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Add forms for Omega(e,d,q), except when d odd, q even; some cleanup #2577

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merged 5 commits into from
Jun 28, 2018
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Add forms for Omega(e,d,q), except when d odd, q even; some cleanup #2577

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b7e497b
grp: fix a typo in a comment
fingolfin Jun 26, 2018
d40abd8
grp: add SetInvariantQuadraticFormFromMatrix helper
fingolfin Jun 26, 2018
7c84682
grp: extend tests to verify sizes of small classic groups
fingolfin Jun 26, 2018
4909658
grp: extend tests for classic groups to more fields
fingolfin Jun 26, 2018
8fe7f65
grp: add forms for Omega(e,d,q), except for d odd, q even
fingolfin Jun 26, 2018
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213 changes: 83 additions & 130 deletions grp/classic.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -341,6 +341,20 @@ InstallMethod( SpecialUnitaryGroupCons,
end );


#############################################################################
##
#M SetInvariantQuadraticFormFromMatrix( <g>, <mat> )
##
## Set the invariant quadratic form of <g> to the matrix <mat>, and also
## set the bilinear form to the value required by the documentation, i.e.,
# to <mat> + <mat>^T.
##
BindGlobal( "SetInvariantQuadraticFormFromMatrix", function( g, mat )
SetInvariantQuadraticForm( g, rec( matrix:= mat ) );
SetInvariantBilinearForm( g, rec( matrix:= mat+TransposedMat(mat) ) );
end );


#############################################################################
##
#F Oplus45() . . . . . . . . . . . . . . . . . . . . . . . . . . . . O+_4(5)
Expand Down Expand Up @@ -388,13 +402,9 @@ BindGlobal( "Oplus45", function()
# set the size
SetSize( g, 28800 );

# construct the form
SetInvariantBilinearForm( g, rec( matrix:= ImmutableMatrix( f,
[[0,1,0,0],[1,0,0,0],[0,0,2,0],[0,0,0,2]] * One( f ), true ) ) );

# and the quadratic form
SetInvariantQuadraticForm( g, rec( matrix:= ImmutableMatrix( f,
[[0,1,0,0],[0,0,0,0],[0,0,1,0],[0,0,0,1]] * One( f ), true ) ) );
# construct the forms
SetInvariantQuadraticFormFromMatrix( g, ImmutableMatrix( f,
[[0,1,0,0],[0,0,0,0],[0,0,1,0],[0,0,0,1]] * One( f ), true ) );

# and return
return g;
Expand Down Expand Up @@ -450,19 +460,12 @@ BindGlobal( "Opm3", function( s, d )
SetDimensionOfMatrixGroup( g, d );
SetFieldOfMatrixGroup( g, f );

# construct the form
delta := List( 2*id, ShallowCopy );
delta{[1,2]}{[1,2]} := [[0,1],[1,0]]*One( f );
delta[3][3] := 2*One( f )*2;
SetInvariantBilinearForm( g,
rec( matrix:= ImmutableMatrix( f, delta, true ) ) );

# construct quadratic form
# construct the forms
delta := List( id, ShallowCopy );
delta{[1,2]}{[1,2]} := [[0,1],[0,0]]*One( f );
delta[3][3] := One( f )*2;
SetInvariantQuadraticForm( g,
rec( matrix:= ImmutableMatrix( f, delta, true ) ) );
delta := ImmutableMatrix( f, delta, true );
SetInvariantQuadraticFormFromMatrix( g, delta );

# set the size
delta := 1;
Expand Down Expand Up @@ -527,19 +530,12 @@ BindGlobal( "OpmSmall", function( s, d, q )
SetDimensionOfMatrixGroup( g, d );
SetFieldOfMatrixGroup( g, f );

# construct the form
delta := List( 2*id, ShallowCopy );
delta{[1,2]}{[1,2]} := [[0,1],[1,0]]*One( f );
delta[3][3] := 2*One( f );
SetInvariantBilinearForm( g,
rec( matrix:= ImmutableMatrix( f, delta, true ) ) );

# construct quadratic form
# construct the forms
delta := List( id, ShallowCopy );
delta{[1,2]}{[1,2]} := [[0,1],[0,0]]*One( f );
delta[3][3] := One( f );
SetInvariantQuadraticForm( g,
rec( matrix:= ImmutableMatrix( f, delta, true ) ) );
delta := ImmutableMatrix( f, delta, true );
SetInvariantQuadraticFormFromMatrix( g, delta );

