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Fix Frank's SL2 in nat.rep. code in even char #394

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Merged
fingolfin merged 1 commit into from
Mar 16, 2026
Merged

Fix Frank's SL2 in nat.rep. code in even char #394

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15 changes: 11 additions & 4 deletions contrib/frank/sl2/NatSL2q.g
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Original file line number Diff line number Diff line change
Expand Up @@ -40,7 +40,7 @@ RecogNaturalSL2 := function(G, q)
mat, tm, ym, y, ymat, tr, d, cm, r1, r2, r, log, i, trupm,
smm, trlowm, F, a2, bas, e, l, emax, tmp;
GM := GroupWithMemory(G);
one := One(G.1[1,1]);
one := OneOfBaseDomain(G.1);
zero := Zero(one);

# find element x of order q-1
Expand Down Expand Up @@ -100,6 +100,8 @@ RecogNaturalSL2 := function(G, q)
# now y^(tm * mat) = diag(a, a^-1)
tr := tm*mat;

Assert(3, y^(tm * mat) = DiagonalMat([a, a^-1]) );

# a-eigenvector of x in new basis
d := tr^-1 * [mat[1,1],mat[2,1]];
# can be scaled to [1,d]
Expand All @@ -125,7 +127,7 @@ RecogNaturalSL2 := function(G, q)
# this will in most cases be a transvection normalized by x
trupm := Comm(xm, ym^i*cm);
smm := trupm^mat;
if smm[1,2] = zero then
if smm[1,2] = zero or smm[2,1] <> zero then
i := false;
else
# rescale first column of mat such that trupm^mat = [[1,1],[0,1]]
Expand All @@ -136,6 +138,8 @@ RecogNaturalSL2 := function(G, q)
fi;
until IsInt(i);

Assert(3, trupm^mat = [[1,1],[0,1]] * one);

# same for the other eigenvector of x:
# 1/a-eigenvector of x in new basis
d := tr^-1 * [mat[1,2],mat[2,2]];
Expand Down Expand Up @@ -164,7 +168,7 @@ RecogNaturalSL2 := function(G, q)
# that the conjugate matrix is [[1,0],[1,1]]).
trlowm := Comm(xm, ym^i*cm);
smm := trlowm^mat;
if smm[2,1] = zero then
if smm[2,1] = zero or smm[1,2] <> zero then
i := false;
fi;
fi;
Expand Down Expand Up @@ -201,15 +205,18 @@ RecogNaturalSL2 := function(G, q)
od;
fi;

Assert(3, trlowm^mat = [[1,0],[1,1]] * one);

# finally power x to change a to 1/Z(q)
if a <> 1/Z(q) then
log := DLog(a, 1/Z(q), qm1fac);
xm := xm^log;
fi;

Assert(3, xm^mat = DiagonalMat([Z(q)^-1, Z(q)]));

# return SLPs of elements mapped by mat to
# diag(1/Z(q),Z(q))[[1,0],[1,1]], [[1,1],[0,1]],
# and mat
return [List([xm, trlowm, trupm], SLPOfElm), mat];
end;

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