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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}. 2
 3, 7, 13, 13, 19, 31 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The sequence is inherently unordered, because there is no standard ordering of these groups. - R. J. Mathar, Dec 04 2011 LINKS FORMULA P[n]=Poincare-Polynomial[n]=Product[1+t^A129766[m],{m,1,n}] DimG[n]=Length[CoefficientList[P[n],t]]-1 Pc[n]=CharacteristicPolynomial[M[n],x] DimC[n]=Length[CoefficientList[Pc[n],x]]-1 a[n]=DimG[n]/DimC[n] MATHEMATICA (* Cartan Matrices: *) e[3] = {{2}}; e[4] = {{2, -3}, {-1, 2}}; e[5] = {{2, -1, 0, 0}, {-1, 2, -2, 0}, {0, -1, 2, -1}, {0, 0, -1, 2}}; 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}}; 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 }}; 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}} ; a0 = Table[Length[CoefficientList[CharacteristicPolynomial[e[n], x], x]] - 1, {n, 3, 8}]; (* PoincarĂ© Polynomials*) (*PoincarĂ© 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]]*) 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}; b0 = Table[Length[CoefficientList[Expand[Product[(1 + t^(2*a[i][[n]] + 1)), {n, 1, Length[a[i]]}]], t]] - 1, {i, 0, 5}]; Table[b0[[n]]/a0[[n]], {n, 1, Length[a0]} CROSSREFS Cf. A117133, A129766, A005556, A005763, A005776. Sequence in context: A128156 A108768 A238476 * A077149 A064829 A290642 Adjacent sequences:  A118886 A118887 A118888 * A118890 A118891 A118892 KEYWORD nonn,fini,full,less,uned AUTHOR Roger L. Bagula, May 17 2007 STATUS approved

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Last modified April 16 07:59 EDT 2021. Contains 343030 sequences. (Running on oeis4.)