Weight (representation theory) - Definition and Overview

Given a set S of complex matrices, each of which is diagonalizable and any two of which commute under multiplication, it is always possible to diagonalize all the elements of S simultaneously. In basis-free terms, for any set of mutually commuting semisimple operators on a finite-dimensional complex vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. To each such basis vector, there is a function assigning each element of S its corresponding eigenvalue. Such a function is called a weight.

Examples

Suppose the elements of S form a topological group isomorphic to the real numbers under addition. A weight is then a continuous additive-to-multiplicative homomorphism φ: R→C^&times. It is easy to see that all such homomorphisms are of the form φ = φ_y for some y in C, where

<math>\phi_y(x) = e^{2\pi i xy}.<math>

More generally, if S is a real vector space W, any continuous homomorphism from S to C^&times is given by a vector y in the complexification of the dual space W^* of W. The homomophism φ_y will be unitary (i.e., have absolute value 1 for all x in W) if any only if y lies in W^* itself.

This situation arises typically in the representation theory of Lie algebras. If S is an abelian subalgebra of a real Lie algebra g (i.e., the Lie bracket of any two elements of S is 0) and V is a representation space of g, we obtain a set of mutually commuting operators on V indexed by S. For example, S could be a Cartan subalgebra. If we choose S judiciously, we can arrange that these operators should be semi-simple. Therefore, V determines a set of weights (with multiplicities) in the (possibly complexified) dual space of S.

Alternatively, if S is the topological group S¹, i.e., a circle, which we identify with the unit circle in the complex plane, a weight on S is given by an integer m: φ_m(s) = s^m. More generally, if S is a compact connected commutative Lie group (and therefore isomorphic to the n-torus (S¹)ⁿ for some n), the possible weights of S are given by n-tuples of integers. This situation arises typically in the representation theory of compact Lie groups, where S is typically taken to be a maximal torus, i.e., a maximal compact connected commutative Lie group.