1,1

Establishes a direct link between the exponents m(i) in the Poincare polynomial representations the group dimension and the dimension of the Cartan Matrices.

(*http : // www.valdostamuseum.org/hamsmith/Weyl.html*) (*http : // www3.baylor.edu/~Mark_Sepanski/e8.html*)

(*Poincare polynomial exponents for G2, E6, E7, E8 from A005556, A005763, A005776 and Armand Borel's Essays in History of Lie Groups and Algebraic Groups*)

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]

(* Cartan Matrices*) (*http : // www.valdostamuseum.org/hamsmith/Weyl.html*) (*http : // www3.baylor.edu/~Mark_Sepanski/e8.html*) 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}]; (* Poincare Polynomials*) (*Poincare 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]}

Cf. A129766, A005556, A005763, A005776.

Sequence in context: A010260 A128156 A108768 this_sequence A077149 A064829 A035496

Adjacent sequences: A118886 A118887 A118888 this_sequence A118890 A118891 A118892

nonn,uned

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 17 2007