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.

A121732 Dimensions of the irreducible representations of the simple Lie algebra of type E8 over the complex numbers, listed in increasing order.

Original entry on oeis.org

1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000, 26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500, 820260000, 1094951000, 2172667860
Offset: 1

Views

Author

Skip Garibaldi (skip(AT)mathcs.emory.edu), Aug 18 2006

Keywords

Comments

We include "1" for the 1-dimensional trivial representation and we list each dimension once, ignoring the possibility that inequivalent representations may have the same dimension.
Inequivalent representations can have the same dimension. For example, the highest weights 10100000 and 10000011 (with fundamental weights numbered as in Bourbaki) both correspond to irreducible representations of dimension 8634368000.

Examples

			The highest weight 00000000 corresponds to the 1-dimensional module on which E8 acts trivially. The smallest faithful representation of E8 is the adjoint representation of dimension 248 (the second term in the sequence), with highest weight 00000001. The smallest non-fundamental representation has dimension 27000 (the fourth term), corresponding to the highest weight 00000002.

References

  • J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1997.

Links

Crossrefs

Cf. A121736, A121737, A121738, A121739, A104599, A121741, A121214, A030650.

Programs

  • GAP
    # see program given in link.

Formula

Given a vector of 8 nonnegative integers, the Weyl dimension formula tells you the dimension of the corresponding irreducible representation. The list of such dimensions is then sorted numerically.

A133238 Dimensions of certain Lie algebra (see reference for precise definition).

Original entry on oeis.org

1, 52, 1053, 12376, 100776, 627912, 3187041, 13748020, 51949755, 175847880, 542393670, 1544927904, 4107092288, 10278624864, 24388573014, 55188666312, 119696471453, 249869263644, 503865726155, 984563860280, 1869304764600, 3456658569000, 6238533257775
Offset: 0

Views

Author

N. J. A. Sloane, Oct 15 2007

Keywords

Links

Crossrefs

The cases a = -4/3, -1, -2/3, 0, 1, 2, 4, 6, 8 of Th. 7.1 of Landsberg and Manivel give sequences A005408, A000578, A085462, A107942, A133238 (this entry), A133239, A133240, A133241 and A030650 respectively. See also triangle in A128894.

Programs

  • Maple
    b:=binomial; t71:= proc(a,k) ((3*a+2*k+5)/(3*a+5)) * b(k+2*a+3,k)*b(k+5*a/2+3,k)*b(k+3*a+4,k)/(b(k+a/2+1,k)*b(k+a+1,k)); end; [seq(t71(1,k),k=0..30)];
  • Mathematica
    t71[a_, k_] := (3a+2k+5) / (3a+5) Binomial[k+2a+3,k] Binomial[k+5/2a+3,k] Binomial[k+3a+4,k] / (Binomial[k+a/2+1,k] Binomial[k+a+1,k]);
Array[t71[1,#]&,30,0] (* Paolo Xausa, Jan 11 2024 *)

Formula

Empirical g.f.: (x^8+36*x^7+341*x^6+1208*x^5+1820*x^4+1208*x^3+341*x^2+36*x+1) / (x-1)^16. - Colin Barker, Jul 27 2013

A124680 Heights of irreducible representations of E_8.

Original entry on oeis.org

58, 92, 114, 136, 168, 182, 220, 270
Offset: 1

Views

Author

Roger L. Bagula, Dec 25 2005

Keywords

Examples

			The factored sequence, [2*29, 2^2*23, 2*3*19, 2^3*17, 2^3*3*7, 2*7*13, 2^2*5*11, 2*3^3*5], shows a close relationship to A005776. - _N. J. A. Sloane_, Dec 25 2006

References

  • R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

Crossrefs

Cf. A005776, A030650, A121732.
Showing 1-3 of 3 results.

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

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