# set the size
delta := 1;
Expand Down Expand Up @@ -600,20 +596,16 @@ BindGlobal( "OpmOdd", function( s, d, q )
g := GroupWithGenerators( [
[[1,0,0,0],[0,1,2,1],[2,0,2,0],[1,0,0,1]]*One( f ),
[[0,2,2,2],[0,1,1,2],[1,0,2,0],[1,2,2,0]]*One( f ) ] );
SetInvariantBilinearForm( g, rec( matrix:= ImmutableMatrix( f,
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,2]]*One( f ), true ) ) );
SetInvariantQuadraticForm( g, rec( matrix:= ImmutableMatrix( f,
[[0,1,0,0],[0,0,0,0],[0,0,2,0],[0,0,0,1]]*One( f ), true ) ) );
SetInvariantQuadraticFormFromMatrix( g, ImmutableMatrix( f,
[[0,1,0,0],[0,0,0,0],[0,0,2,0],[0,0,0,1]]*One( f ), true ) );
SetSize( g, 1152 );
return g;
elif q = 3 and d = 4 and s = -1 then
g := GroupWithGenerators( [
[[0,2,0,0],[2,1,0,1],[0,2,0,1],[0,0,1,0]]*One( f ),
[[2,0,0,0],[1,2,0,2],[1,0,0,1],[0,0,1,0]]*One( f ) ] );
SetInvariantBilinearForm( g, rec( matrix:= ImmutableMatrix( f,
[[0,1,0,0],[1,0,0,0],[0,0,2,0],[0,0,0,2]]*One( f ), true ) ) );
SetInvariantQuadraticForm( g, rec( matrix:= ImmutableMatrix( f,
[[0,1,0,0],[0,0,0,0],[0,0,1,0],[0,0,0,1]]*One( f ), true ) ) );
SetInvariantQuadraticFormFromMatrix( g, ImmutableMatrix( f,
[[0,1,0,0],[0,0,0,0],[0,0,1,0],[0,0,0,1]]*One( f ), true ) );
SetSize( g, 1440 );
return g;
elif q = 5 and d = 4 and s = +1 then
Expand Down Expand Up @@ -674,19 +666,12 @@ BindGlobal( "OpmOdd", function( s, d, q )
SetDimensionOfMatrixGroup( g, d );
SetFieldOfMatrixGroup( g, f );

# construct the form
delta := List( 2*id, ShallowCopy );
delta{[1,2]}{[1,2]} := [[0,1],[1,0]]*One( f );
delta[3][3] := 2*beta;
SetInvariantBilinearForm( g,
rec( matrix:= ImmutableMatrix( f, delta, true ) ) );

# construct quadratic form
# construct the forms
delta := List( id, ShallowCopy );
delta{[1,2]}{[1,2]} := [[0,1],[0,0]]*One( f );
delta[3][3] := beta;
SetInvariantQuadraticForm( g,
rec( matrix:= ImmutableMatrix( f, delta, true ) ) );
delta := ImmutableMatrix( f, delta, true );
SetInvariantQuadraticFormFromMatrix( g, delta );

# set the size
delta := 1;
Expand Down Expand Up @@ -723,10 +708,8 @@ BindGlobal( "Oplus2", function( q )
m2:= ImmutableMatrix( f, m2, true );
# construct the group, set the order, and return
g := GroupWithGenerators( [ m1, m2 ] );
SetInvariantBilinearForm(g,
rec( matrix:= ImmutableMatrix( f, m2, true ) ) );
SetInvariantQuadraticForm( g, rec( matrix:= ImmutableMatrix( f,
[ [ 0, 1 ], [ 0, 0 ] ] * z^0, true ) ) );
SetInvariantQuadraticFormFromMatrix( g, ImmutableMatrix( f,
[ [ 0, 1 ], [ 0, 0 ] ] * z^0, true ) );
SetSize( g, 2*(q-1) );
return g;
end );
Expand Down Expand Up @@ -774,13 +757,9 @@ BindGlobal( "Oplus4Even", function( q )
# set the size
SetSize( g, 2*q^2*(q^2-1)^2 );

# construct the form
SetInvariantBilinearForm( g, rec( matrix:= ImmutableMatrix( f,
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] * One( f ), true ) ) );

# and the quadratic form
SetInvariantQuadraticForm( g, rec( matrix:= ImmutableMatrix( f,
[[0,1,0,0],[0,0,0,0],[0,0,0,1],[0,0,0,0]] * One( f ), true ) ) );
# construct the forms
SetInvariantQuadraticFormFromMatrix( g, ImmutableMatrix( f,
[[0,1,0,0],[0,0,0,0],[0,0,0,1],[0,0,0,0]] * One( f ), true ) );

