Construction of A-Series Root Systems
We start with the -dimensional space with orthonormal basis , and cut out the -dimensional subspace orthogonal to the vector . This consists of those vectors for which the coefficients sum to zero: . We let , consisting of the lattice vectors whose (integer) coefficients sum to zero. Finally, we define our root system to consist of those vectors such that .
From this construction it should be clear that consists of the vectors . The vectors are independent, and thus form a basis of the -dimensional space . This establishes that spans . In particular, if we can write
showing that forms a base for .
We calculate the Cartan integers for this base
For we get the value ; for or we get the value ; otherwise we get the value . This clearly gives us the Dynkin diagram .
Finally, the reflections with respect to the should generate the entire Weyl group. We must verify that these leave the lattice invariant to be sure that we have a root system. We calculate
That is, it swaps the coefficients of and , and thus sends the lattice back to itself, as we need.
We can also see from this effect that any combination of the serves to permute the coefficients of a given vector. That is, the Weyl group of the system is naturally isomorphic to the symmetric group .