Springer Science+Business Media New York (2016)
COMPONENTS OF V (ρ) ⊗ V (ρ)
Matematica e Fisica
Universit`a del Salento
73047 Lecce, Italy
University of North Carolina
Chapel Hill, NC 27599-3250, USA
Universit`a di Pisa
56127 Pisa, Italy
Abstract. Let ρ be half the sum of the positive roots of a root system. We prove
that if λ is a dominant weight, λ ≤ 2ρ with respect to the dominance order, and d is
a saturation factor for the complex Lie algebra associated to the root system, then the
irreducible representation V (dλ) appears in the tensor product V (dρ) ⊗ V (dρ).
Let g be any simple Lie algebra over C. We ﬁx a Borel subalgebra b and a
Cartan subalgebra t ⊂ b and let ρ be the half sum of positive roots, where the
roots of b are called the positive roots. For any dominant integral weight λ ∈ t
let V (λ) be the corresponding irreducible representation of g with highest weight
λ. B. Kostant initiated (and popularized) the study of the irreducible components
of the tensor product V (ρ) ⊗ V (ρ). In fact, he conjectured the following.
Conjecture 1 (Kostant). Let λ be a dominant integral weight. Then, V (λ) is a
component of V (
ρ) ⊗ V (ρ) if and only if λ ≤ 2ρ under the usual Bruhat–Chevalley
order on the set of weights.
It is, of course, clear that if V (λ
) is a component of V (ρ) ⊗ V (ρ), then λ ≤ 2ρ.
One of the main motivations behind Kostant’s conjecture was his result that
the exterior algebra ∧g, as a g-module under the adjoint action, is isomorphic with
copies of V (ρ) ⊗ V (ρ), where r is the rank of g (cf. [Ko]). Recall that ∧g is the
Received August 27, 2015. Accepted January 24, 2016.
Corresponding Author: S. Kumar, e-mail: email@example.com.
-01 -93 -6 75 8
Vol. 22, No.
, 2017, pp.