OFFSET
1,1
COMMENTS
Terms which would be repeated in A121737.
There are infinitely many terms in this sequence; see A181746.
By symmetry of the Dynkin diagram, with fundamental weights numbered as in Bourbaki there is a duality of highest weights [1,0,0,0,0,0] and [0,0,0,0,0,1]. Similarly, there is a duality of highest weights [0,0,0,0,1,0] and [0,0,1,0,0,0]. Note that E6 is the only exceptional Lie algebra with such a duality. However this duality is not responsible for all pairs of non-isomorphic irreducible E6 representations of equal dimension--see example.
There are 6 non-isomorphic irreducible E6 representations of dimension 7183313280, and 8 non-isomorphic irreducible E6 representations of dimension 7980534952482277785600. Both dimensions are minimal with respect to that property. I do not know if such dimensions exist for 9 or more irreducible representations.
REFERENCES
N. Bourbaki, Lie groups and Lie algebras, Chapters 4-6, Springer, 2002.
J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1997.
LINKS
Andy Huchala, Table of n, a(n) for n = 1..20000
Andy Huchala, Java program
Wikipedia, E6 (mathematics)
EXAMPLE
With the fundamental weights numbered as in Bourbaki, the irreducible E6-modules with highest weights [1,0,0,0,0,0] and [0,0,0,0,0,1] both have dimension 77. The vectors [0,0,0,0,1,0], [0,0,1,0,0,0], [2,0,0,0,0,0], and [0,0,0,0,0,2] are the four highest weights which correspond to irreducible representations of dimension 351.
CROSSREFS
KEYWORD
nonn
AUTHOR
Andy Huchala, Apr 02 2021
STATUS
approved