OFFSET
1,2
COMMENTS
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.
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 (terms 1..2856 from Skip Garibaldi)
Andy Huchala, Java program
Wikipedia, E_7 (mathematics)
FORMULA
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.
EXAMPLE
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.
PROG
(GAP) # see program given in sequence A121732
CROSSREFS
KEYWORD
nonn
AUTHOR
Skip Garibaldi (skip(AT)member.ams.org), Aug 18 2006
STATUS
approved