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Search: id:A121214
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| A121214 |
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Dimensions of the irreducible representations of the algebraic group SL4 (equivalently, simple Lie algebra of type A3) over the complex numbers, listed in increasing order. |
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+0
3
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| 1, 4, 6, 10, 15, 20, 35, 36, 45, 50, 56, 60, 64, 70, 84, 105, 120, 126, 140, 160, 165, 175, 189, 196, 216, 220, 224, 256, 270, 280, 286, 300, 315, 336, 360, 364, 384, 396, 420, 440, 455, 480, 500, 504, 540, 560, 594, 616, 630, 640, 680, 715, 729, 735, 750, 756
(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 fact 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 the special linear group
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FORMULA
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Given a vector of 3 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 000 corresponds to the 1-dimensional module on which SL4 acts trivially. The standard representation and its dual have dimension 4 (the second term in the sequence) and highest weights 100 and 001. The third term in the sequence, 6, is the dimension of the representation of SL4 on the second exterior power of the standard representation; it has highest weight 010. The fourth term, 10, is the dimension of the second symmetric power of the standard representation or its dual, with highest weight 200 or 002. The fifth term, 15, corresponds to the adjoint representation with highest weight 101.
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PROGRAM
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(GAP) # see program at A121732
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CROSSREFS
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Cf. A121732, A121741.
Sequence in context: A084372 A140611 A076957 this_sequence A116996 A004399 A028282
Adjacent sequences: A121211 A121212 A121213 this_sequence A121215 A121216 A121217
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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 20 2006
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