Affine action explained
Let
be the
Weyl group of a
semisimple Lie algebra
(associate to fixed choice of a
Cartan subalgebra
). Assume that a set of simple roots in
is chosen.
The affine action (also called the dot action) of the Weyl group on the space
is
w ⋅ λ:=w(λ+\delta)-\delta
where
is the sum of all fundamental weights, or, equivalently, the half of the sum of all positive roots.
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