cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A129844 Triangular sequence constructed from heights of irreducible representations of semi-simple Lie algebras (groups A1, G2, F4, E6, E7, E8).

Original entry on oeis.org

1, 6, 10, 16, 22, 30, 42, 16, 16, 22, 30, 30, 42, 27, 34, 49, 52, 66, 75, 96, 58, 92, 114, 136, 168, 182, 220, 270
Offset: 1

Views

Author

Roger L. Bagula, May 22 2007

Keywords

Examples

			Triangle begins:
  {1},
  {6, 10},
  {16, 22, 30, 42},
  {16, 16, 22, 30, 30, 42},
  {27, 34, 49, 52, 66, 75, 96},
  {58, 92, 114, 136, 168, 182, 220, 270}.

References

  • R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.
  • E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Translations, Vol. 6, 1957, pages 111-244.

Crossrefs

Cf. A120668 (rows sums), A124680, A129766.

Programs

  • Mathematica
    a = {{1}, {6, 10}, {16, 22, 30, 42}, {16, 16, 22, 30, 30, 42}, {27, 34, 49, 52, 66, 75, 96}, {58, 92, 114, 136, 168, 182, 220, 270}}; Flatten[a]

Extensions

Edited by N. J. A. Sloane, Dec 30 2008
a(26) corrected by Andrey Zabolotskiy, Jul 23 2024

A117133 Dimensions of the traditional Cartan exceptional group sequence A1, G2, F4, E6, E7, E8.

Original entry on oeis.org

3, 14, 52, 78, 133, 248
Offset: 1

Views

Author

Roger L. Bagula, May 17 2007

Keywords

Crossrefs

Cf. A129766.

Programs

  • 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};
b0 = Table[Length[CoefficientList[Expand[Product[(1 + t^(2*a[i][[n]] + 1)), {n, 1, Length[a[i]]}]], t]] - 1, {i, 0, 5}]

Formula

P(n) = Poincare-Polynomial(n) = Product_{m=1..n} (1 + t^A129766(m)); a(n) = Length(CoefficientList(P(n),t)) - 1.

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}.

Original entry on oeis.org

3, 7, 13, 13, 19, 31
Offset: 1

Views

Author

Roger L. Bagula, May 17 2007

Keywords

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]
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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