# and return
return g;
Expand Down Expand Up @@ -888,22 +867,12 @@ BindGlobal( "OplusEven", function( d, q )
SetDimensionOfMatrixGroup( g, d );
SetFieldOfMatrixGroup( g, f );

# construct the form
delta := List( 0*id, ShallowCopy );
for i in [ 1 .. d/2 ] do
delta[2*i-1][2*i] := One( f );
delta[2*i][2*i-1] := One( f );
od;
SetInvariantBilinearForm( g,
rec( matrix:= ImmutableMatrix( f, delta, true ) ) );

# construct quadratic form
# construct the forms
delta := List( 0*id, ShallowCopy );
for i in [ 1 .. d/2 ] do
delta[2*i-1][2*i] := One( f );
od;
SetInvariantQuadraticForm( g,
rec( matrix:= ImmutableMatrix( f, delta, true ) ) );
SetInvariantQuadraticFormFromMatrix( g, ImmutableMatrix( f, delta, true ) );

# set the size
delta := 1;
Expand Down Expand Up @@ -954,10 +923,8 @@ BindGlobal( "Ominus2", function( q )
m1:=ImmutableMatrix(GF(q),m1,true);
m2:=ImmutableMatrix(GF(q),m2,true);
g := GroupWithGenerators( [ m1, m2 ] );
SetInvariantBilinearForm( g, rec( matrix:= ImmutableMatrix( f,
[ [ 2, 1 ], [ 1, 2*t ] ] * z^0, true ) ) );
SetInvariantQuadraticForm( g, rec( matrix:= ImmutableMatrix( f,
[ [ 1, 1 ], [ 0, t ] ] * z^0, true ) ) );
SetInvariantQuadraticFormFromMatrix( g, ImmutableMatrix( f,
[ [ 1, 1 ], [ 0, t ] ] * z^0, true ) );
SetSize( g, 2*(q+1) );

return g;
Expand Down Expand Up @@ -1016,13 +983,9 @@ BindGlobal( "Ominus4Even", function( q )
# set the size
SetSize( g, 2*q^2*(q^2+1)*(q^2-1) );

# construct the form
SetInvariantBilinearForm( g, rec( matrix:= ImmutableMatrix( f,
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] * One( f ), true ) ) );

# and the quadratic form
SetInvariantQuadraticForm( g, rec( matrix:= ImmutableMatrix( f,
[[0,1,0,0],[0,0,0,0],[0,0,t,1],[0,0,0,t]] * One( f ), true ) ) );
# construct the forms
SetInvariantQuadraticFormFromMatrix( g, ImmutableMatrix( f,
[[0,1,0,0],[0,0,0,0],[0,0,t,1],[0,0,0,t]] * One( f ), true ) );

# and return
return g;
Expand Down Expand Up @@ -1138,24 +1101,14 @@ BindGlobal( "OminusEven", function( d, q )
SetDimensionOfMatrixGroup( g, d );
SetFieldOfMatrixGroup( g, f );

# construct the form
delta := List( 0*id, ShallowCopy );
for i in [ 1 .. d/2 ] do
delta[2*i-1][2*i] := One( f );
delta[2*i][2*i-1] := One( f );
od;
SetInvariantBilinearForm( g,
rec( matrix:= ImmutableMatrix( f, delta, true ) ) );

# construct quadratic form
# construct the forms
delta := List( 0*id, ShallowCopy );
for i in [ 1 .. d/2 ] do
delta[2*i-1][2*i] := One( f );
od;
delta[3][3] := t;
delta[4][4] := t;
SetInvariantQuadraticForm( g,
rec( matrix:= ImmutableMatrix( f, delta, true ) ) );
SetInvariantQuadraticFormFromMatrix( g, ImmutableMatrix( f, delta, true ) );

# set the size
delta := 1;
Expand Down Expand Up @@ -1233,17 +1186,10 @@ BindGlobal( "OzeroOdd", function( d, q, b )
od;
SetSize( g, 2 * q^((d-1)^2/4) * s );

# construct the form
s := List( 2*b*id, ShallowCopy );
s{[1,2]}{[1,2]} := [[0,1],[1,0]]*One( f );
SetInvariantBilinearForm( g,
rec( matrix:= ImmutableMatrix( f, s, true ) ) );

# and the quadratic form
# construct the forms
s := List( b*id, ShallowCopy );
s{[1,2]}{[1,2]} := [[0,1],[0,0]]*One( f );
SetInvariantQuadraticForm( g,
rec( matrix:= ImmutableMatrix( f, s, true ) ) );
SetInvariantQuadraticFormFromMatrix( g, ImmutableMatrix( f, s, true ) );

