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}.

3, 7, 13, 13, 19, 31

Offset: 1

Author

Roger L. Bagula, May 17 2007

Keywords

• nonn
• fini
• full
• less
• uned

Comments

The sequence is inherently unordered, because there is no standard ordering of these groups. - R. J. Mathar, Dec 04 2011

Crossrefs

Cf. A117133, A129766, A005556, A005763, A005776.

Programs

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}]; (* Poincaré Polynomials*)
(*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]]*)
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]}]

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]