grp: add forms for Omega(e,d,q), except for d odd, q even

gap-system · Jun 26, 2018 · 15110f3 · 15110f3
1 parent 113f861
commit 15110f3
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Showing 2 changed files with 42 additions and 26 deletions.
diff --git a/grp/classic.gi b/grp/classic.gi
@@ -1589,11 +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;
@@ -1700,11 +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;
@@ -1783,11 +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;

diff --git a/tst/testinstall/grp/classic-forms.tst b/tst/testinstall/grp/classic-forms.tst
@@ -1,6 +1,5 @@
 #
 # Tests invariant forms of classic groups
-# TODO: also test quadratic forms
 #
 gap> START_TEST("classic-forms.tst");
 
@@ -117,7 +116,6 @@ true
 #
 # Omega subgroups of special orthogonal groups
 #
-# TODO: add forms to Omega, check them here
 
 # odd-dimensional
 gap> grps:=[];;
@@ -129,10 +127,13 @@ gap> for d in [3,5,7] do
 gap> ForAll(grps, CheckGeneratorsSpecial);
 true
 
-#gap> ForAll(grps, CheckBilinearForm);
-#true
-#gap> ForAll(grps, CheckQuadraticForm);
-#true
+# FIXME: forms are not implemented for odd d, even q
+# gap> ForAll(grps, CheckBilinearForm);
+# true
+gap> ForAll(Filtered(grps, g -> Characteristic(g)<>2), CheckBilinearForm);
+true
+gap> ForAll(grps, CheckQuadraticForm);
+true
 gap> ForAll(grps, CheckSize);
 true
 
@@ -146,11 +147,10 @@ gap> for d in [2,4,6,8] do
 > od;
 gap> ForAll(grps, CheckGeneratorsSpecial);
 true
-
-#gap> ForAll(grps, CheckBilinearForm);
-#true
-#gap> ForAll(grps, CheckQuadraticForm);
-#true
+gap> ForAll(grps, CheckBilinearForm);
+true
+gap> ForAll(grps, CheckQuadraticForm);
+true
 gap> ForAll(grps, CheckSize);
 true