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

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

Author

Roger L. Bagula, May 24 2007

Keywords

• nonn
• fini
• full
• tabf
• uned

Comments

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}

Examples

			{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}
		

References

• 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

Crossrefs

Cf. A118889, A117133.

Programs

Mathematica

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]

Formula

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};