A117133 -id:A117133 - cp's OEIS frontend

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

Roger L. Bagula, May 17 2007

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

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

(* 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]}]

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]

A129001 Heights of roots in Cartan root systems for exceptional groups: A1, G2, F4, E6, E7, E8.

Original entry on oeis.org

1, 3, 2, 2, 3, 4, 2, 1, 2, 2, 3, 2, 1, 2, 2, 3, 4, 3, 2, 1, 2, 3, 4, 6, 5, 4, 3, 2

Offset: 1

Roger L. Bagula, May 24 2007

Roots heights N(i) Helgasson has in his table for the Cartan roots a(i): delta(n)==Sum[N(i)*a(i),{i,1,n}] h(n)=row sum=Sum[N(i),{i,1,n}] What I found was that my dimension ratio: Dimgroup/DimCartan=h(n )+2 which is not in any of my books. Since exponent sum: Dimgroup=Sum[2*m(i)+1,{i,1,n}] That gives a relationship of sorts between the Poincaré polynomials and the Cartan roots systems: Sum[2*m(i)+1,{i,1,n}]/n=Sum[N(i),{i,1,n}]+2 Table[Apply[Plus, a[n]], {n, 1, 6}] {1, 5, 11, 11, 17, 29} A118889: Table[Apply[Plus, a[n]] + 2, {n, 1, 6}] {3, 7, 13, 13, 19, 31}

			{1},
{3, 2},
{2, 3, 4, 2},
{1, 2, 2, 3, 2, 1},
{2, 2, 3, 4, 3, 2, 1},
{2, 3, 4, 6, 5, 4, 3, 2}

Sigurdur Helgasson, Differential Geometry, Lie Groups and Symmetric Spaces, Graduate Studies in Mathematics, volume 34. A. M. S.: ISBN 0-8218-2848-7, 1978, pp. 460, 476

Cf. A118889, A117133.

a[1] = {1}; a[2] = {3, 2}; a[3] = {2, 3, 4, 2}; a[4] = {1, 2, 2, 3, 2, 1}; a[5] = {2, 2, 3, 4, 3, 2, 1}; a[6] = {2, 3, 4, 6, 5, 4, 3, 2}; b = Table[a[n], {n, 1, 6}]; Flatten[b]

a(1) = {1}; a(2) = {3, 2}; a(3) = {2, 3, 4, 2}; a(4) = {1, 2, 2, 3, 2, 1}; a(5) = {2, 2, 3, 4, 3, 2, 1}; a(6) = {2, 3, 4, 6, 5, 4, 3, 2};

cp's OEIS Frontend

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}.

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