforded by the regular            *)
(*                  representation of G.                                      *)
(*     detRepr rG == the linear character afforded by the determinant of rG.  *)
(*      cfDet phi == the linear character afforded by the determinant of a    *)
(*                   representation affording phi.                            *)
(*        'o(phi) == the "determinential order" of phi (the multiplicative    *)
(*                   order of cfDet phi.                                      *)
(* phi \is a character <=> phi : 'CF(G) is a character of G or 0.             *)
(* i \in irr_constt phi <=> 'chi_i is an irreducible constituent of phi: phi  *)
(*                  has a non-zero coordinate on 'chi_i over the basis irr G. *)
(* xi \is a linear_char xi <=> xi : 'CF(G) is a linear character of G.        *)
(*    'Z(chi)%CF == the center of chi when chi is a character of G, i.e.,     *)
(*                  rcenter rG where rG is a representation that affords phi. *)
(*                  If phi is not a character then 'Z(chi)%CF = cfker phi.    *)
(*  aut_Iirr u i == the index of cfAut u 'chi_i in irr G.                     *)
(*  conjC_Iirr i == the index of 'chi_i^*%CF in irr G.                        *)
(*  morph_Iirr i == the index of cfMorph 'chi[f @* G]_i in irr G.             *)
(* isom_Iirr isoG i == the index of cfIsom isoG 'chi[G]_i in irr R.           *)
(*    mod_Iirr i == the index of ('chi[G / H]_i %% H)%CF in irr G.            *)
(*    quo_Iirr i == the index of ('chi[G]_i / H)%CF in irr (G / H).           *)
(*  Ind_Iirr G i == the index of 'Ind[G, H] 'chi_i, provided it is an         *)
(*                  irreducible character (such as when if H is the inertia   *)
(*                  group of 'chi_i).                                         *)
(*  Res_Iirr H i == the index of 'Res[H, G] 'chi_i, provided it is an         *)
(*                  irreducible character (such as when 'chi_i is linear).    *)
(* sdprod_Iirr defG i == the index of cfSdprod defG 'chi_i in irr G, given    *)
(*                       defG : K ><| H = G.                                  *)
(* And, for KxK : K \x H = G.                                                 *)
(*      dprodl_Iirr KxH i == the index of cfDprodl KxH 'chi[K]_i in irr G.    *)
(*      dprodr_Iirr KxH j == the index of cfDprodr KxH 'chi[H]_j in irr G.    *)
(*  dprod_Iirr KxH (i, j) == the index of cfDprod KxH 'chi[K]_i 'chi[H]_j.    *)
(*     inv_dprod_Iirr KxH == the inverse of dprod_Iirr KxH.                   *)
(* The following are used to define and exploit the character table:          *)
(*  character_table G == the character table of G, whose i-th row lists the   *)
(*                       values taken by 'chi_i on the conjugacy classes      *)
(*                       of G; this is a square Nirr G x NirrG matrix.        *)
(*        irr_class i == the conjugacy class of G with index i : Iirr G.      *)
(*      class_Iirr xG == the index of xG \in classes G, in Iirr G.            *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Notation algCF := [fieldType of algC].
Section AlgC.
Variable (gT : finGroupType).

Lemma trowbE n1 m2 n2 A B : trowb B A = @trow n1 A m2 n2 B.
Canonical Structure trowb_linear n1 m2 n2 B :=
Linear (@trowb_is_linear n1 m2 n2 B).