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\Chapter{Tables}
Finite dimensional algebras can be described by structure contants tables.
For nilpotent algebras it is not neccessary to store a full structure
contants table. To use this feature, we introduce *nilpotent structure
constants tables* or just *nilpotent tables* for short. These are used
heavily throughout the package.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Nilpotent tables}
Let $A$ be a finite-dimensional nilpotent associative algebra over a
field $F$. Let $(b_1, \ldots, b_d)$ be a *weighted basis* of $A$; that
is, a basis with weights $(w_1, \ldots, w_d)$ satifying that $A^j =
\langle b_i \mid w_i \geq j \rangle$. Let
$$ b_i b_j = \sum_k a_{i,j,k} b_k.$$
The nilpotent table $T$ for $A$ (with respect to the basis $(b_1,
\ldots, b_d)$) is a record with the following entries.
\beginitems
<dim> &
the dimension $d$ of $A$;
<fld> &
the field $F$ of $A$;
<wgs> &
the weights $(w_1, \ldots, w_d)$;
<rnk> &
the rank $e$ of $A$ (i.e. the dimension of $A/A^2$).
<wds> &
a list of length $d$ with holes; If the $i$th entry is bounded, then
it is of the form $[k,l]$. In this case, $w_i > 1$ and $b_i = b_k b_l$
and $w_k = 1$ and $w_l = w_i-1$ holds.
<tab> &
a partial structure contants table for $A$; If $tab[i][j][k]$ is bounded,
then it is $a_{i,j,k}$. Note that either a full vector $tab[i][j]$ is
given or $tab[i][j]$ is unbounded. The entry $tab[i][j][k]$ is available
for $1 \leq i,j \leq e$ and if $wds[i]$ is unbounded.
<com> &
optional; If this is bounded, then it is a boolean. If this boolean
is true, then the algebra is assumed to be commutative.
\enditems
In a nilpotent table not all structure constants are readily available.
The following function determines the structure constants for the
product $b_i b_j$. If the global variable $STORE$ is true, then the
function stores the computed entry in the table.
\> GetEntryTable( T, i, j ) F
The result of the multiplication of the elements $v$ and $w$ in $T$
can be obtained using the following function. An example of its use is
provided below.
\> MultByTable( T, v, w ) F
We consider two nilpotent tables as equal, if they would be equal if
the full set of structure constants tables would be bound. The following
function provides an effective check for this.
\> CompareTables( T1, T2 ) F
A nilpotent table contains redundant information and hence can be
inconsistent. The next functions can be used to check this to some
extend.
\> CheckAssociativity( T ) F
Checks that $(b_i b_j) b_k = b_i (b_j b_k)$ for all $i,j,k$. Note that this
may be time-consuming.
\> CheckCommutativity( T ) F
Checks whether $T$ defines a commutative algebra and sets the entry $com$
accordingly.
\> CheckConsistency( T ) F
Checks that $wds$ and $tab$ are compatible. This assumes that
CheckAssociativity returns true.
All later described algorithms of this package assume that the tables
considered are fully consistent.
\beginexample
gap> T := rec( dim := 3,
fld := GF(2),
rnk := 2,
wgs := [ 1, 1, 2 ],
wds := [ ,, [ 2, 1 ] ],
tab := [] );;
gap> T.tab[1] := [[0,0,0],[0,0,1]] * One(T.fld);;
gap> T.tab[2] := [[0,0,1],[0,0,0]] * One(T.fld);;
gap> GetEntryTable( T, 3, 1 );
[ 0*Z(2), 0*Z(2), 0*Z(2) ]
\endexample
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\Section{Algebras in the GAP sense}
We provide functions to convert back and forth between algebras in the GAP
sense and nilpotent tables.
\> AlgebraByTable( T ) F
\> NilpotentTable( A ) F
Note that the second function fails if $A$ is not nilpotent.
For modular group algebras of $p$-groups, the group algebra itself is
not nilpotent (as it contains a unit), but its Jacobson radial is. The
following function determines a nilpotent table for the Jacobson radical.
