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Search: id:A121737
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| A121737 |
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Dimensions of the irreducible representations of the simple Lie algebra of type E6 over the complex numbers, listed in increasing order. |
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7
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| 1, 27, 78, 351, 650, 1728, 2430, 2925, 3003, 5824, 7371, 7722, 17550, 19305, 34398, 34749, 43758, 46332, 51975, 54054, 61425, 70070, 78975, 85293, 100386, 105600, 112320, 146432, 252252, 314496, 359424, 371800, 386100, 393822, 412776, 442442
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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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Wikipedia article on E6
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FORMULA
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Given a vector of 6 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 000000 corresponds to the 1-dimensional module on which E6 acts trivially. The smallest faithful representations of E6 have dimension 27, highest weight 000001 or 100000 and are minuscule. The adjoint representation of dimension 78 (the third term in the sequence) has highest weight 010000.
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PROGRAM
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(GAP) # see program at sequence A121732
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CROSSREFS
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Cf. A121732, A121736, A121738, A121739, A104599, A121741.
Sequence in context: A163506 A044165 A044546 this_sequence A124726 A126381 A129254
Adjacent sequences: A121734 A121735 A121736 this_sequence A121738 A121739 A121740
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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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