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Extend meataxe to be able to detect invariant forms of reducible modules (just one possible form is returned) #5803

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Extend meataxe to be able to detect invariant forms of reducible modules (just one possible form is returned) #5803

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634d1fc
meataxe: extend invariant form detection
fingolfin Nov 7, 2024
c075d23
Add example / test case to documentation
fingolfin Nov 7, 2024
3147c2b
Fix SMTX.InvariantSesquilinearForm for reducible case
fingolfin Nov 8, 2024
44802ad
Apply suggestions from code review
fingolfin Dec 5, 2024
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75 changes: 72 additions & 3 deletions doc/ref/meataxe.xml
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Original file line number Diff line number Diff line change
Expand Up @@ -582,15 +582,47 @@ on all composition factors except number <A>nr</A>.
<Section Label="meataxe:Invariant Forms">
<Heading>MeatAxe Functionality for Invariant Forms</Heading>

The functions in this section can only be applied to an absolutely irreducible
MeatAxe module.

<ManSection>
<Func Name="MTX.InvariantBilinearForm" Arg='module'/>

<Description>
returns an invariant bilinear form, which may be symmetric or anti-symmetric,
of <A>module</A>, or <K>fail</K> if no such form exists.

<Example><![CDATA[
gap> g:= SO(-1, 4, 5);;
gap> m:= NaturalGModule( g );;
gap> form:= MTX.InvariantBilinearForm( m );;
gap> Display( form );
. 3 . .
3 . . .
. . 2 .
. . . 1
gap> ForAll(MTX.Generators(m), x -> x*form*TransposedMat(x) = form);
true
]]></Example>

Since GAP 4.14 modules that are not absolutely irreducible are also supported:
<Example><![CDATA[
gap> g1:= GeneratorsOfGroup(SO(-1, 4, 5));;
gap> g2:= GeneratorsOfGroup(SO(+1, 4, 5));;
gap> m:= GModuleByMats(SMTX.MatrixSum(g1,g2), GF(5));;
gap> MTX.IsIrreducible(m);
false
gap> form:= MTX.InvariantBilinearForm( m );;
gap> Display( form );
. 3 . . . . . .
3 . . . . . . .
. . 2 . . . . .
. . . 1 . . . .
. . . . . 3 . .
. . . . 3 . . .
. . . . . . 1 .
. . . . . . . 1
gap> ForAll(MTX.Generators(m), x -> x*form*TransposedMat(x) = form);
true
]]></Example>

</Description>
</ManSection>

Expand All @@ -603,6 +635,42 @@ returns an invariant hermitian (= self-adjoint) sesquilinear form of
<A>module</A>,
which must be defined over a finite field whose order is a square,
or <K>fail</K> if no such form exists.

<Example><![CDATA[
gap> g:= SU(4, 5);;
gap> m:= NaturalGModule( g );;
gap> form:= MTX.InvariantSesquilinearForm( m );;
gap> Display( form );
. . . 1
. . 1 .
. 1 . .
1 . . .
gap> frob5 := g -> List(g,row->List(row,x->x^5));; # field involution
gap> ForAll(MTX.Generators(m), x -> x*form*TransposedMat(frob5(x)) = form);
true
]]></Example>

Since GAP 4.14 modules that are not absolutely irreducible are also supported:
<Example><![CDATA[
gap> g1:= GeneratorsOfGroup(SU(3, 5));;
gap> g2:= GeneratorsOfGroup(SU(4, 5));;
gap> m:= GModuleByMats(SMTX.MatrixSum(g1,g2), GF(25));;
gap> MTX.IsIrreducible(m);
false
gap> form:= MTX.InvariantSesquilinearForm( m );;
gap> Display( form );
. . 1 . . . .
. 1 . . . . .
1 . . . . . .
. . . . . . 1
. . . . . 1 .
. . . . 1 . .
. . . 1 . . .
gap> frob5 := g -> List(g,row->List(row,x->x^5));; # field involution
gap> ForAll(MTX.Generators(m), x -> x*form*TransposedMat(frob5(x)) = form);
true
]]></Example>

</Description>
</ManSection>

Expand All @@ -617,6 +685,7 @@ or <K>fail</K> if no such form exists.
returns a basis of the underlying vector space of <A>module</A> which is contained
in an orbit of the action of the generators of module on that space.
This is used by <Ref Func="MTX.InvariantQuadraticForm"/> in characteristic 2.
Requires <A>module</A> to be irreducible.
</Description>
</ManSection>

Expand Down
43 changes: 12 additions & 31 deletions lib/meataxe.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -3311,21 +3311,19 @@
##
#F InvariantBilinearForm( module ) . . . .
##
## Look for an invariant bilinear form of the absolutely irreducible
## GModule module. Return fail, or the matrix of the form.
## Look for an invariant bilinear form of the GModule module.
## Return fail, or the matrix of the form.
SMTX.InvariantBilinearForm:=function( module )
local DM, iso;

if not SMTX.IsMTXModule(module) or
not SMTX.IsAbsolutelyIrreducible(module) then
Error(
"Argument of InvariantBilinearForm is not an absolutely irreducible module");
if not SMTX.IsMTXModule(module) then
Error("Argument of InvariantBilinearForm is not a module");

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fi;
if IsBound(module.InvariantBilinearForm) then
return module.InvariantBilinearForm;
fi;
DM:=SMTX.DualModule(module);
iso:=MTX.IsomorphismIrred(module,DM);
iso:=MTX.IsomorphismModules(module,DM);
if iso = fail then
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@fingolfin fingolfin Jan 15, 2025 •
edited
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Here, too, we might get additional problems, i.e., iso may not be symmetric -- we ought to add a check for that.

