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## convert the element el of the group algebra kG to an element of the table T. Of course T and kG should have the same underlying group
MIPElementAlgebraToTable := function(el, kG, T)
local k, maxweight, i, lim, sup, coeffs, v0, res, exps, nonzeroexps, sum, S, initprod, prod, s, es, innersum, fac, l;
k := T.fld;
if Augmentation(el) <> Zero(k) then
Error("<el> must be an element in the augmentation ideal");
fi;
maxweight := Maximum(T.wgs);
# we find the limit of the group elements we need for the case T does not see the whole group basis
if maxweight < Maximum(T.pre.jen.weights) then
for i in [1..Size(T.pre.jen.pcgs)] do
if T.pre.jen.weights[i] > maxweight then
lim := i - 1;
fi;
od;
else
lim := Size(T.pre.jen.pcgs);
fi;
sup := Support(el);
coeffs := CoefficientsBySupport(el);
v0 := ListWithIdenticalEntries(T.dim, Zero(k));
res := ShallowCopy(v0);
## for each element g1^e1...gm^em we convert g1^e1...gm^em to an expression in the Jennings basis using the formulas known for this
for i in [1..Size(sup)] do
exps := ExponentsOfPcElement(T.pre.jen.pcgs, sup[i]){[1..lim]};
nonzeroexps := PosNonzero(exps);
sum := ShallowCopy(v0);
# run over all supsets of non-zero exponents. Each gives a summand
for S in Combinations( nonzeroexps ) do
initprod := false;
prod := ShallowCopy(v0);
# this is the factor (g_s - 1)^es which is converted using binomial coefficients
for s in S do
es := exps[s];
innersum := ShallowCopy(v0);
innersum[T.pre.poswone[s]] := One(k)*Binomial(es, 1);
fac := ShallowCopy(v0); # corresponding to g_s - 1
fac[T.pre.poswone[s]] := One(k);
for l in [2..es] do
fac := MultByTable(T, fac, fac); # obtain (g_s - 1)^l
innersum := innersum + One(k)*Binomial(es, l)*fac;
od;
if initprod then
prod := MultByTable(T, prod, innersum);
else
prod := innersum;
initprod := true;
fi;
od;
sum := sum + prod;
od;
res := res + coeffs[i]*sum;
od;
return res;
end;
### convert from element of table to a representative in the group algebra
MIPElementTableToAlgebra := function(v, T, kG)
local k, res, i, exps, prod, j, G, iota, basisels;
k := T.fld;
res := Zero(kG);
G := UnderlyingMagma(kG);
iota := Embedding(G, kG);
basisels := List(T.pre.jen.pcgs, x -> x^iota - One(kG));
for i in [1..T.dim] do
if v[i] <> Zero(k) then
exps := T.pre.exps[i];
prod := One(kG);
for j in [1..Size(exps)] do
prod := prod*(basisels[j]^exps[j]);
od;
prod := v[i]*prod;
res := res + prod;
fi;
od;
return res;
end;
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