A121736 Dimensions of the irreducible representations of the simple Lie algebra of type E7 over the complex numbers, listed in increasing order.

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

Offset: 1

Author

Skip Garibaldi (skip(AT)member.ams.org), Aug 18 2006

Keywords

• nonn

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.

Examples

			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.
		

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)

Crossrefs

Cf. A121732, A121737, A121738, A121739, A104599, A121741, A030649.

Programs

GAP

# see program given in sequence A121732

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.