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Wikipedia, <a href="http://en.wikipedia.org/wiki/Triality">Triality</a>

Wikipedia, <a href="http://en.wikipedia.org/wiki/Triality">Triality</a>

Andy Huchala, <a href="/A121739/b121739.txt">Table of n, a(n) for n = 1..20000</a>

Andy Huchala, <a href="/A121739/a121739.java.txt">Java program</a>

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Wikipedia, <a href="http://en.wikipedia.org/wiki/Triality">Wikipedia articleTriality</a> on triality

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Dimensions of the irreducible representations of the simple Lie algebra of type D4 over the complex numbers, listed in increasing order.

1, 8, 28, 35, 56, 112, 160, 224, 294, 300, 350, 567, 672, 840, 1296, 1386, 1400, 1568, 1680, 1925, 2400, 2640, 2800, 3675, 3696, 4096, 4312, 4536, 4719, 5775, 6160, 6600, 7392, 7776, 7840, 8008, 8800, 8910, 8918, 10752, 12320, 12936, 13013, 13728, 15015

1,2

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.

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

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

<a href="http://en.wikipedia.org/wiki/Triality">Wikipedia article</a> on triality

Given a vector of 4 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 0000 corresponds to the 1-dimensional module on which D4 acts trivially. The second second term in the sequence is 8, corresponding to the three inequivalent representations with highest weights 1000, 0010 and 0001 respectively. The third term in the sequence is 28, corresponding to the adjoint representation, which has highest weight 0100.

(GAP) # see program at sequence A121732

Cf. A121732, A121736, A121737, A121738, A104599.

nonn

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

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