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#############################################################################
##
#W examples.gi NilMat Alla Detinko
#W Bettina Eick
#W Dane Flannery
##
##
## This file contains methods to construct examples of some interesting nilpotent
## matrix groups over Q and over finite fields.
##
#############################################################################
##
#F MonomialNilpotentMatGroup( n )
##
## This function constructs Kronecker products of all MonomialSylow(pi,ai),
## where n = p1^a1...pr^ar is factorization of n.
##
InstallGlobalFunction( MonomialNilpotentMatGroup, function(n)
local l1,l2,k,r,H,W,i,j;
# First construct prime factorization of n.
l1 := Filtered([2..n],x -> IsPrimeInt(x)); #list of primes <=n
l2 := Filtered(l1, x -> IsInt(n/x)); #list of prime x dividing n
r := List(l2, x -> PLength(n,x)); #max powers of x in n
k := Length(l2);
if k=1 then return Group(MonomialSylow(l2[1],r[1]));fi;
H :=[]; #list of lists of generators
for i in [1..k] do
H[i] := MonomialSylow(l2[i],r[i]);
od;
W := H[1];
for j in [2..k] do
W := KroneckerProductLists(W,H[j]);
od;
return Group(W);
end );
#############################################################################
##
#F ReducibleNilpotentMatGroupRN( m, k [,l] ) . .a nilpotent mat group over Q
##
## NilpotentReducibleMatGroupRN constructs a nilpotent subgroup of GL(n, Q)
## for n = mk which is reducible, but not completely reducible (and is not
## represented in the block upper triangular form). If a third arguement is
## given, then the entries in the initial matrices are choosen from [1..l],
## otherwise l=1 is taken. The constructed group is a Kronecker product of a
## subgroup of UT(m,Q) and MonomialNilpotent(k).
##
BindGlobal( "ReducibleNilpotentMatGroupRN", function(arg)
local m, k, e, L, M, i;
# catch arguments
m := arg[1];
k := arg[2];
e := IdentityMat(m, Rationals);
# an easy case
if m = 1 then return MonomialNilpotentMatGroup(k); fi;
# a list of matrices of UT(m,Q).
L := [];
for i in [1..(m-1)] do
L[i] := ShallowCopy(e);
if IsBound(arg[3]) then
L[i][i][m] := Random([1..arg[3]]);
else
L[i][i][m] := 1;
fi;
od;
# monomial subgroups
M := GeneratorsOfGroup(MonomialNilpotentMatGroup(k));
# Kronecker products
return Group(KroneckerProductLists(L,M));
end );
#############################################################################
##
#F ReducibleNilpotentMatGroupFF( m,k,p,l ) . .a nilpotent mat group over FF
##
## Constructs a nilpotent subgroup of GL(n, q)
## for n = mk and q = p^l which is reducible, but not completely reducible
## (and is not represented in the block upper triangular form).
##
## The group is a Kronecker product of a subgroup of UT(m,q) and a
## NilpotentMaxAbsIrreducible(k,po,l); it is supposed that (k,po,l) are
## such that the latter exists.
##
BindGlobal( "ReducibleNilpotentMatGroupFF", function(m,k,po,l)
local U, q, w, e, L, i;
U := MaximalAbsolutelyIrreducibleNilpotentMatGroup(k,po,l);
if U = fail then return fail;fi;
if m = 1 then return U; fi;
q := po^l;
w := Z(q);
e := IdentityMat(m, w);
L := [];
# a list of matrices of UT(m,w).
for i in [1..(m-1)] do
L[i] := ShallowCopy(e);
L[i][i][m] := w;
od;
return Group(KroneckerProductLists(L,GeneratorsOfGroup(U)));
end );
#############################################################################
##
#F ReducibleNilpotentMatGroup( m,k,[p,l] )
##
InstallGlobalFunction(ReducibleNilpotentMatGroup, function(arg)
if Length(arg) = 4 then
return ReducibleNilpotentMatGroupFF(arg[1],arg[2],arg[3],arg[4]);
elif Length(arg) = 3 then
return ReducibleNilpotentMatGroupRN(arg[1],arg[2],arg[3]);
else
return ReducibleNilpotentMatGroupRN(arg[1],arg[2]);
fi;
end );
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