E₇ (mathematics) - Important Subalgebras and Representations

Important Subalgebras and Representations

E₇ has an SU(8) subalgebra, as is evident by noting that in the 8-dimensional description of the root system, the first group of roots are identical to the roots of SU(8) (with the same Cartan subalgebra as in the E₇).

In addition to the 133-dimensional adjoint representation, there is a 56-dimensional "vector" representation, to be found in the E₈ adjoint representation.

The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A121736 in OEIS):

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...

The underlined terms in the sequence above are the dimensions of those irreducible representations possessed by the adjoint form of E₇ (equivalently, those whose weights belong to the root lattice of E₇), whereas the full sequence gives the dimensions of the irreducible representations of the simply connected form of E₇. There exist non-isomorphic irreducible representation of dimensions 1903725824, 16349520330, etc.

The fundamental representations are those with dimensions 133, 8645, 365750, 27664, 1539, 56 and 912 (corresponding to the seven nodes in the Dynkin diagram in the order chosen for the Cartan matrix above, i.e., the nodes are read in the six-node chain first, with the last node being connected to the third).