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A121736 Dimensions of the irreducible representations of the simple Lie algebra of type E7 over the complex numbers, listed in increasing order. 11
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 (list; graph; refs; listen; history; text; internal format)
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

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

Sequence in context: A253418 A204840 A204833 * A107937 A044307 A044688

Adjacent sequences: A121733 A121734 A121735 * A121737 A121738 A121739

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified November 27 06:50 EST 2022. Contains 358362 sequences. (Running on oeis4.)