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Representations of the exceptional compact Lie group F4

Let G be the real compact simple Lie group of exceptional type F4. Choose four fundamental weights w1,w2,w3,w4 as in Bourbaki's book Groupes et Alg├Ębres de Lie. By Weyl's character formula, we can compute the dimensions of irreducible representations corrsponding to these four dominant weights: dim V1=52, dim V2=1274, dim V3=273, dim V4=26, where Vi has the highest weight wi. It's well know that V4 is the trace zero part of the 27-dimensional exceptional Jordan algebra, and V1 is the adjoint representation of G.

In Exceptioinal Lie Group F4 and its Representation Rings, I.Yokota proves that the representation ring of G is a polynomial ring

Z[V4, wedge^2 V4, wedge^3 V4, V1]=Z[V1,V2,V3,V4].

By Freudenthal's multiplicity formula, we obtain the following direct sum decompositions:

wedge^2 V4=V3+V1,

wedge^3 V4=V2+V3+V_{w1+w4},

the tensor product of V1 and V4=V_{w1+w4}+V3+V4.