\> NilpotentTableOfRad( FG ) F
\beginexample
gap> A := GroupRing(GF(2), SmallGroup(8,3));
<algebra-with-one over GF(2), with 3 generators>
gap> NilpotentTableOfRad(A);
rec( dim := 7, fld := GF(2), rnk := 2,
tab :=
[
[ [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ],
[ [ 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ],,
[ [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ] ],
wds := [ ,, [ 1, 2 ],, [ 1, 4 ], [ 2, 4 ], [ 1, 6 ] ],
wgs := [ 1, 1, 2, 2, 3, 3, 4 ] )
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\Section{Tables for the Modular Isomorphism Problem}
A special kind of nilpotent table is available in the context of the
Modular Isomorphism Problem. Let $G$ be a finite group, $F$ a field of
characteristic $p$ and $I(FG)$ the augmentation ideal of $FG$ which equals
its radical and is a nilpotent ideal. Two functions allow to compute
class-$n$ quotient of $I(FG)$, i.e. $I(FG)/I(FG)^{n+1}$. The output is a nilpotent
table for this quotient, but in addition to the standard entries of
a nilpotent table it contains further entries, which allow more efficient
computations and can also facilitate manual calculations.
This allows to determine the class-$n$ quotient of the augmentation ideal
without computing the full augmentation ideal using $NilpotentTableOfRad$.
The corresponding table can be computed by
\> TableOfRadQuotient( FG, n ) F
or
\> ModIsomTable( G, n, [f] ) F
Here $ModIsomTable(G, n)$ will produce the quotient with respect to the algebra
$\F_pG$, while $ModIsomTable( G, n, f )$ will do the same for the algebra $\F_{p^f}G$.
If $ModIsomTable$ is executed as $ModIsomTable(G, n)$ it provides the table with respect to the algebra $\F_pG$.
\medskip
The components $dim$, $fld$, $rnk$, $tab$, $wgs$, $wds$
remain unchanged from a usual nilpotent table. The additional components are
$commwords$, $powwords$ and $pre$. These new components
contain additional information on precisely which basis of $I(FG)/I(FG)^{n+1}$ is used
and what the result of multiplying basis elements is. We explain how users
can understand how the basis looks and how they can multiply two elements in the algebra.
The components $T\.commords$ and $T\.powwords$ contain information on how the elements of
the basis behave with respect to commutators and $p$-th powers. The component $T\.pre$
contains information on the construction of the basis and we describe it in more detail.
\medskip
The dimension of $I(FG)/I(FG)^{n+1}$ is recorded in $T\.dim$.
The basis of $I(FG)/I(FG)^{n+1}$ is found as in the theory of Jennings going back
to \cite{Jen41}, cf. \cite{MM22} for the information needed here. The elements
of $G$ chosen to provide the basis of subsequent quotients of dimension subgroups
are recorded in $T\.pre\.jen\.pcgs$. Let us call these elements $g_1,\ldots,g_m$.
Note that $|G| = p^m$. The weights of the elements $g_1-1,\ldots ,g_m-1$ are recorded
in $T\.pre\.jen\.weights$. If now $r$ is an integer smaller than $T\.dim+1$, then the
$r$-th element of the basis of $I(FG)/I(FG)^{n+1}$ is
$(g_1-1)^{e_1} \cdot \ldots (g_m-1)^{e_m}$ where $[e\_1,\ldots,e\_m] = T\.pre\.exps[r]$.
The weight of this element is recorded in $T\.wgs[r]$ and also $T\.pre\.weights[r]$.
Moreover, the positions of $g_1-1,\ldots ,g_m-1$ in the chosen basis of $T$ are recorded
in $T\.pre\.poswone$. We elaborate using an example.
\medskip
We consider the group $G=SmallGroup(3^7, 19)$. The following calculation shows
that $I(FG)/I(FG)^9$ has dimension $135$ and that the full augmentation
ideal $I(FG)$ has dimension $2186$.
\beginexample
gap> G := SmallGroup(3^7, 19);;
gap> T := ModIsomTable(G, 8);;
gap> T.dim;
135
gap> FG := GroupRing(GF(3), G);;
gap> TT := TableOfRadQuotient(FG, 8);;
gap> TT.dim;
135
gap> T := ModIsomTable(G, 38);;
gap> T.dim;
2186
gap> T := ModIsomTable(G, 39);;
gap> T.dim;
2186
\endexample
We next consider an example how the basis used can be recognized.
