A104599 Dimensions of the irreducible representations of the simple Lie algebra of type G2 over the complex numbers, listed in increasing order.

1, 7, 14, 27, 64, 77, 182, 189, 273, 286, 378, 448, 714, 729, 748, 896, 924, 1254, 1547, 1728, 1729, 2079, 2261, 2926, 3003, 3289, 3542, 4096, 4914, 4928, 5005, 5103, 6630, 7293, 7371, 7722, 8372, 9177, 9660, 10206, 10556, 11571, 11648, 12096, 13090

Offset: 1

Author

Skip Garibaldi (skip(AT)member.ams.org), Aug 19 2006

Keywords

• nonn

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.

Examples

			The highest weight 00 corresponds to the 1-dimensional module on which G2 acts trivially. The smallest faithful representation of G2 is the "standard" representation of dimension 7 (the second term in the sequence), with highest weight 10. (This vector space can be viewed as the trace zero elements of an octonion algebra.) The third term in the sequence, 14, is the dimension of the adjoint representation, which has highest weight 01.
		

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

• Andy Huchala, Table of n, a(n) for n = 1..20000
• Andy Huchala, Java program
• Wikipedia, G_2 (mathematics)

Crossrefs

Cf. A121732, A121736, A121737, A121738, A121739.

Programs

GAP

# see program at sequence A121732

Formula

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