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Andy Huchala, <a href="/A104599/a104599.java.txt">Java program for computing a(n)</a>

Wikipedia, <a href="http://en.wikipedia.org/wiki/G2_%28mathematics%29">G_2 (mathematics)</a>

Wikipedia, <a href="http://en.wikipedia.org/wiki/G2_%28mathematics%29">G_2 (mathematics)</a>

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Andy Huchala, <a href="/A104599/b104599.txt">Table of n, a(n) for n = 1..20000</a>

Andy Huchala, <a href="/A104599/a104599.java.txt">Java program for computing a(n)</a>

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Wikipedia, <a href="http://en.wikipedia.org/wiki/G2_%28mathematics%29">Wikipedia article</a> on G<sub>_2 (mathematics)</suba>

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Number Dimensions of the irreducible representations of (0,1)-matrices the simple Lie algebra of any size, with n ones, and no zero row or columntype G2 over the complex numbers, listed in increasing order.

1, 5, 15, 161, 1679, 21457, 317199, 5342017, 100968319, 2116803073, 48754966527, 1223849591361, 33256343226879, 972631093895233, 30463342591779327, 1017339915400133377, 36086366842385043199, 13549594652708275543057, 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

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.

N. Bourbaki, Lie groups and Lie algebras, Chapters 4-6, Springer, 2002.

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

M. Maia and M. Mendez, <a href="http://arXiven.wikipedia.org/abs/math.COwiki/0503436G2_%28mathematics%29">On the arithmetic product of combinatorial speciesWikipedia article</a> on G₂

1/(4n!) * Sum{r, s>=0, (rs)_n / 2^(r+s) }, where (m)_n is the falling factorial m * (m-1) * ... * (m-n+1).

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.

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.

(GAP) # see program at sequence A121732

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

nonn,new

nonn

Ralf Stephan, Mar 27 2005

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