A129766 Triangular array read by rows, made up of traditional exceptional groups plus A1: as A1,G2,F4,E6,E7,E8 as m(i) exponents as in A005556, A005763, A005776.

1, 1, 5, 1, 5, 7, 11, 1, 4, 5, 7, 8, 11, 1, 5, 7, 9, 11, 13, 17, 1, 7, 11, 13, 17, 19, 23, 29

Offset

1,3

Comments

Extra condition of group dimension: b[n] = a[n] + 1 ; DimGroup = Apply[Plus, b[n]]; Table[Apply[Plus, b[n]], {n, 0, 5}] {3, 14, 52, 78, 133, 248} Extra condition of Betti sum: Table[Apply[Plus, CoefficientList[Expand[Product[(1 + t^(2*a[i][[n]] + 1)), {n, 1,Length[a[i]]}]], t]], {i, 0, 5}] {2, 4, 16, 64, 128, 256} These exponents are necessary to the Poincaré polynomials for these exceptional groups.

Links

Table of n, a(n) for n=1..28.

Armand Borel, Essays in History of Lie Groups and Algebraic Groups gives G2 Poincaré polynomial, History of Mathematics, V. 21.

Example

1;
1,5;
1,5,7,11;
1,4,5,7,8,11;
1,5,7,9,11,13,17;
1,7,11,13,17,19,23,29;

Mathematica

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}; Flatten[Table[a[n], {n, 0, 5}]]

Crossrefs

Cf. A005556, A005763, A005776.

Sequence in context: A338167 A372958 A129769 * A120283 A103986 A196404

Adjacent sequences: A129763 A129764 A129765 * A129767 A129768 A129769

Keyword

nonn,fini,full,tabf,uned

Author

Roger L. Bagula, May 16 2007

Status

approved