Weyl character and dimension formulas

Fix a finite-dimensional complex semisimple algebra, Cartan, and positive roots. A finite-dimensional module decomposes into weight spaces Vμ. Its character is the sum of dim(Vμ) times a formal symbol eμ, with multiplication of symbols defined by adding weights. These symbols are not initially ordinary scalar exponentials. Restriction to a torus can turn them into actual functions, but the formal object retains the individual labels and multiplicities. Two different weights can yield equal numerical values at a selected torus point without becoming the same weight.

For a dominant integral λ, Weyl's theorem gives the character of L(λ) as the alternating sum over w(λ+ρ) divided by the alternating sum over wρ. The denominator also equals eρ times the product of 1−e⁻α over positive roots. The result is a finite character even though a quotient notation is used. This is a structural theorem, whose proof requires more than checking its symmetry. An alternating numerator has the right transformation behavior, but that observation alone does not establish all multiplicities or the equality with a representation.

Setting every formal weight value to one sums the multiplicities and gives the dimension. In the uncancelled alternating quotient, however, numerator and denominator both vanish at that specialization for a nontrivial Weyl group. Taking the appropriate limit yields the product below. Each positive coroot pairs positively with ρ, so the denominators are nonzero. For a semisimple direct sum, the positive roots separate by factors and the product matches the tensor-product behavior of simple modules.

The formula counts every weight with its multiplicity, not merely the size of a highest-weight Weyl orbit. The adjoint module of sl(3,C), for example, has six root weights and a two-dimensional zero-weight space, giving dimension eight. Its extreme-root orbit has six elements and does not account for the zero space. An integral answer is a useful arithmetic check, but it is not proof that an arbitrarily chosen nonintegral label defines a finite module. The dominant integral hypothesis is part of the formula's representation-theoretic meaning.

At a nonidentity torus element, a character is a trace of that element's representation and may be smaller than the dimension, zero, or negative in a real-valued example. The standard sl(2) group module restricts to the diagonal torus element diag(exp(iθ),exp(−iθ)) with character 2cosθ. At θ=0 it equals two, its dimension; at θ=π/3 it equals one. Neither value changes the underlying module. Treating one sampled trace as a count of basis vectors would lose the role of the identity specialization.

Characters also support tensor-product calculations because the character of a tensor product is a product of characters. Decomposing that product into irreducible characters requires multiplicity bookkeeping and is not supplied solely by multiplying dimensions. Close to chamber walls, stabilizers and repeated weights make informal orbit counting particularly unreliable. Use a formal expansion, the dimension product, and direct low-rank examples as complementary checks. When all three agree, they test different aspects of a calculation rather than repeating the same numerical shortcut.