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.

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A132210 a(n) = index j such that A121736(j)=A030649(n).

Original entry on oeis.org

1, 2, 5, 10, 19, 31, 54, 77, 108, 155, 206, 274, 356, 458, 579, 715, 884, 1068, 1270, 1531, 1805
Offset: 1

Views

Author

Alexander R. Povolotsky, Nov 06 2007

Keywords

Comments

A030649 is a subset of A121736 (A. R. Povolotsky, Nov 06 2007). - R. J. Mathar, Nov 16 2007

Examples

			For n=3, we list the index a(3)=5 because A121736(5)=1463 equals A030649(3).

Crossrefs

Cf. A030649, A121736.

Formula

A121736(a(n))=A030649(n). - R. J. Mathar, Nov 16 2007

Extensions

Better definition from R. J. Mathar, Nov 16 2007

A121736 Dimensions of the irreducible representations of the simple Lie algebra of type E7 over the complex numbers, listed in increasing order.

Original entry on oeis.org

1, 56, 133, 912, 1463, 1539, 6480, 7371, 8645, 24320, 27664, 40755, 51072, 86184, 150822, 152152, 238602, 253935, 293930, 320112, 362880, 365750, 573440, 617253, 861840, 885248, 915705, 980343, 2273920, 2282280, 2785552, 3424256, 3635840
Offset: 1

Views

Author

Skip Garibaldi (skip(AT)member.ams.org), 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.
See also comments in A030649.

Examples

			The highest weight 0000000 corresponds to the 1-dimensional module on which E7 acts trivially. The smallest faithful representation of E7 is the so-called "standard" representation of dimension 56 (the second term in the sequence), with highest weight 0000001; it is minuscule and supports the famous invariant quartic form. The adjoint representation of dimension 133 (the third term in the sequence), has highest weight 1000000.

References

  • N. Bourbaki, Lie groups and Lie algebras, Chapters 4-6, Springer, 2002.
  • J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1997.

Links

Crossrefs

Cf. A121732, A121737, A121738, A121739, A104599, A121741, A030649.

Programs

  • GAP
    # see program given in sequence A121732

Formula

Given a vector of 7 nonnegative integers, the Weyl dimension formula tells you the dimension of the corresponding irreducible representation. The list of such dimensions is then sorted numerically.
Showing 1-2 of 2 results.

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