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Search: id:A121736
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| A121736 |
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Dimensions of the irreducible representations of the simple Lie algebra of type E7 over the complex numbers, listed in increasing order. |
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+0
9
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| 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
(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.
See also comments in A030649.
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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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Skip Garibaldi, Table of n, a(n) for n = 1..2856
Wikipedia article on E7
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FORMULA
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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.
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EXAMPLE
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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.
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PROGRAM
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(GAP) # see program given in sequence A121732
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CROSSREFS
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Cf. A121732, A121737, A121738, A121739, A104599, A121741.
Cf. A030649.
Adjacent sequences: A121733 A121734 A121735 this_sequence A121737 A121738 A121739
Sequence in context: A030443 A135803 A048452 this_sequence A107937 A044307 A044688
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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 18 2006
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