\beginexample
gap> G := DihedralGroup(8);;
gap> T := ModIsomTable(G, 4);;
gap> T.dim;
7
gap> pcgs := T.pre.jen.pcgs;
Pcgs([ f1, f2, f3 ])
gap> List(pcgs, Order);
[ 2, 4, 2 ]
gap> pcgs[3] in Center(G);
true
gap> T.pre.exps{[1..7]};
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 1, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 1 ], [ 0, 1, 1 ],
[ 1, 1, 1 ]]
\endexample
We conclude that $I(kG)/I(kG)^5$ is $7$-dimensional and if we denote by
$a$ a reflection and by $b$ a non-central rotation in $G$, then the
basis used by $T$ is, in this order: $(a-1)$, $(b-1)$, $(a-1)(b-1)$, $(b^2-1)$,
$(a-1)(b^2-1)$, $(b-1)(b^2-1)$, $(a-1)(b-1)(b^2-1)$.
\medskip
Say continuing the previous example we want to multiply $(b-1)+(a-1)(b-1)+(a-1)(b^2-1)$ and $(a-1)+(b-1)+(b^2-1)$.
\beginexample
gap> v := Z(2)^0*[0,1,1,0,1,0,0];
[ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ]
gap> w := Z(2)^0*[1,1,0,1,0,0,0];
[ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ]
gap> MultByTable(T,v,w);
[ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ]
\endexample
So the result is $(a-1)(b-1) + (a-1)(b^2-1)$.
\medskip
To facilitate the translation of elements of the group algebra
and a corresponding table of a quotient of the augmentation ideal
the functions
\> MIPElementTableToAlgebra( v, T, FG ) F
and
\> MIPElementAlgebraToTable( el, FG, T ) F
can be used. In the second function of course only a possible
representative of $v$ in $FG$ is provided. Also, only elements
from the augmentation ideal of $FG$ can be represented using
$MIPElementAlgebraToTable$. These functions can be used for
instance to obtain representatives in the same class modulo
a power of the augmentation ideal which are more practical to
work with, as the following example shows.
\beginexample
gap> G := SmallGroup(3^7, 19);
<pc group of size 2187 with 7 generators>
gap> T := ModIsomTable(G, 4);;
gap> FG := GroupRing(GF(3), G);
<algebra-with-one over GF(3), with 7 generators>
gap> iota := Embedding(G, FG);
<mapping: Group( [ f1, f2, f3, f4, f5, f6, f7
] ) -> AlgebraWithOne( GF(3), ... ) >
gap> a := (T.pre.jen.pcgs[1])^iota;
(Z(3)^0)*f1
gap> b := (T.pre.jen.pcgs[2])^iota;
(Z(3)^0)*f2
gap> z := One(FG);
(Z(3)^0)*<identity> of ...
gap> r := (z + (a-z)*(b-z) )^-1;;
gap> Size(Support(r-z));
1376
gap> el := MIPElementAlgebraToTable(r-z, FG, T);
[ 0*Z(3), 0*Z(3), 0*Z(3), Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3),
0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3),
0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ]
gap> MIPElementTableToAlgebra(el, T, FG);
(Z(3))*<identity> of ...+(Z(3)^0)*f3+(Z(3)^0)*f1^2+(Z(3))*f1*f2+(Z(3))*f1*f3+(
Z(3)^0)*f2^2+(Z(3))*f2*f3+(Z(3)^0)*f1^2*f2+(Z(3)^0)*f1*f2^2+(Z(3)^
0)*f1*f2*f3+(Z(3)^0)*f1^2*f2^2
\endexample
We illustrate the information in $T\.pre\.poswone$:
\beginexample
gap> d := (T.pre.jen.pcgs[4])^iota;
(Z(3)^0)*f4
gap> el := MIPElementAlgebraToTable(d-z, FG, T);
[ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3),
0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3),
0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ]
gap> Position(last, Z(3)^0);
11
gap> T.pre.poswone[4];
11
\endexample
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