weight (representation theory)

A weight records how a commuting family of symmetries — a torus, or in Lie algebra language a Cartan subalgebra — acts on a representation. Because commuting operators can be simultaneously diagonalized, a representation breaks into eigenlines, and the weights are the labels (the eigenvalue data) attached to those lines. They are the coordinates in which the whole representation is organized.

In the Lie algebra setting, let h be a Cartan subalgebra of a semisimple Lie algebra and V a representation. A weight is a linear functional lambda in h* such that the weight space V_lambda = {v in V : h.v = lambda(h) v for all h in h} is nonzero. The representation decomposes as the direct sum of its weight spaces, V = direct sum over weights lambda of V_lambda. For a torus T acting on V, a weight is instead a character of T, i.e. a homomorphism T -> GL(1), and V splits into the corresponding character eigenspaces.

Weights are partially ordered, and a finite-dimensional irreducible representation of a semisimple Lie algebra is pinned down by a single highest weight, a dominant integral functional, via the theorem of the highest weight. The set of weights is invariant under the Weyl group, and their pattern — together with the root system, which is just the set of nonzero weights of the adjoint representation — encodes the entire combinatorial skeleton of the representation.