relations between root lattice and weight lattice

Let $Q$ and $P$ denote the $\mathbb{Z}$-span of the simple roots and fundamental weights respectively. What are the relations between $Q$ and $P$? Does $P$ contain $Q$? Thank you.

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1 Answer

Dear user, the roots are a "special kind" of weights - corresponding to the adjoint representation. The fundamental representation is the "seed" of all other representations - all others may be obtained as terms in the decomposition of tensor powers of the fundamental ones.

It follows that the root lattice is a subset of the weight lattice. This statement is equivalent to the statement that the weights of representations appearing in tensor powers of the adjoint are a subset of the weights of representations appearing in tensor powers of the fundamental ones - i.e. all representations.

$Q$ being a subset of $P$ is not the only interesting relationship between them. For example, the weight lattice is dual to the coroot lattice - where the latter is closely related to the root lattice. For self-dual lattices such as the $E_8$ root lattice, the weight lattice and root lattice may coincide. In the $E_8$ case, this is equivalent to the fact that the fundamental 248-dimensional representation is the adjoint one, too.

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thanks. How to show that the weight lattice is dual to the coroot lattice? – LJR May 25 '11 at 11:39