SMTX.SetInvariantBilinearForm(module, fail);
return fail;
Expand Down Expand Up @@ -3368,41 +3366,24 @@
##
#F InvariantSesquilinearForm( module ) . . . .
##
## Look for an invariant sesquililinear form of the absolutely irreducible
## GModule module. Return fail, or the matrix of the form.
## Look for an invariant sesquililinear form of the GModule module.
## Return fail, or the matrix of the form.
SMTX.InvariantSesquilinearForm:=function( module )
local DM, q, r, iso, isot, l;
local DM, iso;

if not SMTX.IsMTXModule(module) or
not SMTX.IsAbsolutelyIrreducible(module) then
Error(
"Argument of InvariantSesquilinearForm is not an absolutely irreducible module"
);
if not SMTX.IsMTXModule(module) then
Error("Argument of InvariantSesquilinearForm is not a module");
fi;

if IsBound(module.InvariantSesquilinearForm) then
return module.InvariantSesquilinearForm;
fi;
DM:=SMTX.TwistedDualModule(module);
iso:=MTX.IsomorphismIrred(module,DM);
iso:=MTX.IsomorphismModules(module,DM);
if iso = fail then
SMTX.SetInvariantSesquilinearForm(module, fail);
return fail;
fi;
# Replace iso by a scalar multiple to get iso twisted symmetric
q:=Size(module.field);
r:=RootInt(q,2);
isot:=List( TransposedMat(iso), x -> List(x, y->y^r) );
isot:=iso * isot^-1;
Comment on lines -3392 to -3396
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@fingolfin fingolfin Jan 15, 2025 •
edited
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Looking at this again I realize that of course this code is needed. Not sure why I was too dense to grasp it before: the reason why is right in this comment: to ensure we get a form which is twisted symmetric. I.e. the matrix M representing the isomorphism aka the form should satisfy $M = \overline{M}^T$.

Indeed in the code the matrix M is called iso and it computes iso * isot^-1 which is just $M \cdot (\overline{M}^T)^{-1}$. If it is the identity, we are done. If it is a scalar matrix, we can hopefully scale M to satisfy the property, which is what the below does.

I'll restore the deleted code and add comments with details.

That then gets me back to the tests failing then, because now in the reducible case, iso * isot^-1 is not always diagonal. I did not yet look into the details, but surely this happens if there is a homogeneous reducible summand of the module, i.e., $A \oplus A$ -- then there are many automorphisms which are not "twisted symmetric". Gotta deal with that.

if not IsDiagonalMat(isot) then
Error("Form does not seem to be of the right kind (non-diagonal)!");
fi;
l:=LogFFE(isot[1,1],Z(q));
if l mod (r-1) <> 0 then
Error("Form does not seem to be of the right kind (not (q-1)st root)!");
fi;
iso:=Z(q)^(l/(r-1)) * iso;
iso:=ImmutableMatrix(GF(q), iso);
iso:=ImmutableMatrix(module.field, iso);
SMTX.SetInvariantSesquilinearForm(module, iso);
return iso;
end;
Expand Down
16 changes: 13 additions & 3 deletions tst/testinstall/meataxe.tst
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Original file line number Diff line number Diff line change
@@ -1,4 +1,4 @@
#@local G,M,M2,M3,M4,M5,V,bf,bo,cf,homs,m,mat,qf,randM,res,sf,subs,mats,Q,orig,S
#@local G,M,M2,M3,M4,M5,V,bf,bo,cf,homs,m,mat,qf,randM,res,sf,subs,mats,Q,orig,S,g1,g2,form,frob5
gap> START_TEST("meataxe.tst");

#
Expand Down Expand Up @@ -105,6 +105,16 @@ gap> MTX.OrthogonalSign(M2);
gap> SMTX.RandomIrreducibleSubGModule(M2); # returns false for irreducible module
false

# test invariant form detection on reducible module with two isomorphic
# components (hence many automorphisms exist)
gap> g1:= GeneratorsOfGroup(SU(4, 5));;
gap> g2:= GeneratorsOfGroup(SU(4, 5));;
gap> m:= GModuleByMats(SMTX.MatrixSum(g1,g2), GF(25));;
gap> form:= MTX.InvariantSesquilinearForm( m );;
gap> frob5 := g -> List(g,row->List(row,x->x^5));; # field involution
gap> ForAll(MTX.Generators(m), x -> x*form*TransposedMat(frob5(x)) = form);
true

#
gap> Display(MTX.IsomorphismModules(M,M));
1 . . . .
Expand Down Expand Up @@ -254,8 +264,8 @@ gap> MTX.InvariantQuadraticForm( m );
Error, Argument of InvariantQuadraticForm is not an absolutely irreducible mod\
ule
gap> MTX.OrthogonalSign( m );
Error, Argument of InvariantBilinearForm is not an absolutely irreducible modu\
le
Error, Argument of InvariantQuadraticForm is not an absolutely irreducible mod\
ule
gap> mats:= GeneratorsOfGroup( SP( 4, 2 ) );;
gap> m:= GModuleByMats( mats, GF(2) );;
gap> Q:= MTX.InvariantQuadraticForm( m );
Expand Down
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