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A104599
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Dimensions of the irreducible representations of the simple Lie algebra of type G2 over the complex numbers, listed in increasing order.
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6
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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
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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N. Bourbaki, Lie groups and Lie algebras, Chapters 4-6, Springer, 2002.
J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1997.
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LINKS
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Table of n, a(n) for n=1..45.
Wikipedia article on G<sub>2</sub>
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FORMULA
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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.
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EXAMPLE
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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.
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PROG
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(GAP) # see program at sequence A121732
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CROSSREFS
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Cf. A121732, A121736, A121737, A121738, A121739.
Sequence in context: A030414 A095894 A225227 * A115874 A083495 A185066
Adjacent sequences: A104596 A104597 A104598 * A104600 A104601 A104602
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KEYWORD
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nonn
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AUTHOR
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Skip Garibaldi (skip(AT)member.ams.org), Aug 19 2006
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STATUS
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approved
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