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Also not an answer, just a hint on how the GAP code can be made a bit faster, for future reference: GAP actually "knows" the character tables for the alternating groups. For some reason I don't know this is not used here, but we can ask GAP directly for that character table, bypassing all computations in groups, to get a faster computation, like so:

Phenomenon:=function(n)
    local ct, irr,r,L,i,j,k;
    ct:=CharacterTable(Concatenation("A",String(n)));
    irr:=Irr(ct);
    r:=Size(irr);
    for i in [2..r] do
        for j in [i..r] do
            if IsIrreducible(irr[i]*irr[j]) then
                Print([i,j],"\n");; 
            fi;
        od;
    od;
end;;

for n in [5..20] do Print(n,"\n");; Phenomenon(n);; od;