%I #6 Jul 19 2015 10:13:48
%S 1,1,5,1,5,7,11,1,4,5,7,8,11,1,5,7,9,11,13,17,1,7,11,13,17,19,23,29
%N Triangular array read by rows, made up of traditional exceptional groups plus A1: as A1,G2,F4,E6,E7,E8 as m(i) exponents as in A005556, A005763, A005776.
%C Extra condition of group dimension: b[n] = a[n] + 1 ; DimGroup = Apply[Plus, b[n]]; Table[Apply[Plus, b[n]], {n, 0, 5}] {3, 14, 52, 78, 133, 248} Extra condition of Betti sum: Table[Apply[Plus, CoefficientList[Expand[Product[(1 + t^(2*a[i][[n]] + 1)), {n, 1,Length[a[i]]}]], t]], {i, 0, 5}] {2, 4, 16, 64, 128, 256} These exponents are necessary to the PoincarĂ© polynomials for these exceptional groups.
%H Armand Borel, <a href="http://www.amazon.com/EssaysHistoryGroupsAlgebraicMathematics/dp/0821802887/ref=pd_rhf_p_3/10400296170633535">Essays in History of Lie Groups and Algebraic Groups</a> gives G2 PoincarĂ© polynomial, History of Mathematics, V. 21.
%e 1;
%e 1,5;
%e 1,5,7,11;
%e 1,4,5,7,8,11;
%e 1,5,7,9,11,13,17;
%e 1,7,11,13,17,19,23,29;
%t a[0] = {1}; a[1] = {1, 5}; a[2] = {1, 5, 7, 11}; a[3] = {1, 4, 5, 7, 8, 11}; a[4] = {1, 5, 7, 9, 11, 13, 17}; a[5] = {1, 7, 11, 13, 17, 19, 23, 29}; Flatten[Table[a[n], {n, 0, 5}]]
%Y Cf. A005556, A005763, A005776.
%K nonn,fini,full,tabf,uned
%O 1,3
%A _Roger L. Bagula_, May 16 2007
