Glossary of Semisimple Groups - F

F

• F4 (mathematics)
• Flag manifold
• Fundamental representation

For the irreducible representations of a simply-connected compact Lie group there exists a set of fundamental weights, indexed by the vertices of the Dynkin diagram of G, such that dominant weights are simply non-negative integer linear combinations of the fundamental weights.

The corresponding irreducible representations are the fundamental representations of the Lie group. In particular, from the expansion of a dominant weight in terms of the fundamental weights, one can take a corresponding tensor product of the fundamental representations and extract one copy of the irreducible representation corresponding to that dominant weight.

In the case of the special unitary group SU(n), the n − 1 fundamental representations are the wedge products

consisting of alternating tensors, for k=1,2,...,n-1.

• Fundamental Weyl chamber