A118889 -id:A118889 - cp's OEIS frontend

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

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};

A129520 Orders of symmetric groups associated with the exceptional groups G2, F4, E6, E7, E7.5, E8, E9 (or better? E11): A_3, S_3, A_4, S_5, "? ", A_5, S_5.

Original entry on oeis.org

3, 6, 24, 40, 60, 120

Offset: 1

Roger L. Bagula, May 28 2007

The calculation by hand based on dimensions of: Dim_Ex(i)-n(i)*Dim_Symm(i)=small_integer The groups E7.5 and E9 are better thought of as E7.6 and E11 when integer results are wanted. The "?" question mark group is a symmetrical group between s4 and A5 with dimension 40; it is not in any of the books I have or on the web as far as I can find by searching.

			14-4*3=2
52-8*6=4
78-5*12=6
133-5*24=13
190-4.6*40=6
248-4*60=8
484-4*120=4
Example:
DimEx(E7.6)/Dim_cartan=190/7.6=25
DimEx(E11)/Dim_cartan=484/11=44
which fit in the sequence of the traditional exceptional groups A118889
as:3, 7, 13, 13, 19, 25, 31,44

Gareth Jones and David Singerman, Bull. London Math Soc. 28, (1996) pages 561-590 ( A_4, S_4, A_5 on p585)

Cf. A118889.

Partially edited by N. J. A. Sloane, May 15 2010

cp's OEIS Frontend

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

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