# and return
return g;
Expand Down Expand Up @@ -1277,7 +1223,7 @@ end );
## one-to-one in characteristic $2$.
##
BindGlobal( "OzeroEven", function( d, q )
local f, z, o, n, mat1, mat2, i, g, size, qi, c, s;
local f, z, o, n, mat1, mat2, i, g, size, qi, s;

# <d> must be odd, <q> must be even
if d mod 2 = 0 then
Expand Down Expand Up @@ -1346,6 +1292,7 @@ BindGlobal( "OzeroEven", function( d, q )
g:= GroupWithGenerators( [ mat1, mat2 ] );
SetDimensionOfMatrixGroup( g, Length( mat1 ) );
SetFieldOfMatrixGroup( g, f );
SetIsSubgroupSL( g, true );

# add the size
size := 1;
Expand All @@ -1356,24 +1303,14 @@ BindGlobal( "OzeroEven", function( d, q )
od;
SetSize( g, q^(((d-1)/2)^2) * size );

# construct the form
c := List( 0 * One( g ), ShallowCopy );
for i in [ 2 .. (d+1)/2 ] do
c[(d-1)/2+i][i] := o;
c[i][(d-1)/2+i] := o;
od;
SetInvariantBilinearForm( g,
rec( matrix:= ImmutableMatrix( f, c, true ) ) );
SetIsSubgroupSL( g, true );

# and the quadratic form
# construct the forms
s := List( 0 * One( g ), ShallowCopy );
s[1][1]:= o;
for i in [ 2 .. (d+1)/2 ] do
s[(d-1)/2+i][i]:= o;
od;
SetInvariantQuadraticForm( g,
rec( matrix:= ImmutableMatrix( f, s, true ) ) );
s:= ImmutableMatrix( f, s, true );
SetInvariantQuadraticFormFromMatrix( g, s );

# and return
return g;
Expand Down Expand Up @@ -1652,11 +1589,18 @@ BindGlobal( "OmegaZero", function( d, q )
fi;
SetSize( g, q^(m^2) * s );

# construct the bilinear form
#T add the form!

# and the quadratic form
#T add the form!
# construct the forms
if q mod 2 = 0 then
# FIXME: add forms for even characteristic, if that is possible at all
# (it doesn't seem to be)
else
x:= NullMat( d, d, f );
for i in [ 1 .. m ] do
x[i,d-i+1] := o;
od;
x[m+1,m+1] := (Characteristic(f)+1)/4*o;
SetInvariantQuadraticFormFromMatrix(g, ImmutableMatrix( f, x, true ) );
fi;

# and return
return g;
Expand All @@ -1665,7 +1609,7 @@ BindGlobal( "OmegaZero", function( d, q )

#############################################################################
##
#F OmegaPlus( <d>, <q> ) . . . . . . . . . . . . . . . . \Omega^-_{<d>}(<q>)
#F OmegaPlus( <d>, <q> ) . . . . . . . . . . . . . . . . \Omega^+_{<d>}(<q>)
##
BindGlobal( "OmegaPlus", function( d, q )
local f, o, m, xi, g, a, mo, n, i, x1, x2, x, h, s, q2, q2i;
Expand Down Expand Up @@ -1763,11 +1707,13 @@ BindGlobal( "OmegaPlus", function( d, q )
fi;
SetSize( g, q^(m*(m-1)) * (q^m-1) * s );

# construct the bilinear form
#T add the form!

# and the quadratic form
#T add the form!
# construct the forms
x:= NullMat( d, d, f );
for i in [ 1 .. m ] do
x[i,d-i+1] := o;
od;
x:= ImmutableMatrix( f, x, true );
SetInvariantQuadraticFormFromMatrix( g, x );

# and return
return g;
Expand Down Expand Up @@ -1846,11 +1792,18 @@ BindGlobal( "OmegaMinus", function( d, q )
fi;
SetSize( g, q^(m*(m-1)) * (q^m+1) * s );

# construct the bilinear form
#T add the form!

# and the quadratic form
#T add the form!
# construct the forms
x:= NullMat( d, d, f );
for i in [ 1 .. m-1 ] do
x[i,d-i+1] := o;
od;
x[m,d-m+1] := -nu - nubar;
if q mod 2 = 1 then
x[m,d-m] := -o;
x[m+1,d-m+1] := xi^( (q+1)/2 );
fi;
x:= ImmutableMatrix( f, x, true );
SetInvariantQuadraticFormFromMatrix( g, x );

# and return
return g;
Expand Down
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