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A118889 Ratio of Dimensions of the traditional Cartan exceptional group sequence A1,G2,F4,E6,E7,E8 to the Cartan matrix Dimension: Dimc={1, 2, 4, 6, 7, 8} DimG={3, 14, 52, 78, 133, 248} DimG/DimC={3, 7, 13, 13, 19, 31}.

%I A118889 #13 Jun 24 2025 10:01:44
%S A118889 3,7,13,13,19,31
%N A118889 Ratio of Dimensions of the traditional Cartan exceptional group sequence A1,G2,F4,E6,E7,E8 to the Cartan matrix Dimension: Dimc={1, 2, 4, 6, 7, 8} DimG={3, 14, 52, 78, 133, 248} DimG/DimC={3, 7, 13, 13, 19, 31}.
%C A118889 The sequence is inherently unordered, because there is no standard ordering of these groups. - _R. J. Mathar_, Dec 04 2011
%F A118889 P[n]=Poincare-Polynomial[n]=Product[1+t^A129766[m],{m,1,n}]
%F A118889 DimG[n]=Length[CoefficientList[P[n],t]]-1
%F A118889 Pc[n]=CharacteristicPolynomial[M[n],x]
%F A118889 DimC[n]=Length[CoefficientList[Pc[n],x]]-1
%F A118889 a[n]=DimG[n]/DimC[n]
%t A118889 (* Cartan Matrices: *)
%t A118889 e[3] = {{2}};
%t A118889 e[4] = {{2, -3}, {-1, 2}};
%t A118889 e[5] = {{2, -1, 0, 0}, {-1, 2, -2, 0}, {0, -1, 2, -1}, {0, 0, -1, 2}};
%t A118889 e[6] = {{2, 0, -1, 0, 0, 0}, {0, 2, 0, -1, 0, 0}, {-1, 0, 2, -1, 0, 0}, { 0, -1, -1, 2, -1, 0}, { 0, 0, 0, -1, 2, -1}, { 0, 0, 0, 0, -1, 2}};
%t A118889 e[7] = {{2, 0, -1, 0, 0, 0, 0}, {0, 2, 0, -1, 0, 0, 0}, {-1, 0, 2, -1, 0, 0, 0}, {0, -1, -1, 2, -1, 0, 0}, {0, 0, 0, -1, 2, -1, 0}, { 0, 0, 0, 0, -1, 2, -1 }, { 0, 0, 0, 0, 0, -1, 2 }};
%t A118889 e[8] = { {2, 0, -1, 0, 0, 0, 0, 0}, { 0, 2, 0, -1, 0, 0, 0, 0}, {-1, 0, 2, -1, 0, 0, 0, 0}, {0, -1, -1, 2, -1, 0, 0, 0}, {0, 0, 0, -1, 2, -1, 0, 0}, { 0, 0, 0, 0, -1, 2, -1, 0}, { 0, 0, 0, 0, 0, -1, 2, -1}, {0, 0, 0, 0, 0, 0, -1, 2}} ;
%t A118889 a0 = Table[Length[CoefficientList[CharacteristicPolynomial[e[n], x], x]] - 1, {n, 3, 8}]; (* Poincaré Polynomials*)
%t A118889 (*Poincaré polynomial exponents for G2, E6, E7, E8 from A005556, A005763, A005776 and Armand Borel's Essays in History of Lie Groups and Algebraic Groups*) (* b[n] = a[n] + 1 : DimGroup = Apply[Plus, b[n]]*)
%t A118889 a[0] = {1};
%t A118889 a[1] = {1, 5};
%t A118889 a[2] = {1, 5, 7, 11};
%t A118889 a[3] = {1, 4, 5, 7, 8, 11};
%t A118889 a[4] = {1, 5, 7, 9, 11, 13, 17};
%t A118889 a[5] = {1, 7, 11, 13, 17, 19, 23, 29};
%t A118889 b0 = Table[Length[CoefficientList[Expand[Product[(1 + t^(2*a[i][[n]] + 1)), {n, 1, Length[a[i]]}]], t]] - 1, {i, 0, 5}];
%t A118889 Table[b0[[n]]/a0[[n]], {n, 1, Length[a0]}]
%Y A118889 Cf. A117133, A129766, A005556, A005763, A005776.
%K A118889 nonn,fini,full,less,uned
%O A118889 1,1
%A A118889 _Roger L. Bagula_, May 17 2007