#############################################################################
##
## This file is part of recog, a package for the GAP computer algebra system
## which provides a collection of methods for the constructive recognition
## of groups.
##
## This files's authors include Max Neunhöffer, Ákos Seress.
##
## Copyright of recog belongs to its developers whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-3.0-or-later
##
##
## Generic code for projective groups.
##
#############################################################################
# Assumes <m> is a non-empty square matrix defined over a field. <m> defines a
# projective one iff it is a non-zero scalar multiple of the identity matrix.
InstallGlobalFunction( IsOneProjective,
function(m)
Assert(1, NrRows(m) > 0 and NrRows(m) = NrCols(m));
return not IsZero(m[1,1]) and RECOG.IsScalarMat(m) <> false;
end );
# Assumes <a> and <b> are non-empty equal-dimension square matrices over equal
# fields, and that the first row of <a> contains a non-zero entry.
#
# <a> and <b> define equal projective elements, or are "equal modulo scalars",
# iff there exists a (non-zero) scalar <s> in the field such that <s * a = b>.
InstallGlobalFunction( IsEqualProjective,
function(a, b)
local n, p, s;
n := NrRows(a);
Assert(1, n > 0 and n = NrRows(b) and n = NrCols(a) and n = NrCols(b));
p := First([1..n], i -> not IsZero(a[1,i])); # Find non-zero entry in <a[1]>
Assert(1, p <> fail);
s := b[1,p] / a[1,p]; # The unique scalar <s> with <s * a[1,p] = b[1,p]>.
if IsZero(s) then
return false; # <a[1,p] <> 0 = b[1,p]>, so <a> and <b> are not equal.
elif IsRowListMatrix(a) and IsRowListMatrix(b) then
# Separate case for performance reasons
return ForAll([1 .. n], i -> s * a[i] = b[i]);
fi;
return s * a = b;
end );
# The projective order of an invertible square matrix <m> over a finite field is
# the least positive integer <k> such that <m ^ k> is a projective one.
RECOG.ProjectiveOrder := m -> ProjectiveOrder(m)[1];
# Assumes <m> is a non-empty square matrix over a field.
#
# <m> is a scalar matrix iff it is a scalar multiple of the identity matrix,
# i.e. a diagonal square matrix with the same value along its main diagonal.
#
# If <m> is a scalar matrix, then this function returns the scalar that appears
# on its main diagonal. Otherwise, this function returns <false>.
RECOG.IsScalarMat := function(m)
local x;
Assert(1, NrRows(m) > 0 and NrRows(m) = NrCols(m));
x := m[1,1];
if ForAny([2..NrRows(m)], i -> m[i,i] <> x) or not IsDiagonalMat(m) then
return false;
fi;
return x;
end;
