Resolve some reported minor issues (#3879)

* CLEANUP: Minor MeatAxe quirks

This addresses several points in #3764

Remove obsolete BasisRadical method Dualizing the socle is better since
maximals aleady are computed through dualization.

Documentation now states `InducedAction` only for nontrivial result.
Make result of BasisSocle immutable.

* FIX: AssociatedPartition for empty partition

This closes #3854

* Added test.

doc/ref/meataxe.xml

@@ -393,6 +393,7 @@ and <A>change</A> is the base change from <A>bas</A> to <A>sub</A>
 
 <Description>
 creates a new module corresponding to the action of <A>module</A> on
+the non-trivial submodule
 <A>sub</A>.
 In the <C>NB</C> version the basis <A>sub</A> must be normed.
 (That is it must be in echelon form with pivots normed to 1,
@@ -405,7 +406,7 @@ see&nbsp;<Ref Func="MTX.NormedBasisAndBaseChange"/>.)
 
 <Description>
 creates a new module corresponding to the action of <A>module</A> on the
-factor of <A>sub</A>. If <A>compl</A> is given, it has to be a basis of a
+factor of the proper submodule <A>sub</A>. If <A>compl</A> is given, it has to be a basis of a
 (vector space-)complement of <A>sub</A>. The action then will correspond to
 <A>compl</A>.
 <P/>

lib/combinat.gi

@@ -2466,6 +2466,7 @@ InstallGlobalFunction(AssociatedPartition,function(lambda)
   local res, k, j;
   res := [];
   k := Length(lambda);
+  if k=0 then return res;fi; # empty partition
   for j in [1..lambda[1]] do
     if j <= lambda[k] then
       res[j] := k;

lib/meataxe.gi

@@ -3128,17 +3128,6 @@ local a,u,i,nb;
 end;
 SMTX.BasesMinimalSupermodules:=SMTX_BasesMinimalSupermodules;
 
-SMTX_BasisRadical:=function(module)
-local m,i,r;
-  m:=SMTX.BasesMaximalSubmodules(module);
-  r:=m[1];
-  for i in [2..Length(m)] do
-    r:=SumIntersectionMat(r,m[i])[2];
-  od;
-  return r;
-end;
-SMTX.BasisRadical:=SMTX_BasisRadical;
-
 #############################################################################
 ##
 #F SMTX.SpanOfMinimalSubGModules(m1, m2) . .
@@ -3182,7 +3171,7 @@ local cf, mat, i;
    cf:=SMTX.CollectedFactors(module);
    mat:=Concatenation(List(cf,i->SMTX_SpanOfMinimalSubGModules(i[1],module)));
    if Length(cf) = 1 then
-     return mat;
+     return ImmutableMatrix(module.field,mat);
    fi;
    TriangulizeMat(mat);
    mat:=ImmutableMatrix(module.field,mat);

tst/testinstall/combinat.tst

@@ -498,5 +498,11 @@ gap> Print(List( [0..14], Bernoulli ),"\n");
 gap> Bernoulli( 80 );
 -4603784299479457646935574969019046849794257872751288919656867/230010
 
+# AssociatedPartition
+gap> AssociatedPartition([]);
+[  ]
+gap> AssociatedPartition(Concatenation([7],ListWithIdenticalEntries(99,1)));
+[ 100, 1, 1, 1, 1, 1, 1 ]
+
 # thats it for the combinatorical package  ##################################
 gap> STOP_TEST( "combinat.tst", 1);