Highest Weight

If we have a representation with a weight μ = (m₁, m₂), then by applying the root vectors X_α X₂, X₃, Y₁, Y₂, Y₃, we can get some new weights of the form μ + α, where α is the root (recall that π(H₁)π(Z_α)v = (m₁+ a₁)π(Z_α)v). If π(Z_α)v = 0, then μ + α is not necessarily a weight. In analogy to the classification of the representation sl(2;C) : In each irreducible representation of sl(2;C), π(H) is diagonalizable, and there is a largest eigenvalue of π(H). Two irreducible representations of sl(2;C) with the same largest eigenvalue are equivalent. The highest eigenvalue is always a non-negative integer, and, conversely, for every non-negative integer m, there is an irreducible representation with highest eigenvalue m. 

Let α₁ = (2, -1) and  α₂ = (-1, 2) be the roots introduced in "Weights & Roots". Let μ₁  and μ₂ be two weights. Then, μ₁is higher than μ₂ if μ₁- μ₂ can be written in the form  μ₁ - μ₂ = aα₁ + bα₂ with a ≥ 0 and b ≥ 0. If π is a representation of sl(3;C), then a weight μ₀ for π is said to be a highest weight if for all weights μ of π, μ ≤ μ₀.

Note that the relation of "higher" is only a partial ordering because μ₁ is neither higher nor lower than μ₂. For example, {0, α₁ - α₂} has no highest element. Moreover, the coefficients a and b do not have to be integers, even if both μ₁ and μ₂ have integer entries. For example, (1,0) is higher than (0,0) since (1,0) = 2/3α₁ + 1/3α₂.

Theorem of Highest Weight

The theorem of highest weight is a main theorem regarding the irreducible representation of sl(3;C).

1. Every irreducible representation π of sl(3;C) is the direct sum of its weight spaces; that is π(H₁) and  π(H₂) are simultaneously diagonalizable in every irreducible representation.

2. Every irreducible representation of sl(3;C) has a unique highest weight μ₀, and two equivalent irreducible representations have the same highest weight.

3. two irreducible representations of sl(3;C) with the same highest weight are equivalent.

4. If π is an irreducible representation of sl(3;C), then the highest weight μ₀ of π is of the form μ₀ = (m₁, m₂) with m₁ and m₂ being non-negative integers. 

An ordered pair (m₁, m₂) with m₁ and m₂ being non-negative integers is called a dominant integral element. The theorem says that the highest weight of each irreducible representation of sl(3;C) is a dominant integral element and, conversely, that every dominant integral element occurs as the highest weight of some irreducible representation. 

However, if μ has integer coefficients and is higher than zero, this does not necessarily mean that μ is dominant integral. For example, α₁ = (2, -1) is higher than zero but is not dominant integral. Note that the condition on which weights can be highest weights is m₁ and m₂ being non-negative integer.

The dimension of the irreducible representation with highest weight (m₁, m₂) is

1/2(m₁ + 1)(m₂ + 1)(m₁ + m₂ + 2).