I'll start by saying that the reference dancing here makes my eyes cross, so take my remarks with some salt. Also, since I don't have your input file or a good idea of what this is to produce, I can't really test. That said, look here...

    my @parents = @population[40 .. 49];
    shuffle @parents;

    my @children;
    push @children, crossover( pop @parents, pop @parents )
        while @parents;

    @population[0 .. 4] = map { mutate($_) } @children;
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The first thing I notice is that crossover returns two things, so @children will have the same length as @parents, but you're discarding half of them when you put them back into @population. I can't tell if that's deliberate or not, but it easily could be.

What I don't think is deliberate is that mutate is operating directly on the items it gets.

sub mutate{
    my @string = @{$_[0]};
    for my $i (0.. $#string){
        my $aref = $string[$i];
        #print "\n";
        if (rand() < $mutation_rate){
            my $position1 = int(rand($#{$aref}));
            my $position2 = int(rand($#{$aref}));
            if ($position1 ne $position2){
                my $temp = $string[$i][$position1];
                $string[$i][$position1] = $string[$i][$position2];
                $string[$i][$position2] = $temp;
            }        
        }
    }
    return \@string;
}
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You copy into @string, which is good, and then pass out a reference to it, which is also good, but in the meantime you do this:

                my $temp = $string[$i][$position1];
                $string[$i][$position1] = $string[$i][$position2];
                $string[$i][$position2] = $temp;
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I think $string[$i] is a copy and so safe to modify, but $string[$i][$position1] is a reference that also exists in the original array. What you pass in are references from @children, but those also come directly from @parents.

To fix this, it might help to make a deep copy of the data structures involved before you operate on them. These might help with that:

• deep copy of nested data structure
• deepcopy
• perlfaq4: How do I print out or copy a recursive data structure?

Again, I'm not sure that's the problem, but it might be worth looking into.